Cards Preview

Type: Basic Math::Card 1::Back

Deck: Math::Algebra

Tags: Math Algebra Identity

Quadratic Formula

a x^{2} + b x + c = 0

then

x = \frac{- b \pm b ^{2} - 4 a c}{2 a}

Type: Basic Math (and reversed card)::Card 1::Back

Deck: Math::Algebra

Tags: Math Algebra Identity

(a + b)^{2}

a^{2} + 2 ab + b^{2}

(a + b)^{2} (a - b)^{2} = a^{2} + 2 ab + b^{2} = a^{2} - 2 ab + b^{2}

Type: Basic Math (and reversed card)::Card 2::Back

Deck: Math::Algebra

Tags: Math Algebra Identity

a^{2} + 2 ab + b^{2}

(a + b)^{2}

(a + b)^{2} (a - b)^{2} = a^{2} + 2 ab + b^{2} = a^{2} - 2 ab + b^{2}

Type: Basic Math (and reversed card)::Card 1::Back

Deck: Math::Algebra

Tags: Math Algebra Identity

(a - b)^{2}

a^{2} - 2 ab + b^{2}

(a + b)^{2} (a - b)^{2} = a^{2} + 2 ab + b^{2} = a^{2} - 2 ab + b^{2}

Type: Basic Math (and reversed card)::Card 2::Back

Deck: Math::Algebra

Tags: Math Algebra Identity

a^{2} - 2 ab + b^{2}

(a - b)^{2}

(a + b)^{2} (a - b)^{2} = a^{2} + 2 ab + b^{2} = a^{2} - 2 ab + b^{2}

Type: Basic Math (and reversed card)::Card 1::Back

Deck: Math::Algebra

Tags: Math Algebra Identity

a^{2} + b^{2}

(a - ib) (a + ib)

a^{2} + b^{2} a^{2} - b^{2} = (a - ib) (a + ib) = (a - b) (a + b)

Type: Basic Math (and reversed card)::Card 2::Back

Deck: Math::Algebra

Tags: Math Algebra Identity

(a - ib) (a + ib)

a^{2} + b^{2}

a^{2} + b^{2} a^{2} - b^{2} = (a - ib) (a + ib) = (a - b) (a + b)

Type: Basic Math (and reversed card)::Card 1::Back

Deck: Math::Algebra

Tags: Math Algebra Identity

a^{2} - b^{2}

(a - b) (a + b)

a^{2} + b^{2} a^{2} - b^{2} = (a - ib) (a + ib) = (a - b) (a + b)

Type: Basic Math (and reversed card)::Card 2::Back

Deck: Math::Algebra

Tags: Math Algebra Identity

(a - b) (a + b)

a^{2} - b^{2}

a^{2} + b^{2} a^{2} - b^{2} = (a - ib) (a + ib) = (a - b) (a + b)

Type: Basic Math (and reversed card)::Card 1::Back

Deck: Math::Algebra

Tags: Math Algebra Identity

a^{3} + b^{3}

(a + b) (a^{2} - ab + b^{2})

a^{3} + b^{3} a^{3} - b^{3} = (a + b) (a^{2} - ab + b^{2}) = (a - b) (a^{2} + ab + b^{2})

Type: Basic Math (and reversed card)::Card 2::Back

Deck: Math::Algebra

Tags: Math Algebra Identity

(a + b) (a^{2} - ab + b^{2})

a^{3} + b^{3}

a^{3} + b^{3} a^{3} - b^{3} = (a + b) (a^{2} - ab + b^{2}) = (a - b) (a^{2} + ab + b^{2})

Type: Basic Math (and reversed card)::Card 1::Back

Deck: Math::Algebra

Tags: Math Algebra Identity

a^{3} - b^{3}

(a - b) (a^{2} + ab + b^{2})

a^{3} + b^{3} a^{3} - b^{3} = (a + b) (a^{2} - ab + b^{2}) = (a - b) (a^{2} + ab + b^{2})

Type: Basic Math (and reversed card)::Card 2::Back

Deck: Math::Algebra

Tags: Math Algebra Identity

(a - b) (a^{2} + ab + b^{2})

a^{3} - b^{3}

a^{3} + b^{3} a^{3} - b^{3} = (a + b) (a^{2} - ab + b^{2}) = (a - b) (a^{2} + ab + b^{2})

Type: Basic Math (and reversed card)::Card 1::Back

Deck: Math::Algebra

Tags: Math Algebra Identity

(a + b)^{3}

a^{3} + 3 a^{2} b + 3 a b^{2} + b^{3}

(a - b)^{3} = a^{3} - 3 a^{2} b + 3 a b^{2} - b^{3}

Type: Basic Math (and reversed card)::Card 2::Back

Deck: Math::Algebra

Tags: Math Algebra Identity

a^{3} + 3 a^{2} b + 3 a b^{2} + b^{3}

(a + b)^{3}

(a - b)^{3} = a^{3} - 3 a^{2} b + 3 a b^{2} - b^{3}

Type: Basic Math (and reversed card)::Card 1::Back

Deck: Math::Algebra

Tags: Math Algebra Identity

(a - b)^{3}

a^{3} - 3 a^{2} b + 3 a b^{2} - b^{3}

(a + b)^{3} = a^{3} + 3 a^{2} b + 3 a b^{2} + b^{3}

Type: Basic Math (and reversed card)::Card 2::Back

Deck: Math::Algebra

Tags: Math Algebra Identity

a^{3} - 3 a^{2} b + 3 a b^{2} - b^{3}

(a - b)^{3}

(a + b)^{3} = a^{3} + 3 a^{2} b + 3 a b^{2} + b^{3}

Type: Basic Math::Card 1::Back

Deck: Math::Algebra

Tags: Math Algebra Identity

a^{2} + b^{2} + c^{2} + 2 (ab + b c + c a)

(a + b + c)^{2}

Type: Basic Math::Card 1::Back

Deck: Math::Algebra

Tags: Math Algebra Identity

a^{2} + b^{2} + c^{2} - ab - b c - c a

\frac{1}{2} [(a - b)^{2} + (b - c)^{2} + (c - a)^{2}]

Type: Basic Math::Card 1::Back

Deck: Math::Algebra

Tags: Math Algebra Identity

a^{3} + b^{3} + c^{3} - 3 ab c

(a + b + c) (a^{2} + b^{2} + c^{2} - ab - b c - c a)

Type: Basic Math::Card 1::Back

Deck: Math::Algebra

Tags: Math Algebra Identity

a^{4} - b^{4}

(a - b) (a + b) (a^{2} + b^{2})

Type: Basic Math::Card 1::Back

Deck: Math::Algebra

Tags: Math Algebra Identity

a^{4} + 4 b^{4}

(a^{2} - 2 ab + 2 b^{2}) (a^{2} + 2 ab + 2 b^{2})

Sophie Germain Identity

Type: Basic Math::Card 1::Back

Deck: Math::Algebra

Tags: Math Algebra Identity

(a + b + c)^{3}

a^{3} + b^{3} + c^{3} + 3 (a + b) (b + c) (c + a)

Type: Basic Math::Card 1::Back

Deck: Math::Algebra

Tags: Math Algebra Identity

\frac{1}{a} + \frac{1}{b} + \frac{1}{c}

\frac{ab + b c + c a}{ab c}

Type: Basic Math::Card 1::Back

Deck: Math::Algebra

Tags: Math Algebra Identity

(x - a) (x - b)

x^{2} - (a + b) x + ab

Type: Basic Math::Card 1::Back

Deck: Math::Algebra

Tags: Math Algebra Identity

x^{2} + (a + b) x + ab

(x + a) (x + b)

Type: Basic Math::Card 1::Back

Deck: Math::Algebra

Tags: Math Algebra Identity

Discriminant

Δ = b^{2} - 4 a c

Type: Basic Math::Card 1::Back

Deck: Math::Algebra

Tags: Math Algebra Inequality

Explain how $∣ x ∣ \leq c$ , $c \geq 0$ is a stronger statement (gives more information) than $x \leq c$ , $c \geq 0$ .

$∣ x ∣ \leq c$ means $- c \leq x \leq c$ , that is lower and upper bounds, while $x \leq c$ only means the upper bound.

Type: Basic Math::Card 1::Back

Deck: Math::Algebra

Tags: Math Algebra Inequality

Explain why if $∣ x ∣ \leq c$ , $c \geq 0$ is true then $x \leq c$ is true as well .

If $x$ is non-negative, then $∣ x ∣ = x$ , so $x \leq c$ .

If $x$ is negative, then $∣ x ∣ = - x$ , so $- x \leq c$ which implies $x \geq - c$ . Since $c \geq 0$ , we have $- c \leq c$ , so $x \leq c$ .

Type: Basic Math::Card 1::Back

Deck: Math::Algebra

Tags: Math Algebra Inequality

Explain why if $x \leq c$ , $c \geq 0$ is true then $∣ x ∣ \leq c$ is not necessarily true as well .

Counterexample: Let $x = - 2$ , $∣ x ∣ = 2$ and $c = 1$ . Then $- 2 \leq 1$ is true, but $2 \neq \leq 1$ .

Type: Basic Math::Card 1::Back

Deck: Math::Algebra

Tags: Math Algebra Inequality

Explain why $- ∣ x ∣ \leq x \leq ∣ x ∣$ .

If $x$ is non-negative, then $∣ x ∣ = x$ , so $- ∣ x ∣ = - x \leq x \leq x = ∣ x ∣$ .

If $x$ is negative, then $∣ x ∣ = - x$ , so $- ∣ x ∣ = x \leq x \leq - x = ∣ x ∣$ .

Type: Basic Math::Card 1::Back

Deck: Math::Algebra

Tags: Math Algebra Inequality

Solve $∣ x ∣ \leq c$ , $c \geq 0$ .

- c \leq x \leq c

x \in [- c, c]

Type: Basic Math::Card 1::Back

Deck: Math::Algebra

Tags: Math Algebra Inequality

Solve $∣ x ∣ \geq c$ , $c \geq 0$ .

x \leq - c or x \geq c

x \in (- \infty, - c] \cup [c, \infty)

Type: Basic Math::Card 1::Back

Deck: Math::Algebra

Tags: Math Algebra Inequality

Solve $x^{2} \leq c$ , $c \geq 0$ .

x^{2} x^{2} ∣ x ∣ \leq c \leq c \leq c

The solution is

- c \leq x \leq c

x \in [- c, c]

Type: Basic Math::Card 1::Back

Deck: Math::Algebra

Tags: Math Algebra Inequality

Solve $x^{2} \geq c$ , $c \geq 0$ .

x^{2} x^{2} ∣ x ∣ \geq c \geq c \geq c

The solution is

x \leq - c or x \geq c

x \in (- \infty, - c] \cup [c, \infty)

Type: Basic Math::Card 1::Back

Deck: Math::Algebra

Tags: Math Algebra Inequality

Solve $x^{2} = c$ , $c \geq 0$ .

x^{2} x^{2} ∣ x ∣ = c = c = c

The solution is

x = - c or x = c

x \in {- c, c}

Type: Basic Math::Card 1::Back

Deck: Math::Algebra

Tags: Math Algebra Inequality

When solving an inequality $a \leq b$ , what happens with the inequality direction if we add a constant $c$ to both sides?

Inequality direction stays the same.

Type: Basic Math::Card 1::Back

Deck: Math::Algebra

Tags: Math Algebra Inequality

When solving an inequality $a \leq b$ , what happens with the inequality direction if we multiply both sides by a positive constant $c$ ?

Inequality direction stays the same.

Type: Basic Math::Card 1::Back

Deck: Math::Algebra

Tags: Math Algebra Inequality

When solving an inequality $a \leq b$ , what happens with the inequality direction if we multiply both sides by a negative constant $c$ ?

Inequality direction flips.

Type: Basic Math::Card 1::Back

Deck: Math::Algebra

Tags: Math Algebra Inequality

If $a \leq b$ and $f$ is an increasing function, what happens to the inequality after applying $f$ ?

It stays the same:

f (a) \leq f (b)

Type: Basic Math::Card 1::Back

Deck: Math::Algebra

Tags: Math Algebra Inequality

If $a \leq b$ and $f$ is a decreasing function, what happens to the inequality after applying $f$ ?

The direction flips:

f (a) \geq f (b)

Type: Basic Math::Card 1::Back

Deck: Math::Algebra

Tags: Math Algebra Inequality

If $0 < a \leq b$ , what inequality holds between $1/ a$ and $1/ b$ ?

$1/ a \geq 1/ b$ (reciprocal is decreasing for positive numbers).

Type: Basic Math::Card 1::Back

Deck: Math::Algebra

Tags: Math Algebra Inequality

Is it true that if $a > b$ then $ln (a) > ln (b)$ ? Why?

Yes, if $a > 0$ and $b > 0$ then

a > b ⟹ ln (a) > ln (b)

because $ln (x)$ is monotonic and strictly increasing.

Type: Basic Math::Card 1::Back

Deck: Math::Algebra

Tags: Math Algebra Inequality

Is it true that if $a > b$ then $e^{a} > e^{b}$ ? Why?

Yes, for any $a$ and $b$

a > b ⟹ e^{a} > e^{b}

because $ln (x)$ is monotonic and strictly increasing.

Type: Basic Math::Card 1::Back

Deck: Math::Algebra

Tags: Math Algebra Inequality

True or false: From $(a + b)^{2} \geq 0$ can we conclude that $a + b \geq 0$ .

False. The correct conclusion is $∣ x + y ∣ \geq 0$ .

Type: Basic Math::Card 1::Back

Deck: Math::Algebra

Tags: Math Algebra Inequality

Solve: If $x^{2} \leq 9$ , what range does $x$ lie in?

- 3 \leq x \leq 3

Type: Basic Math::Card 1::Back

Deck: Math::Algebra

Tags: Math Algebra Inequality

Solve: If $x^{2} \geq 9$ , what range does $x$ lie in?

x \leq - 3 or x \geq 3

Type: Basic Math::Card 1::Back

Deck: Math::Algebra

Tags: Math Algebra Inequality

True or false: If $a \leq b$ , then $a^{2} \leq b^{2}$

False in general.

Counterexample: Let $a = - 3$ , $b = 2$ , then $- 3 \leq 2$ but $9 \neq \leq 4$ .

Type: Basic Math::Card 1::Back

Deck: Math::Algebra

Tags: Math Algebra Inequality

If $a^{2} \leq b^{2}$ , what inequality does this imply about $∣ a ∣$ and $∣ b ∣$ ?

∣ a ∣ \leq ∣ b ∣

Type: Basic Math::Card 1::Back

Deck: Math::Algebra

Tags: Math Algebra Inequality

Prove the inequality $ab \leq ∣ a ∣∣ b ∣$ .

Case 1: $a, b \geq 0$ . Then $∣ a ∣ = a$ , $∣ b ∣ = b$ , so $ab = ∣ a ∣∣ b ∣$ .
Case 2: $a, b \leq 0$ . Then $∣ a ∣ = - a$ , $∣ b ∣ = - b$ , so $ab = (- a) (- b) = ∣ a ∣∣ b ∣$ .
Case 3: $a$ and $b$ have opposite signs. Say $a \geq 0$ , $b \leq 0$ . Then $ab \leq 0$ , but $∣ a ∣∣ b ∣ \geq 0$ . So automatically $ab \leq ∣ a ∣∣ b ∣$ . Same if $a \leq 0$ , $b \geq 0$ .

Type: Basic Math::Card 1::Back

Deck: Math::Algebra

Tags: Math Algebra Inequality

What is the Triangle Inequality?

The Tirangle Inequality states that for any real numbers $a$ and $b$ :

∣ a + b ∣ \leq ∣ a ∣ + ∣ b ∣

Type: Basic Math::Card 1::Back

Deck: Math::Algebra

Tags: Math Algebra Inequality

Prove the Triangle Inequality $∣ a + b ∣ \leq ∣ a ∣ + ∣ b ∣$ .

Let's square both sides:

∣ a + b ∣^{2} \leq (∣ a ∣ + ∣ b ∣)^{2}

We know that

∣ a + b ∣^{2} ∣ a ∣^{2} ∣ b ∣^{2} = (a + b)^{2} = a^{2} = b^{2}

a^{2} + 2 ab + b^{2} ab \leq a^{2} + 2∣ a ∣∣ b ∣ + b^{2} \leq ∣ a ∣∣ b ∣

The last inequality is true for any $a$ and $b$ , so the Triangle Inequality is true as well.

Type: Basic Math::Card 1::Back

Deck: Math::Algebra

Tags: Math Algebra Inequality

Solve $∣ x - a ∣ \leq c$ , $c \geq 0$ .

- c \leq x - a \leq c

a - c \leq x \leq a + c

x \in [a - c, a + c]

Type: Basic Math::Card 1::Back

Deck: Math::Algebra

Tags: Math Algebra Inequality

Solve $∣ x - a ∣ \geq c$ , $c \geq 0$ .

x - a \leq - c or x - a \geq c

x \leq a - c or x \geq a + c

x \in (- \infty, a - c] \cup [a + c, \infty)

Type: Basic Math::Card 1::Back

Deck: Math::Algebra

Tags: Math Algebra Inequality

Prove that $∣ x ∣ \geq 0$ for all real $x$ .

By definition,

∣ x ∣ = {x, - x, x \geq 0 x < 0

In both cases the value is non-negative, hence

∣ x ∣ \geq 0

Type: Basic Math::Card 1::Back

Deck: Math::Algebra

Tags: Math Algebra Inequality

When does equality hold in the triangle inequality $∣ a + b ∣ \leq ∣ a ∣ + ∣ b ∣$ ?

Equality holds when $a$ and $b$ have the same sign or one of them is zero.

Type: Basic Math::Card 1::Back

Deck: Math::Algebra

Tags: Math Algebra Inequality

Prove the reverse triangle inequality.

The reverse triangle inequality states:

∣ a ∣ - ∣ b ∣ \leq ∣ a - b ∣

Proof:

Apply triangle inequality to $(a - b) + b = a$ :

∣ a ∣ = ∣ (a - b) + b ∣ \leq ∣ a - b ∣ + ∣ b ∣

Rearranging:

∣ a ∣ - ∣ b ∣ \leq ∣ a - b ∣

Swap $a, b$ to get:

∣ b ∣ - ∣ a ∣ \leq ∣ a - b ∣

Combining both gives:

∣ a ∣ - ∣ b ∣ \leq ∣ a - b ∣

Type: Basic Math::Card 1::Back

Deck: Math::Algebra

Tags: Math Algebra Inequality

If $∣ x ∣ < ε$ , what bound does this give on $x$ ?

- ε < x < ε

Type: Basic Math::Card 1::Back

Deck: Math::Algebra

Tags: Math Algebra Inequality

If $∣ x - a ∣ < ε$ , what does this mean geometrically?

$x$ lies within distance $ε$ of $a$ :

a - ε < x < a + ε

Type: Basic Math::Card 1::Back

Deck: Math::Algebra

Tags: Math Algebra Inequality

True or false: If $∣ a ∣ \leq ∣ b ∣$ then $a^{2} \leq b^{2}$ .

True, because

∣ a ∣ \leq ∣ b ∣ \Rightarrow ∣ a ∣^{2} \leq ∣ b ∣^{2} \Rightarrow a^{2} \leq b^{2}

Type: Basic Math::Card 1::Back

Deck: Math::Algebra

Tags: Math Algebra Inequality

If $0 \leq a \leq b$ , what can we say about $a^{n}$ and $b^{n}$ for integer $n \geq 1$ ?

0 \leq a^{n} \leq b^{n}

Powers preserve order for non-negative numbers.

Type: Basic Math::Card 1::Back

Deck: Math::Algebra

Tags: Math Algebra Inequality

True or false: If $a \leq b$ then $a^{3} \leq b^{3}$ .

True, because $f (x) = x^{3}$ is strictly increasing on $R$ .

Type: Basic Math::Card 1::Back

Deck: Math::Algebra

Tags: Math Algebra Inequality

True or false: If $a \leq b$ then $a^{4} \leq b^{4}$ .

False in general.

Counterexample: $a = - 2$ , $b = - 1$ gives $- 2 \leq - 1$ but $16 \neq \leq 1$ .

Type: Basic Math::Card 1::Back

Deck: Math::Algebra

Tags: Math Algebra Inequality

State the AM–GM inequality for two non-negative numbers.

For $a, b \geq 0$ :

\frac{a + b}{2} \geq ab

Equality holds iff $a = b$ .

Type: Basic Math::Card 1::Back

Deck: Math::Algebra

Tags: Math Algebra Inequality

Prove that $x^{2} + 1 \geq 2 x$ for all real $x$ .

Rearrange:

x^{2} - 2 x + 1 \geq 0

(x - 1)^{2} \geq 0

which is always true.

Type: Basic Math::Card 1::Back

Deck: Math::Algebra

Tags: Math Algebra Inequality

Prove that $a^{2} + b^{2} \geq 2 ab$ .

a^{2} + b^{2} - 2 ab = (a - b)^{2} \geq 0

Type: Basic Math::Card 1::Back

Deck: Math::Algebra

Tags: Math Algebra Inequality

Prove that $(a - b)^{2} \geq 0$ implies $a^{2} + b^{2} \geq 2 ab$ .

Expand:

(a - b)^{2} = a^{2} - 2 ab + b^{2} \geq 0

Rearrange:

a^{2} + b^{2} \geq 2 ab

Type: Basic Math::Card 1::Back

Deck: Math::Algebra

Tags: Math Algebra Inequality

Show that $∣ a - b ∣ \geq ∣∣ a ∣ - ∣ b ∣∣$ .

This is the reverse triangle inequality:

∣∣ a ∣ - ∣ b ∣∣ \leq ∣ a - b ∣

Type: Basic Math (and reversed card)::Card 1::Back

Deck: Math::Trigonometry

Tags: Math Trigonometric Identity

sin (α + β)

sin α cos β + cos α sin β

sin (α + β) sin (α - β) cos (α + β) cos (α - β) = sin α cos β + cos α sin β = sin α cos β - cos α sin β = cos α cos β - sin α sin β = cos α cos β + sin α sin β

Type: Basic Math (and reversed card)::Card 2::Back

Deck: Math::Trigonometry

Tags: Math Trigonometric Identity

sin α cos β + cos α sin β

sin (α + β)

sin (α + β) sin (α - β) cos (α + β) cos (α - β) = sin α cos β + cos α sin β = sin α cos β - cos α sin β = cos α cos β - sin α sin β = cos α cos β + sin α sin β

Type: Basic Math (and reversed card)::Card 1::Back

Deck: Math::Trigonometry

Tags: Math Trigonometric Identity

sin (α - β)

sin α cos β - cos α sin β

sin (α + β) sin (α - β) cos (α + β) cos (α - β) = sin α cos β + cos α sin β = sin α cos β - cos α sin β = cos α cos β - sin α sin β = cos α cos β + sin α sin β

Type: Basic Math (and reversed card)::Card 2::Back

Deck: Math::Trigonometry

Tags: Math Trigonometric Identity

sin α cos β - cos α sin β

sin (α - β)

sin (α + β) sin (α - β) cos (α + β) cos (α - β) = sin α cos β + cos α sin β = sin α cos β - cos α sin β = cos α cos β - sin α sin β = cos α cos β + sin α sin β

Type: Basic Math (and reversed card)::Card 1::Back

Deck: Math::Trigonometry

Tags: Math Trigonometric Identity

cos (α + β)

cos α cos β - sin α sin β

sin (α + β) sin (α - β) cos (α + β) cos (α - β) = sin α cos β + cos α sin β = sin α cos β - cos α sin β = cos α cos β - sin α sin β = cos α cos β + sin α sin β

Type: Basic Math (and reversed card)::Card 2::Back

Deck: Math::Trigonometry

Tags: Math Trigonometric Identity

cos α cos β - sin α sin β

cos (α + β)

sin (α + β) sin (α - β) cos (α + β) cos (α - β) = sin α cos β + cos α sin β = sin α cos β - cos α sin β = cos α cos β - sin α sin β = cos α cos β + sin α sin β

Type: Basic Math (and reversed card)::Card 1::Back

Deck: Math::Trigonometry

Tags: Math Trigonometric Identity

cos (α - β)

cos α cos β + sin α sin β

sin (α + β) sin (α - β) cos (α + β) cos (α - β) = sin α cos β + cos α sin β = sin α cos β - cos α sin β = cos α cos β - sin α sin β = cos α cos β + sin α sin β

Type: Basic Math (and reversed card)::Card 2::Back

Deck: Math::Trigonometry

Tags: Math Trigonometric Identity

cos α cos β + sin α sin β

cos (α - β)

sin (α + β) sin (α - β) cos (α + β) cos (α - β) = sin α cos β + cos α sin β = sin α cos β - cos α sin β = cos α cos β - sin α sin β = cos α cos β + sin α sin β

Type: Basic Math (and reversed card)::Card 1::Back

Deck: Math::Trigonometry

Tags: Math Trigonometric Identity

tan (α + β)

\frac{tan α + tan β}{1 - tan α tan β}

tan (α + β) tan (α - β) = \frac{tan α + tan β}{1 - tan α tan β} = \frac{tan α - tan β}{1 + tan α tan β}

Type: Basic Math (and reversed card)::Card 2::Back

Deck: Math::Trigonometry

Tags: Math Trigonometric Identity

\frac{tan α + tan β}{1 - tan α tan β}

tan (α + β)

tan (α + β) tan (α - β) = \frac{tan α + tan β}{1 - tan α tan β} = \frac{tan α - tan β}{1 + tan α tan β}

Type: Basic Math (and reversed card)::Card 1::Back

Deck: Math::Trigonometry

Tags: Math Trigonometric Identity

tan (α - β)

\frac{tan α - tan β}{1 + tan α tan β}

tan (α + β) tan (α - β) = \frac{tan α + tan β}{1 - tan α tan β} = \frac{tan α - tan β}{1 + tan α tan β}

Type: Basic Math (and reversed card)::Card 2::Back

Deck: Math::Trigonometry

Tags: Math Trigonometric Identity

\frac{tan α - tan β}{1 + tan α tan β}

tan (α - β)

tan (α + β) tan (α - β) = \frac{tan α + tan β}{1 - tan α tan β} = \frac{tan α - tan β}{1 + tan α tan β}

Type: Basic Math (and reversed card)::Card 1::Back

Deck: Math::Trigonometry

Tags: Math Trigonometric Identity

sin \frac{θ}{2}

\pm \frac{1 - cos θ}{2}

sin \frac{θ}{2} cos \frac{θ}{2} = \pm \frac{1 - cos θ}{2} = \pm \frac{1 + cos θ}{2}

Type: Basic Math (and reversed card)::Card 2::Back

Deck: Math::Trigonometry

Tags: Math Trigonometric Identity

\pm \frac{1 - cos θ}{2}

sin \frac{θ}{2}

sin \frac{θ}{2} cos \frac{θ}{2} = \pm \frac{1 - cos θ}{2} = \pm \frac{1 + cos θ}{2}

Type: Basic Math (and reversed card)::Card 1::Back

Deck: Math::Trigonometry

Tags: Math Trigonometric Identity

cos \frac{θ}{2}

\pm \frac{1 + cos θ}{2}

sin \frac{θ}{2} cos \frac{θ}{2} = \pm \frac{1 - cos θ}{2} = \pm \frac{1 + cos θ}{2}

Type: Basic Math (and reversed card)::Card 2::Back

Deck: Math::Trigonometry

Tags: Math Trigonometric Identity

\pm \frac{1 + cos θ}{2}

cos \frac{θ}{2}

sin \frac{θ}{2} cos \frac{θ}{2} = \pm \frac{1 - cos θ}{2} = \pm \frac{1 + cos θ}{2}

Type: Basic Math (and reversed card)::Card 1::Back

Deck: Math::Trigonometry

Tags: Math Trigonometric Identity

tan \frac{θ}{2}

\pm \frac{1 - cos θ}{1 + cos θ}

\frac{sin θ}{1 + cos θ}

\frac{1 - cos θ}{sin θ}

Type: Basic Math (and reversed card)::Card 2::Back

Deck: Math::Trigonometry

Tags: Math Trigonometric Identity

\pm \frac{1 - cos θ}{1 + cos θ}

\frac{sin θ}{1 + cos θ}

\frac{1 - cos θ}{sin θ}

tan \frac{θ}{2}

Type: Basic Math (and reversed card)::Card 1::Back

Deck: Math::Trigonometry

Tags: Math Trigonometric Identity

sin^{2} θ

\frac{1 - cos 2 θ}{2}

sin^{2} θ cos^{2} θ = \frac{1 - cos 2 θ}{2} = \frac{1 + cos 2 θ}{2}

Type: Basic Math (and reversed card)::Card 2::Back

Deck: Math::Trigonometry

Tags: Math Trigonometric Identity

\frac{1 - cos 2 θ}{2}

sin^{2} θ

sin^{2} θ cos^{2} θ = \frac{1 - cos 2 θ}{2} = \frac{1 + cos 2 θ}{2}

Type: Basic Math (and reversed card)::Card 1::Back

Deck: Math::Trigonometry

Tags: Math Trigonometric Identity

cos^{2} θ

\frac{1 + cos 2 θ}{2}

sin^{2} θ cos^{2} θ = \frac{1 - cos 2 θ}{2} = \frac{1 + cos 2 θ}{2}

Type: Basic Math (and reversed card)::Card 2::Back

Deck: Math::Trigonometry

Tags: Math Trigonometric Identity

\frac{1 + cos 2 θ}{2}

cos^{2} θ

sin^{2} θ cos^{2} θ = \frac{1 - cos 2 θ}{2} = \frac{1 + cos 2 θ}{2}

Type: Basic Math (and reversed card)::Card 1::Back

Deck: Math::Trigonometry

Tags: Math Trigonometric Identity

tan^{2} θ

\frac{1 - cos 2 θ}{1 + cos 2 θ}

Type: Basic Math (and reversed card)::Card 2::Back

Deck: Math::Trigonometry

Tags: Math Trigonometric Identity

\frac{1 - cos 2 θ}{1 + cos 2 θ}

tan^{2} θ

Type: Basic Math (and reversed card)::Card 1::Back

Deck: Math::Trigonometry

Tags: Math Trigonometric Identity

sin 2 θ

2 sin θ cos θ

Type: Basic Math (and reversed card)::Card 2::Back

Deck: Math::Trigonometry

Tags: Math Trigonometric Identity

2 sin θ cos θ

sin 2 θ

Type: Basic Math (and reversed card)::Card 1::Back

Deck: Math::Trigonometry

Tags: Math Trigonometric Identity

cos 2 θ

cos^{2} θ - sin^{2} θ

1 - 2 sin^{2} θ

2 cos^{2} θ - 1

Type: Basic Math (and reversed card)::Card 2::Back

Deck: Math::Trigonometry

Tags: Math Trigonometric Identity

cos^{2} θ - sin^{2} θ

1 - 2 sin^{2} θ

2 cos^{2} θ - 1

cos 2 θ

Type: Basic Math (and reversed card)::Card 1::Back

Deck: Math::Trigonometry

Tags: Math Trigonometric Identity

tan 2 θ

\frac{2 tan θ}{1 - tan ^{2} θ}

\frac{2}{cot θ - tan θ}

Type: Basic Math (and reversed card)::Card 2::Back

Deck: Math::Trigonometry

Tags: Math Trigonometric Identity

\frac{2 tan θ}{1 - tan ^{2} θ}

\frac{2}{cot θ - tan θ}

tan 2 θ

Type: Basic Math (and reversed card)::Card 1::Back

Deck: Math::Trigonometry

Tags: Math Trigonometric Identity

sin α \cdot sin β

\frac{1}{2} [cos (α - β) - cos (α + β)]

sin α \cdot sin β cos α \cdot cos β sin α \cdot cos β cos α \cdot sin β = \frac{1}{2} [cos (α - β) - cos (α + β)] = \frac{1}{2} [cos (α - β) + cos (α + β)] = \frac{1}{2} [sin (α + β) + sin (α - β)] = \frac{1}{2} [sin (α + β) - sin (α - β)]

Type: Basic Math (and reversed card)::Card 2::Back

Deck: Math::Trigonometry

Tags: Math Trigonometric Identity

\frac{1}{2} [cos (α - β) - cos (α + β)]

sin α \cdot sin β

sin α \cdot sin β cos α \cdot cos β sin α \cdot cos β cos α \cdot sin β = \frac{1}{2} [cos (α - β) - cos (α + β)] = \frac{1}{2} [cos (α - β) + cos (α + β)] = \frac{1}{2} [sin (α + β) + sin (α - β)] = \frac{1}{2} [sin (α + β) - sin (α - β)]

Type: Basic Math (and reversed card)::Card 1::Back

Deck: Math::Trigonometry

Tags: Math Trigonometric Identity

cos α \cdot cos β

\frac{1}{2} [cos (α - β) + cos (α + β)]

sin α \cdot sin β cos α \cdot cos β sin α \cdot cos β cos α \cdot sin β = \frac{1}{2} [cos (α - β) - cos (α + β)] = \frac{1}{2} [cos (α - β) + cos (α + β)] = \frac{1}{2} [sin (α + β) + sin (α - β)] = \frac{1}{2} [sin (α + β) - sin (α - β)]

Type: Basic Math (and reversed card)::Card 2::Back

Deck: Math::Trigonometry

Tags: Math Trigonometric Identity

\frac{1}{2} [cos (α - β) + cos (α + β)]

cos α \cdot cos β

sin α \cdot sin β cos α \cdot cos β sin α \cdot cos β cos α \cdot sin β = \frac{1}{2} [cos (α - β) - cos (α + β)] = \frac{1}{2} [cos (α - β) + cos (α + β)] = \frac{1}{2} [sin (α + β) + sin (α - β)] = \frac{1}{2} [sin (α + β) - sin (α - β)]

Type: Basic Math (and reversed card)::Card 1::Back

Deck: Math::Trigonometry

Tags: Math Trigonometric Identity

sin α \cdot cos β

\frac{1}{2} [sin (α + β) + sin (α - β)]

sin α \cdot sin β cos α \cdot cos β sin α \cdot cos β cos α \cdot sin β = \frac{1}{2} [cos (α - β) - cos (α + β)] = \frac{1}{2} [cos (α - β) + cos (α + β)] = \frac{1}{2} [sin (α + β) + sin (α - β)] = \frac{1}{2} [sin (α + β) - sin (α - β)]

Type: Basic Math (and reversed card)::Card 2::Back

Deck: Math::Trigonometry

Tags: Math Trigonometric Identity

\frac{1}{2} [sin (α + β) + sin (α - β)]

sin α \cdot cos β

sin α \cdot sin β cos α \cdot cos β sin α \cdot cos β cos α \cdot sin β = \frac{1}{2} [cos (α - β) - cos (α + β)] = \frac{1}{2} [cos (α - β) + cos (α + β)] = \frac{1}{2} [sin (α + β) + sin (α - β)] = \frac{1}{2} [sin (α + β) - sin (α - β)]

Type: Basic Math (and reversed card)::Card 1::Back

Deck: Math::Trigonometry

Tags: Math Trigonometric Identity

cos α \cdot sin β

\frac{1}{2} [sin (α + β) - sin (α - β)]

sin α \cdot sin β cos α \cdot cos β sin α \cdot cos β cos α \cdot sin β = \frac{1}{2} [cos (α - β) - cos (α + β)] = \frac{1}{2} [cos (α - β) + cos (α + β)] = \frac{1}{2} [sin (α + β) + sin (α - β)] = \frac{1}{2} [sin (α + β) - sin (α - β)]

Type: Basic Math (and reversed card)::Card 2::Back

Deck: Math::Trigonometry

Tags: Math Trigonometric Identity

\frac{1}{2} [sin (α + β) - sin (α - β)]

cos α \cdot sin β

sin α \cdot sin β cos α \cdot cos β sin α \cdot cos β cos α \cdot sin β = \frac{1}{2} [cos (α - β) - cos (α + β)] = \frac{1}{2} [cos (α - β) + cos (α + β)] = \frac{1}{2} [sin (α + β) + sin (α - β)] = \frac{1}{2} [sin (α + β) - sin (α - β)]

Type: Basic Math (and reversed card)::Card 1::Back

Deck: Math::Trigonometry

Tags: Math Trigonometric Identity

sin α + sin β

2 sin (\frac{α + β}{2}) cos (\frac{α - β}{2})

sin α + sin β sin α - sin β cos α + cos β cos α - cos β = 2 sin (\frac{α + β}{2}) cos (\frac{α - β}{2}) = 2 cos (\frac{α + β}{2}) sin (\frac{α - β}{2}) = 2 cos (\frac{α + β}{2}) cos (\frac{α - β}{2}) = 2 sin (\frac{α + β}{2}) sin (\frac{α - β}{2})

Type: Basic Math (and reversed card)::Card 2::Back

Deck: Math::Trigonometry

Tags: Math Trigonometric Identity

2 sin (\frac{α + β}{2}) cos (\frac{α - β}{2})

sin α + sin β

sin α + sin β sin α - sin β cos α + cos β cos α - cos β = 2 sin (\frac{α + β}{2}) cos (\frac{α - β}{2}) = 2 cos (\frac{α + β}{2}) sin (\frac{α - β}{2}) = 2 cos (\frac{α + β}{2}) cos (\frac{α - β}{2}) = 2 sin (\frac{α + β}{2}) sin (\frac{α - β}{2})

Type: Basic Math (and reversed card)::Card 1::Back

Deck: Math::Trigonometry

Tags: Math Trigonometric Identity

sin α - sin β

2 cos (\frac{α + β}{2}) sin (\frac{α - β}{2})

sin α + sin β sin α - sin β cos α + cos β cos α - cos β = 2 sin (\frac{α + β}{2}) cos (\frac{α - β}{2}) = 2 cos (\frac{α + β}{2}) sin (\frac{α - β}{2}) = 2 cos (\frac{α + β}{2}) cos (\frac{α - β}{2}) = 2 sin (\frac{α + β}{2}) sin (\frac{α - β}{2})

Type: Basic Math (and reversed card)::Card 2::Back

Deck: Math::Trigonometry

Tags: Math Trigonometric Identity

2 cos (\frac{α + β}{2}) sin (\frac{α - β}{2})

sin α - sin β

sin α + sin β sin α - sin β cos α + cos β cos α - cos β = 2 sin (\frac{α + β}{2}) cos (\frac{α - β}{2}) = 2 cos (\frac{α + β}{2}) sin (\frac{α - β}{2}) = 2 cos (\frac{α + β}{2}) cos (\frac{α - β}{2}) = 2 sin (\frac{α + β}{2}) sin (\frac{α - β}{2})

Type: Basic Math (and reversed card)::Card 1::Back

Deck: Math::Trigonometry

Tags: Math Trigonometric Identity

cos α + cos β

2 cos (\frac{α + β}{2}) cos (\frac{α - β}{2})

sin α + sin β sin α - sin β cos α + cos β cos α - cos β = 2 sin (\frac{α + β}{2}) cos (\frac{α - β}{2}) = 2 cos (\frac{α + β}{2}) sin (\frac{α - β}{2}) = 2 cos (\frac{α + β}{2}) cos (\frac{α - β}{2}) = 2 sin (\frac{α + β}{2}) sin (\frac{α - β}{2})

Type: Basic Math (and reversed card)::Card 2::Back

Deck: Math::Trigonometry

Tags: Math Trigonometric Identity

2 cos (\frac{α + β}{2}) cos (\frac{α - β}{2})

cos α + cos β

sin α + sin β sin α - sin β cos α + cos β cos α - cos β = 2 sin (\frac{α + β}{2}) cos (\frac{α - β}{2}) = 2 cos (\frac{α + β}{2}) sin (\frac{α - β}{2}) = 2 cos (\frac{α + β}{2}) cos (\frac{α - β}{2}) = 2 sin (\frac{α + β}{2}) sin (\frac{α - β}{2})

Type: Basic Math (and reversed card)::Card 1::Back

Deck: Math::Trigonometry

Tags: Math Trigonometric Identity

cos α - cos β

2 sin (\frac{α + β}{2}) sin (\frac{α - β}{2})

sin α + sin β sin α - sin β cos α + cos β cos α - cos β = 2 sin (\frac{α + β}{2}) cos (\frac{α - β}{2}) = 2 cos (\frac{α + β}{2}) sin (\frac{α - β}{2}) = 2 cos (\frac{α + β}{2}) cos (\frac{α - β}{2}) = 2 sin (\frac{α + β}{2}) sin (\frac{α - β}{2})

Type: Basic Math (and reversed card)::Card 2::Back

Deck: Math::Trigonometry

Tags: Math Trigonometric Identity

2 sin (\frac{α + β}{2}) sin (\frac{α - β}{2})

cos α - cos β

sin α + sin β sin α - sin β cos α + cos β cos α - cos β = 2 sin (\frac{α + β}{2}) cos (\frac{α - β}{2}) = 2 cos (\frac{α + β}{2}) sin (\frac{α - β}{2}) = 2 cos (\frac{α + β}{2}) cos (\frac{α - β}{2}) = 2 sin (\frac{α + β}{2}) sin (\frac{α - β}{2})

Type: Basic Math::Card 1::Back

Deck: Math::Trigonometry

Tags: Math Trigonometric Hyperbolic Identity

sinh x

The definition of

\frac{e ^{x} - e ^{- x}}{2}

Type: Basic Math::Card 1::Back

Deck: Math::Trigonometry

Tags: Math Trigonometric Hyperbolic Identity

cosh x

The definition of

\frac{e ^{x} + e ^{- x}}{2}

Type: Basic Math::Card 1::Back

Deck: Math::Trigonometry

Tags: Math Trigonometric Hyperbolic Identity

tanh x

The definition of

\frac{sinh x}{cosh x} = \frac{e ^{x} - e ^{- x}}{e ^{x} + e ^{- x}}

Type: Basic Math::Card 1::Back

Deck: Math::Trigonometry

Tags: Math Trigonometric Hyperbolic Identity

e^{x} = \dots

Hyperbolic identity

cosh x + sinh x

Type: Basic Math::Card 1::Back

Deck: Math::Trigonometry

Tags: Math Trigonometric Hyperbolic Identity

e^{- x} = \dots

Hyperbolic identity

cosh x - sinh x

Type: Basic Math::Card 1::Back

Deck: Math::Trigonometry

Tags: Math Trigonometric Hyperbolic Identity

1 = \dots

Hyperbolic identity

cosh^{2} x - sinh^{2} x

Type: Basic Math::Card 1::Back

Deck: Math::Trigonometry

Tags: Math Trigonometric Hyperbolic Identity

sin x = \dots

Express as a hyperbolic function

- i sinh (i x) = \frac{e ^{i x} - e ^{- i x}}{2 i}

Type: Basic Math::Card 1::Back

Deck: Math::Trigonometry

Tags: Math Trigonometric Hyperbolic Identity

sinh x = \dots

Express as a trigonometric function

- i sin (i x) = \frac{e ^{x} - e ^{- x}}{2}

Type: Basic Math::Card 1::Back

Deck: Math::Trigonometry

Tags: Math Trigonometric Hyperbolic Identity

cos x = \dots

Express as a hyperbolic function

cosh (i x) = \frac{e ^{i x} + e ^{- i x}}{2}

Type: Basic Math::Card 1::Back

Deck: Math::Trigonometry

Tags: Math Trigonometric Hyperbolic Identity

cosh x = \dots

Express as a trigonometric function

cos (i x) = \frac{e ^{x} + e ^{- x}}{2}

Type: Basic Math::Card 1::Back

Deck: Math::Trigonometry

Tags: Math Trigonometric Hyperbolic Identity

cosh^{2} x + sinh^{2} x = \dots

cosh (2 x)

Type: Basic Math::Card 1::Back

Deck: Math::Trigonometry

Tags: Math Trigonometric Hyperbolic Identity

1 + sinh^{2} x = \dots

cosh^{2} x

Type: Basic Math::Card 1::Back

Deck: Math::Trigonometry

Tags: Math Trigonometric Hyperbolic Identity

1 - tanh^{2} x = \dots

sech^{2} x

Type: Basic Math::Card 1::Back

Deck: Math::Trigonometry

Tags: Math Trigonometric Hyperbolic Identity

sech x

Definition

\frac{1}{cosh x}

Type: Basic Math::Card 1::Back

Deck: Math::Trigonometry

Tags: Math Trigonometric Hyperbolic Identity

Euler's Formula

e^{i x} e^{- i x} = cos x + i sin x = cos x - i sin x

Type: Basic Math::Card 1::Back

Deck: Math::Trigonometry

Tags: Math Trigonometric Hyperbolic Identity

De Moivre's Formula

(cos (x) + i sin (x))^{n} (cosh (x) + i sinh (x))^{n} = cos (n x) + i sin (n x) = cosh (n x) + i sinh (n x)

Type: Basic Math::Card 1::Back

Deck: Math::Calculus::Identities::Limits

Tags: Math Calculus Identity Limits

The definition of $e$ as a limit when $n \to \infty$

e = n \to \infty lim (1 + \frac{1}{n})^{n}

Type: Basic Math::Card 1::Back

Deck: Math::Calculus::Identities::Limits

Tags: Math Calculus Identity Limits

The definition of $e^{x}$ as a limit when $n \to \infty$

e^{x} = n \to \infty lim (1 + \frac{x}{n})^{n}

Type: Basic Math::Card 1::Back

Deck: Math::Calculus::Identities::Limits

Tags: Math Calculus Identity Limits

The definition of $e$ as a limit when $n \to 0$

e = n \to 0 lim (1 + n)^{1/ n}

Type: Basic Math::Card 1::Back

Deck: Math::Calculus::Identities::Limits

Tags: Math Calculus Identity Limits

The definition of $e^{x}$ as a limit when $n \to 0$

e^{x} = n \to 0 lim (1 + n)^{x / n}

Type: Basic Math::Card 1::Back

Deck: Math::Calculus::Identities::Derivatives

Tags: Math Calculus Identity Derivative

\frac{d}{d x} (f (x) g (x))

f^{'} (x) g (x) + f (x) g^{'} (x)

Type: Basic Math::Card 1::Back

Deck: Math::Calculus::Identities::Derivatives

Tags: Math Calculus Identity Derivative

\frac{d}{d x} (\frac{f ( x )}{g ( x )})

\frac{f ^{'} ( x ) g ( x ) - f ( x ) g ^{'} ( x )}{( g ( x ) ^{2} )}

Type: Basic Math::Card 1::Back

Deck: Math::Calculus::Identities::Derivatives

Tags: Math Calculus Identity Derivative

\frac{d}{d x} (x^{n})

n x^{n - 1}

Type: Basic Math::Card 1::Back

Deck: Math::Calculus::Identities::Derivatives

Tags: Math Calculus Identity Derivative

\frac{d}{d x} (x)

\frac{1}{2 x}

Type: Basic Math::Card 1::Back

Deck: Math::Calculus::Identities::Derivatives

Tags: Math Calculus Identity Derivative

\frac{d}{d x} (e^{x})

e^{x}

Type: Basic Math::Card 1::Back

Deck: Math::Calculus::Identities::Derivatives

Tags: Math Calculus Identity Derivative

\frac{d}{d x} (a^{x})

a^{x} ln a

Type: Basic Math::Card 1::Back

Deck: Math::Calculus::Identities::Derivatives

Tags: Math Calculus Identity Derivative

\frac{d}{d x} (ln x)

\frac{1}{x}

Type: Basic Math::Card 1::Back

Deck: Math::Calculus::Identities::Derivatives

Tags: Math Calculus Identity Derivative

\frac{d}{d x} (lo g_{b} x)

\frac{1}{x ln b}

Type: Basic Math::Card 1::Back

Deck: Math::Calculus::Identities::Derivatives

Tags: Math Calculus Derivative Trigonometric Identity

\frac{d}{d x} (sin x)

cos x

\frac{d}{d x} (sin x) \frac{d}{d x} (cos x) \frac{d}{d x} (sinh x) \frac{d}{d x} (cosh x) = cos x = - sin x = cosh x = sinh x

Type: Basic Math::Card 1::Back

Deck: Math::Calculus::Identities::Derivatives

Tags: Math Calculus Derivative Trigonometric Identity

\frac{d}{d x} (cos x)

- sin x

\frac{d}{d x} (sin x) \frac{d}{d x} (cos x) \frac{d}{d x} (sinh x) \frac{d}{d x} (cosh x) = cos x = - sin x = cosh x = sinh x

Type: Basic Math::Card 1::Back

Deck: Math::Calculus::Identities::Derivatives

Tags: Math Calculus Derivative Trigonometric Identity

\frac{d}{d x} (tan x)

sec^{2} x

\frac{d}{d x} (tan x) \frac{d}{d x} (cot x) \frac{d}{d x} (tanh x) \frac{d}{d x} (coth x) = sec^{2} x = - csc^{2} x = sech^{2} x = - csch^{2} x

Type: Basic Math::Card 1::Back

Deck: Math::Calculus::Identities::Derivatives

Tags: Math Calculus Derivative Trigonometric Identity

\frac{d}{d x} (cot x)

- csc^{2} x

\frac{d}{d x} (tan x) \frac{d}{d x} (cot x) \frac{d}{d x} (tanh x) \frac{d}{d x} (coth x) = sec^{2} x = - csc^{2} x = sech^{2} x = - csch^{2} x

Type: Basic Math::Card 1::Back

Deck: Math::Calculus::Identities::Derivatives

Tags: Math Calculus Derivative Trigonometric Identity

\frac{d}{d x} (sec x)

sec x tan x

\frac{d}{d x} (sec x) \frac{d}{d x} (csc x) \frac{d}{d x} (sech x) \frac{d}{d x} (csch x) = sec x tan x = - csc x cot x = - sech x tanh x = - csch x coth x

Type: Basic Math::Card 1::Back

Deck: Math::Calculus::Identities::Derivatives

Tags: Math Calculus Derivative Trigonometric Identity

\frac{d}{d x} (csc x)

- csc x cot x

\frac{d}{d x} (sec x) \frac{d}{d x} (csc x) \frac{d}{d x} (sech x) \frac{d}{d x} (csch x) = sec x tan x = - csc x cot x = - sech x tanh x = - csch x coth x

Type: Basic Math::Card 1::Back

Deck: Math::Calculus::Identities::Derivatives

Tags: Math Calculus Derivative Trigonometric Identity

\frac{d}{d x} (sin^{- 1} x)

\frac{1}{1 - x ^{2}}

\frac{d}{d x} (sin^{- 1} x) \frac{d}{d x} (cos^{- 1} x) \frac{d}{d x} (sinh^{- 1} x) \frac{d}{d x} (cosh^{- 1} x) = \frac{1}{1 - x ^{2}} = \frac{- 1}{1 - x ^{2}} = \frac{1}{x ^{2} + 1} = \frac{1}{x ^{2} - 1}, (x > 1)

Type: Basic Math::Card 1::Back

Deck: Math::Calculus::Identities::Derivatives

Tags: Math Calculus Derivative Trigonometric Identity

\frac{d}{d x} (cos^{- 1} x)

\frac{- 1}{1 - x ^{2}}

\frac{d}{d x} (sin^{- 1} x) \frac{d}{d x} (cos^{- 1} x) \frac{d}{d x} (sinh^{- 1} x) \frac{d}{d x} (cosh^{- 1} x) = \frac{1}{1 - x ^{2}} = \frac{- 1}{1 - x ^{2}} = \frac{1}{x ^{2} + 1} = \frac{1}{x ^{2} - 1}, (x > 1)

Type: Basic Math::Card 1::Back

Deck: Math::Calculus::Identities::Derivatives

Tags: Math Calculus Derivative Trigonometric Identity

\frac{d}{d x} (tan^{- 1} x)

\frac{1}{1 + x ^{2}}

\frac{d}{d x} (tan^{- 1} x) \frac{d}{d x} (cot^{- 1} x) \frac{d}{d x} (tanh^{- 1} x) \frac{d}{d x} (coth^{- 1} x) = \frac{1}{1 + x ^{2}} = \frac{- 1}{1 + x ^{2}} = \frac{1}{1 - x ^{2}}, (∣ x ∣ < 1) = \frac{1}{1 - x ^{2}}, (∣ x ∣ > 1)

Type: Basic Math::Card 1::Back

Deck: Math::Calculus::Identities::Derivatives

Tags: Math Calculus Derivative Trigonometric Identity

\frac{d}{d x} (cot^{- 1} x)

\frac{- 1}{1 + x ^{2}}

\frac{d}{d x} (tan^{- 1} x) \frac{d}{d x} (cot^{- 1} x) \frac{d}{d x} (tanh^{- 1} x) \frac{d}{d x} (coth^{- 1} x) = \frac{1}{1 + x ^{2}} = \frac{- 1}{1 + x ^{2}} = \frac{1}{1 - x ^{2}}, (∣ x ∣ < 1) = \frac{1}{1 - x ^{2}}, (∣ x ∣ > 1)

Type: Basic Math::Card 1::Back

Deck: Math::Calculus::Identities::Derivatives

Tags: Math Calculus Derivative Trigonometric Identity

\frac{d}{d x} (sec^{- 1} x)

\frac{1}{∣ x ∣ x ^{2} - 1}

\frac{d}{d x} (sec^{- 1} x) \frac{d}{d x} (csc^{- 1} x) \frac{d}{d x} (sech^{- 1} x) \frac{d}{d x} (csch^{- 1} x) = \frac{1}{∣ x ∣ x ^{2} - 1} = \frac{- 1}{∣ x ∣ x ^{2} - 1} = \frac{- 1}{x 1 - x ^{2}}, (0 < x < 1) = \frac{- 1}{∣ x ∣ 1 + x ^{2}}, (x \neq = 0)

Type: Basic Math::Card 1::Back

Deck: Math::Calculus::Identities::Derivatives

Tags: Math Calculus Derivative Trigonometric Identity

\frac{d}{d x} (csc^{- 1} x)

\frac{- 1}{∣ x ∣ x ^{2} - 1}

\frac{d}{d x} (sec^{- 1} x) \frac{d}{d x} (csc^{- 1} x) \frac{d}{d x} (sech^{- 1} x) \frac{d}{d x} (csch^{- 1} x) = \frac{1}{∣ x ∣ x ^{2} - 1} = \frac{- 1}{∣ x ∣ x ^{2} - 1} = \frac{- 1}{x 1 - x ^{2}}, (0 < x < 1) = \frac{- 1}{∣ x ∣ 1 + x ^{2}}, (x \neq = 0)

Type: Basic Math::Card 1::Back

Deck: Math::Calculus::Identities::Derivatives

Tags: Math Calculus Derivative Hyperbolic Identity

\frac{d}{d x} (sinh) x

cosh x

\frac{d}{d x} (sin x) \frac{d}{d x} (cos x) \frac{d}{d x} (sinh x) \frac{d}{d x} (cosh x) = cos x = - sin x = cosh x = sinh x

Type: Basic Math::Card 1::Back

Deck: Math::Calculus::Identities::Derivatives

Tags: Math Calculus Derivative Hyperbolic Identity

\frac{d}{d x} (cosh) x

sinh x

\frac{d}{d x} (sin x) \frac{d}{d x} (cos x) \frac{d}{d x} (sinh x) \frac{d}{d x} (cosh x) = cos x = - sin x = cosh x = sinh x

Type: Basic Math::Card 1::Back

Deck: Math::Calculus::Identities::Derivatives

Tags: Math Calculus Derivative Hyperbolic Identity

\frac{d}{d x} (tanh x)

sech^{2} x

\frac{d}{d x} (tan x) \frac{d}{d x} (cot x) \frac{d}{d x} (tanh x) \frac{d}{d x} (coth x) = sec^{2} x = - csc^{2} x = sech^{2} x = - csch^{2} x

Type: Basic Math::Card 1::Back

Deck: Math::Calculus::Identities::Derivatives

Tags: Math Calculus Derivative Hyperbolic Identity

\frac{d}{d x} (coth x)

- csch^{2} x

\frac{d}{d x} (tan x) \frac{d}{d x} (cot x) \frac{d}{d x} (tanh x) \frac{d}{d x} (coth x) = sec^{2} x = - csc^{2} x = sech^{2} x = - csch^{2} x

Type: Basic Math::Card 1::Back

Deck: Math::Calculus::Identities::Derivatives

Tags: Math Calculus Derivative Hyperbolic Identity

\frac{d}{d x} (sech x)

- sech x tanh x

\frac{d}{d x} (sec x) \frac{d}{d x} (csc x) \frac{d}{d x} (sech x) \frac{d}{d x} (csch x) = sec x tan x = - csc x cot x = - sech x tanh x = - csch x coth x

Type: Basic Math::Card 1::Back

Deck: Math::Calculus::Identities::Derivatives

Tags: Math Calculus Derivative Hyperbolic Identity

\frac{d}{d x} (csch x)

- csch x coth x

\frac{d}{d x} (sec x) \frac{d}{d x} (csc x) \frac{d}{d x} (sech x) \frac{d}{d x} (csch x) = sec x tan x = - csc x cot x = - sech x tanh x = - csch x coth x

Type: Basic Math::Card 1::Back

Deck: Math::Calculus::Identities::Derivatives

Tags: Math Calculus Derivative Hyperbolic Identity

\frac{d}{d x} (sinh^{- 1} x)

\frac{1}{x ^{2} + 1}

\frac{d}{d x} (sin^{- 1} x) \frac{d}{d x} (cos^{- 1} x) \frac{d}{d x} (sinh^{- 1} x) \frac{d}{d x} (cosh^{- 1} x) = \frac{1}{1 - x ^{2}} = \frac{- 1}{1 - x ^{2}} = \frac{1}{x ^{2} + 1} = \frac{1}{x ^{2} - 1}, (x > 1)

Type: Basic Math::Card 1::Back

Deck: Math::Calculus::Identities::Derivatives

Tags: Math Calculus Derivative Hyperbolic Identity

\frac{d}{d x} (cosh^{- 1} x)

\frac{1}{x ^{2} - 1}, (x > 1)

\frac{d}{d x} (sin^{- 1} x) \frac{d}{d x} (cos^{- 1} x) \frac{d}{d x} (sinh^{- 1} x) \frac{d}{d x} (cosh^{- 1} x) = \frac{1}{1 - x ^{2}} = \frac{- 1}{1 - x ^{2}} = \frac{1}{x ^{2} + 1} = \frac{1}{x ^{2} - 1}, (x > 1)

Type: Basic Math::Card 1::Back

Deck: Math::Calculus::Identities::Derivatives

Tags: Math Calculus Derivative Hyperbolic Identity

\frac{d}{d x} (tanh^{- 1} x)

\frac{1}{1 - x ^{2}}, (∣ x ∣ < 1)

\frac{d}{d x} (tan^{- 1} x) \frac{d}{d x} (cot^{- 1} x) \frac{d}{d x} (tanh^{- 1} x) \frac{d}{d x} (coth^{- 1} x) = \frac{1}{1 + x ^{2}} = \frac{- 1}{1 + x ^{2}} = \frac{1}{1 - x ^{2}}, (∣ x ∣ < 1) = \frac{1}{1 - x ^{2}}, (∣ x ∣ > 1)

Type: Basic Math::Card 1::Back

Deck: Math::Calculus::Identities::Derivatives

Tags: Math Calculus Derivative Hyperbolic Identity

\frac{d}{d x} (coth^{- 1} x)

\frac{1}{1 - x ^{2}}, (∣ x ∣ > 1)

\frac{d}{d x} (tan^{- 1} x) \frac{d}{d x} (cot^{- 1} x) \frac{d}{d x} (tanh^{- 1} x) \frac{d}{d x} (coth^{- 1} x) = \frac{1}{1 + x ^{2}} = \frac{- 1}{1 + x ^{2}} = \frac{1}{1 - x ^{2}}, (∣ x ∣ < 1) = \frac{1}{1 - x ^{2}}, (∣ x ∣ > 1)

Type: Basic Math::Card 1::Back

Deck: Math::Calculus::Identities::Derivatives

Tags: Math Calculus Derivative Hyperbolic Identity

\frac{d}{d x} (sech^{- 1} x)

\frac{- 1}{x 1 - x ^{2}}, (0 < x < 1)

\frac{d}{d x} (sec^{- 1} x) \frac{d}{d x} (csc^{- 1} x) \frac{d}{d x} (sech^{- 1} x) \frac{d}{d x} (csch^{- 1} x) = \frac{1}{∣ x ∣ x ^{2} - 1} = \frac{- 1}{∣ x ∣ x ^{2} - 1} = \frac{- 1}{x 1 - x ^{2}}, (0 < x < 1) = \frac{- 1}{∣ x ∣ 1 + x ^{2}}, (x \neq = 0)

Type: Basic Math::Card 1::Back

Deck: Math::Calculus::Identities::Derivatives

Tags: Math Calculus Derivative Hyperbolic Identity

\frac{d}{d x} (csch^{- 1} x)

\frac{- 1}{∣ x ∣ 1 + x ^{2}}, (x \neq = 0)

\frac{d}{d x} (sec^{- 1} x) \frac{d}{d x} (csc^{- 1} x) \frac{d}{d x} (sech^{- 1} x) \frac{d}{d x} (csch^{- 1} x) = \frac{1}{∣ x ∣ x ^{2} - 1} = \frac{- 1}{∣ x ∣ x ^{2} - 1} = \frac{- 1}{x 1 - x ^{2}}, (0 < x < 1) = \frac{- 1}{∣ x ∣ 1 + x ^{2}}, (x \neq = 0)

Type: Basic Math::Card 1::Back

Deck: Math::Calculus::Identities::Integrals

Tags: Math Calculus Identity Integral

\int u^{n} d u

\frac{u ^{n + 1}}{n + 1} + C, n \neq = - 1

Type: Basic Math::Card 1::Back

Deck: Math::Calculus::Identities::Integrals

Tags: Math Calculus Identity Integral

\int \frac{1}{u} d u

ln ∣ u ∣ + C

Type: Basic Math::Card 1::Back

Deck: Math::Calculus::Identities::Integrals

Tags: Math Calculus Identity Integral

\int e^{u} d u

e^{u} + C

Type: Basic Math::Card 1::Back

Deck: Math::Calculus::Identities::Integrals

Tags: Math Calculus Identity Integral

\int a^{u} d u

\frac{a ^{u}}{ln a} + C

Type: Basic Math::Card 1::Back

Deck: Math::Calculus::Identities::Integrals

Tags: Math Calculus Integral Trigonometric Identity

\int sin u d u

- cos u + C

Type: Basic Math::Card 1::Back

Deck: Math::Calculus::Identities::Integrals

Tags: Math Calculus Integral Trigonometric Identity

\int cos u d u

sin u + C

Type: Basic Math::Card 1::Back

Deck: Math::Calculus::Identities::Integrals

Tags: Math Calculus Integral Trigonometric Identity

\int sec^{2} u d u

tan u + C

Type: Basic Math::Card 1::Back

Deck: Math::Calculus::Identities::Integrals

Tags: Math Calculus Integral Trigonometric Identity

\int csc^{2} u d u

- cot u + C

Type: Basic Math::Card 1::Back

Deck: Math::Calculus::Identities::Integrals

Tags: Math Calculus Integral Trigonometric Identity

\int sec u tan u d u

sec u + C

Type: Basic Math::Card 1::Back

Deck: Math::Calculus::Identities::Integrals

Tags: Math Calculus Integral Trigonometric Identity

\int csc u cot u d u

- csc u + C

Type: Basic Math::Card 1::Back

Deck: Math::Calculus::Identities::Integrals

Tags: Math Calculus Integral Trigonometric Identity

\int tan u d u

- ln ∣ cos u ∣ + C

ln ∣ sec u ∣ + C

Type: Basic Math::Card 1::Back

Deck: Math::Calculus::Identities::Integrals

Tags: Math Calculus Integral Trigonometric Identity

\int cot u d u

ln ∣ sin u ∣ + C

- ln ∣ csc u ∣ + C

Type: Basic Math::Card 1::Back

Deck: Math::Calculus::Identities::Integrals

Tags: Math Calculus Integral Trigonometric Identity

\int sec u d u

ln ∣ sec u + tan u ∣ + C

Type: Basic Math::Card 1::Back

Deck: Math::Calculus::Identities::Integrals

Tags: Math Calculus Integral Trigonometric Identity

\int csc u d u

ln ∣ csc u - cot u ∣ + C

Type: Basic Math::Card 1::Back

Deck: Math::Calculus::Identities::Integrals

Tags: Math Calculus Integral Trigonometric Identity

\int \frac{1}{a ^{2} - u ^{2}} d u

sin^{- 1} \frac{u}{a} + C

Type: Basic Math::Card 1::Back

Deck: Math::Calculus::Identities::Integrals

Tags: Math Calculus Integral Trigonometric Identity

\int \frac{1}{a ^{2} + u ^{2}} d u

tan^{- 1} \frac{u}{a} + C

Type: Basic Math::Card 1::Back

Deck: Math::Calculus::Identities::Integrals

Tags: Math Calculus Integral Trigonometric Identity

\int \frac{1}{u u ^{2} - a ^{2}} d u

\frac{1}{a} sec^{- 1} \frac{u}{a} + C

Type: Basic Math::Card 1::Back

Deck: Math::Calculus::Identities::Integrals

Tags: Math Calculus Integral Hyperbolic Identity

\int sinh u d u

cosh u + C

Type: Basic Math::Card 1::Back

Deck: Math::Calculus::Identities::Integrals

Tags: Math Calculus Integral Hyperbolic Identity

\int cosh u d u

sinh u + C

Type: Basic Math::Card 1::Back

Deck: Math::Calculus::Identities::Integrals

Tags: Math Calculus Integral Hyperbolic Identity

\int sech^{2} u d u

tanh u + C

Type: Basic Math::Card 1::Back

Deck: Math::Calculus::Identities::Integrals

Tags: Math Calculus Integral Hyperbolic Identity

\int csch^{2} u d u

- coth u + C

Type: Basic Math::Card 1::Back

Deck: Math::Calculus::Identities::Integrals

Tags: Math Calculus Integral Hyperbolic Identity

\int sech u tanh u d u

- sech u + C

Type: Basic Math::Card 1::Back

Deck: Math::Calculus::Identities::Integrals

Tags: Math Calculus Integral Hyperbolic Identity

\int csch u coth u d u

- csch u + C

Type: Basic Math::Card 1::Back

Deck: Math::Calculus::Identities::Integrals

Tags: Math Calculus Integral Identity

\int u e^{u} d u

u e^{u} - e^{u} + C

Type: Basic Math::Card 1::Back

Deck: Math::Calculus::Identities::Integrals

Tags: Math Calculus Integral Identity

\int ln u d u

u ln u - u + C

Type: Basic Math::Card 1::Back

Deck: Math::Calculus::Identities::Taylor Series

Tags: Math Calculus Identity Taylor-Series

Taylor series expansion of $\frac{1}{1 - x}$

\frac{1}{1 - x} = 1 + x + x^{2} + x^{3} + \dots = n = 0 \sum \infty x^{n}

Valid for $∣ x ∣ < 1$

Type: Basic Math::Card 1::Back

Deck: Math::Calculus::Identities::Taylor Series

Tags: Math Calculus Identity Taylor-Series

Taylor series expansion of $\frac{1}{1 + x}$

\frac{1}{1 + x} = 1 - x + x^{2} - x^{3} + \dots = n = 0 \sum \infty (- x)^{n}

Valid for $∣ x ∣ < 1$

Type: Basic Math::Card 1::Back

Deck: Math::Calculus::Identities::Taylor Series

Tags: Math Calculus Identity Taylor-Series

Taylor series expansion of $e^{x}$

e^{x} = 1 + x + \frac{x ^{2}}{2 !} + \frac{x ^{3}}{3 !} + \dots = n = 0 \sum \infty \frac{x ^{n}}{n !}

Type: Basic Math::Card 1::Back

Deck: Math::Calculus::Identities::Taylor Series

Tags: Math Calculus Identity Taylor-Series

Taylor series expansion of $e^{- x}$

e^{- x} = 1 - x + \frac{x ^{2}}{2 !} - \frac{x ^{3}}{3 !} + \dots = n = 0 \sum \infty \frac{( - x ) ^{n}}{n !}

Type: Basic Math::Card 1::Back

Deck: Math::Calculus::Identities::Taylor Series

Tags: Math Calculus Identity Taylor-Series

Taylor series expansion of $ln (x)$

ln (x) = (x - 1) - \frac{( x - 1 ) ^{2}}{2} + \frac{( x - 1 ) ^{3}}{3} - \frac{( x - 1 ) ^{4}}{4} + \dots = n = 1 \sum \infty (- 1)^{n + 1} \frac{( x - 1 ) ^{n}}{n}

Type: Basic Math::Card 1::Back

Deck: Math::Calculus::Identities::Taylor Series

Tags: Math Calculus Identity Taylor-Series

Taylor series expansion of $ln (1 - x)$

ln (1 - x) = - x - \frac{x ^{2}}{2} - \frac{x ^{3}}{3} - \frac{x ^{4}}{4} - \dots = - n = 1 \sum \infty \frac{x ^{n}}{n}

Type: Basic Math::Card 1::Back

Deck: Math::Calculus::Identities::Taylor Series

Tags: Math Calculus Identity Taylor-Series

Taylor series expansion of $ln (1 + x)$

ln (1 + x) = x - \frac{x ^{2}}{2} + \frac{x ^{3}}{3} - \frac{x ^{4}}{4} + \dots = n = 1 \sum \infty (- 1)^{n + 1} \frac{x ^{n}}{n}

Type: Basic Math::Card 1::Back

Deck: Math::Calculus::Identities::Taylor Series

Tags: Math Calculus Taylor-Series Trigonometric Identity

Taylor series expansion of $sin (x)$

sin (x) = x - \frac{x ^{3}}{3 !} + \frac{x ^{5}}{5 !} - \frac{x ^{7}}{7 !} + \dots = n = 0 \sum \infty (- 1)^{n} \frac{x ^{2 n + 1}}{( 2 n + 1 )!}

Type: Basic Math::Card 1::Back

Deck: Math::Calculus::Identities::Taylor Series

Tags: Math Calculus Taylor-Series Trigonometric Identity

Taylor series expansion of $cos (x)$

cos (x) = 1 - \frac{x ^{2}}{2 !} + \frac{x ^{4}}{4 !} - \frac{x ^{6}}{6 !} + \dots = n = 0 \sum \infty (- 1)^{n} \frac{x ^{2 n}}{( 2 n )!}

Type: Basic Math::Card 1::Back

Deck: Math::Calculus::Identities::Taylor Series

Tags: Math Calculus Taylor-Series Trigonometric Identity

Taylor series expansion of $arctan (x)$

arctan (x) = x - \frac{x ^{3}}{3} + \frac{x ^{5}}{5} - \frac{x ^{7}}{7} + \dots = n = 0 \sum \infty (- 1)^{n} \frac{x ^{2 n + 1}}{2 n + 1}

Type: Basic Math::Card 1::Back

Deck: Math::Calculus::Identities::Taylor Series

Tags: Math Calculus Taylor-Series Hyperbolic Identity

Taylor series expansion of $sinh (x)$

sinh (x) = x + \frac{x ^{3}}{3 !} + \frac{x ^{5}}{5 !} + \frac{x ^{7}}{7 !} + \dots = n = 0 \sum \infty \frac{x ^{2 n + 1}}{( 2 n + 1 )!}

Type: Basic Math::Card 1::Back

Deck: Math::Calculus::Identities::Taylor Series

Tags: Math Calculus Taylor-Series Hyperbolic Identity

Taylor series expansion of $cosh (x)$

cosh (x) = 1 + \frac{x ^{2}}{2 !} + \frac{x ^{4}}{4 !} + \frac{x ^{6}}{6 !} + \dots = n = 0 \sum \infty \frac{x ^{2 n}}{( 2 n )!}

Type: Basic Math (and reversed card)::Card 1::Back

Deck: Math::Calculus::Identities::Laplace Transforms

Tags: Math Calculus Identity Laplace-Transform

L {1}

\frac{1}{s}

Type: Basic Math (and reversed card)::Card 2::Back

Deck: Math::Calculus::Identities::Laplace Transforms

Tags: Math Calculus Identity Laplace-Transform

\frac{1}{s}

L {1}

Type: Basic Math (and reversed card)::Card 1::Back

Deck: Math::Calculus::Identities::Laplace Transforms

Tags: Math Calculus Identity Laplace-Transform

L {t^{n}}, n = 1, 2, 3, \dots

\frac{n !}{s ^{n + 1}}

Type: Basic Math (and reversed card)::Card 2::Back

Deck: Math::Calculus::Identities::Laplace Transforms

Tags: Math Calculus Identity Laplace-Transform

\frac{n !}{s ^{n + 1}}

L {t^{n}}, n = 1, 2, 3, \dots

Type: Basic Math (and reversed card)::Card 1::Back

Deck: Math::Calculus::Identities::Laplace Transforms

Tags: Math Calculus Identity Laplace-Transform

L {sin (a t)}

\frac{a}{s ^{2} + a ^{2}}

L {sin (a t)} L {cos (a t)} L {sinh (a t)} L {cosh (a t)} = \frac{a}{s ^{2} + a ^{2}} = \frac{s}{s ^{2} + a ^{2}} = \frac{a}{s ^{2} - a ^{2}} = \frac{s}{s ^{2} - a ^{2}}

Type: Basic Math (and reversed card)::Card 2::Back

Deck: Math::Calculus::Identities::Laplace Transforms

Tags: Math Calculus Identity Laplace-Transform

\frac{a}{s ^{2} + a ^{2}}

L {sin (a t)}

L {sin (a t)} L {cos (a t)} L {sinh (a t)} L {cosh (a t)} = \frac{a}{s ^{2} + a ^{2}} = \frac{s}{s ^{2} + a ^{2}} = \frac{a}{s ^{2} - a ^{2}} = \frac{s}{s ^{2} - a ^{2}}

Type: Basic Math (and reversed card)::Card 1::Back

Deck: Math::Calculus::Identities::Laplace Transforms

Tags: Math Calculus Identity Laplace-Transform

L {cos (a t)}

\frac{s}{s ^{2} + a ^{2}}

L {sin (a t)} L {cos (a t)} L {sinh (a t)} L {cosh (a t)} = \frac{a}{s ^{2} + a ^{2}} = \frac{s}{s ^{2} + a ^{2}} = \frac{a}{s ^{2} - a ^{2}} = \frac{s}{s ^{2} - a ^{2}}

Type: Basic Math (and reversed card)::Card 2::Back

Deck: Math::Calculus::Identities::Laplace Transforms

Tags: Math Calculus Identity Laplace-Transform

\frac{s}{s ^{2} + a ^{2}}

L {cos (a t)}

L {sin (a t)} L {cos (a t)} L {sinh (a t)} L {cosh (a t)} = \frac{a}{s ^{2} + a ^{2}} = \frac{s}{s ^{2} + a ^{2}} = \frac{a}{s ^{2} - a ^{2}} = \frac{s}{s ^{2} - a ^{2}}

Type: Basic Math (and reversed card)::Card 1::Back

Deck: Math::Calculus::Identities::Laplace Transforms

Tags: Math Calculus Identity Laplace-Transform

L {sin (a t + b)}

\frac{s sin ( b ) + a cos ( b )}{s ^{2} + a ^{2}}

Type: Basic Math (and reversed card)::Card 2::Back

Deck: Math::Calculus::Identities::Laplace Transforms

Tags: Math Calculus Identity Laplace-Transform

\frac{s sin ( b ) + a cos ( b )}{s ^{2} + a ^{2}}

L {sin (a t + b)}

Type: Basic Math (and reversed card)::Card 1::Back

Deck: Math::Calculus::Identities::Laplace Transforms

Tags: Math Calculus Identity Laplace-Transform

L {cos (a t + b)}

\frac{s cos ( b ) - a sin ( b )}{s ^{2} + a ^{2}}

Type: Basic Math (and reversed card)::Card 2::Back

Deck: Math::Calculus::Identities::Laplace Transforms

Tags: Math Calculus Identity Laplace-Transform

\frac{s cos ( b ) - a sin ( b )}{s ^{2} + a ^{2}}

L {cos (a t + b)}

Type: Basic Math (and reversed card)::Card 1::Back

Deck: Math::Calculus::Identities::Laplace Transforms

Tags: Math Calculus Identity Laplace-Transform

L {t sin (a t)}

\frac{2 a s}{( s ^{2} + a ^{2} ) ^{2}}

Type: Basic Math (and reversed card)::Card 2::Back

Deck: Math::Calculus::Identities::Laplace Transforms

Tags: Math Calculus Identity Laplace-Transform

\frac{2 a s}{( s ^{2} + a ^{2} ) ^{2}}

L {t sin (a t)}

Type: Basic Math (and reversed card)::Card 1::Back

Deck: Math::Calculus::Identities::Laplace Transforms

Tags: Math Calculus Identity Laplace-Transform

L {t cos (a t)}

\frac{s ^{2} - a ^{2}}{( s ^{2} + a ^{2} ) ^{2}}

Type: Basic Math (and reversed card)::Card 2::Back

Deck: Math::Calculus::Identities::Laplace Transforms

Tags: Math Calculus Identity Laplace-Transform

\frac{s ^{2} - a ^{2}}{( s ^{2} + a ^{2} ) ^{2}}

L {t cos (a t)}

Type: Basic Math (and reversed card)::Card 1::Back

Deck: Math::Calculus::Identities::Laplace Transforms

Tags: Math Calculus Identity Laplace-Transform

L {sinh (a t)}

\frac{a}{s ^{2} - a ^{2}}

L {sin (a t)} L {cos (a t)} L {sinh (a t)} L {cosh (a t)} = \frac{a}{s ^{2} + a ^{2}} = \frac{s}{s ^{2} + a ^{2}} = \frac{a}{s ^{2} - a ^{2}} = \frac{s}{s ^{2} - a ^{2}}

Type: Basic Math (and reversed card)::Card 2::Back

Deck: Math::Calculus::Identities::Laplace Transforms

Tags: Math Calculus Identity Laplace-Transform

\frac{a}{s ^{2} - a ^{2}}

L {sinh (a t)}

L {sin (a t)} L {cos (a t)} L {sinh (a t)} L {cosh (a t)} = \frac{a}{s ^{2} + a ^{2}} = \frac{s}{s ^{2} + a ^{2}} = \frac{a}{s ^{2} - a ^{2}} = \frac{s}{s ^{2} - a ^{2}}

Type: Basic Math (and reversed card)::Card 1::Back

Deck: Math::Calculus::Identities::Laplace Transforms

Tags: Math Calculus Identity Laplace-Transform

L {cosh (a t)}

\frac{s}{s ^{2} - a ^{2}}

L {sin (a t)} L {cos (a t)} L {sinh (a t)} L {cosh (a t)} = \frac{a}{s ^{2} + a ^{2}} = \frac{s}{s ^{2} + a ^{2}} = \frac{a}{s ^{2} - a ^{2}} = \frac{s}{s ^{2} - a ^{2}}

Type: Basic Math (and reversed card)::Card 2::Back

Deck: Math::Calculus::Identities::Laplace Transforms

Tags: Math Calculus Identity Laplace-Transform

\frac{s}{s ^{2} - a ^{2}}

L {cosh (a t)}

L {sin (a t)} L {cos (a t)} L {sinh (a t)} L {cosh (a t)} = \frac{a}{s ^{2} + a ^{2}} = \frac{s}{s ^{2} + a ^{2}} = \frac{a}{s ^{2} - a ^{2}} = \frac{s}{s ^{2} - a ^{2}}

Type: Basic Math (and reversed card)::Card 1::Back

Deck: Math::Calculus::Identities::Laplace Transforms

Tags: Math Calculus Identity Laplace-Transform

L {e^{c t}}

\frac{1}{s - c}

Type: Basic Math (and reversed card)::Card 2::Back

Deck: Math::Calculus::Identities::Laplace Transforms

Tags: Math Calculus Identity Laplace-Transform

\frac{1}{s - c}

L {e^{c t}}

Type: Basic Math (and reversed card)::Card 1::Back

Deck: Math::Calculus::Identities::Laplace Transforms

Tags: Math Calculus Identity Laplace-Transform

L {e^{c t} t^{n}}, n = 1, 2, 3, \dots

\frac{n !}{( s - c ) ^{n + 1}}

Type: Basic Math (and reversed card)::Card 2::Back

Deck: Math::Calculus::Identities::Laplace Transforms

Tags: Math Calculus Identity Laplace-Transform

\frac{n !}{( s - c ) ^{n + 1}}

L {e^{c t} t^{n}}, n = 1, 2, 3, \dots

Type: Basic Math (and reversed card)::Card 1::Back

Deck: Math::Calculus::Identities::Laplace Transforms

Tags: Math Calculus Identity Laplace-Transform

L {e^{c t} sin (a t)}

\frac{a}{( s - c ) ^{2} + a ^{2}}

L {e^{c t} sin (a t)} L {e^{c t} cos (a t)} L {e^{c t} sinh (a t)} L {e^{c t} cosh (a t)} = \frac{a}{( s - c ) ^{2} + a ^{2}} = \frac{s - c}{( s - c ) ^{2} + a ^{2}} = \frac{a}{( s - c ) ^{2} - a ^{2}} = \frac{s - c}{( s - c ) ^{2} - a ^{2}}

Type: Basic Math (and reversed card)::Card 2::Back

Deck: Math::Calculus::Identities::Laplace Transforms

Tags: Math Calculus Identity Laplace-Transform

\frac{a}{( s - c ) ^{2} + a ^{2}}

L {e^{c t} sin (a t)}

L {e^{c t} sin (a t)} L {e^{c t} cos (a t)} L {e^{c t} sinh (a t)} L {e^{c t} cosh (a t)} = \frac{a}{( s - c ) ^{2} + a ^{2}} = \frac{s - c}{( s - c ) ^{2} + a ^{2}} = \frac{a}{( s - c ) ^{2} - a ^{2}} = \frac{s - c}{( s - c ) ^{2} - a ^{2}}

Type: Basic Math (and reversed card)::Card 1::Back

Deck: Math::Calculus::Identities::Laplace Transforms

Tags: Math Calculus Identity Laplace-Transform

L {e^{c t} cos (a t)}

\frac{s - c}{( s - c ) ^{2} + a ^{2}}

L {e^{c t} sin (a t)} L {e^{c t} cos (a t)} L {e^{c t} sinh (a t)} L {e^{c t} cosh (a t)} = \frac{a}{( s - c ) ^{2} + a ^{2}} = \frac{s - c}{( s - c ) ^{2} + a ^{2}} = \frac{a}{( s - c ) ^{2} - a ^{2}} = \frac{s - c}{( s - c ) ^{2} - a ^{2}}

Type: Basic Math (and reversed card)::Card 2::Back

Deck: Math::Calculus::Identities::Laplace Transforms

Tags: Math Calculus Identity Laplace-Transform

\frac{s - c}{( s - c ) ^{2} + a ^{2}}

L {e^{c t} cos (a t)}

L {e^{c t} sin (a t)} L {e^{c t} cos (a t)} L {e^{c t} sinh (a t)} L {e^{c t} cosh (a t)} = \frac{a}{( s - c ) ^{2} + a ^{2}} = \frac{s - c}{( s - c ) ^{2} + a ^{2}} = \frac{a}{( s - c ) ^{2} - a ^{2}} = \frac{s - c}{( s - c ) ^{2} - a ^{2}}

Type: Basic Math (and reversed card)::Card 1::Back

Deck: Math::Calculus::Identities::Laplace Transforms

Tags: Math Calculus Identity Laplace-Transform

L {e^{c t} sinh (a t)}

\frac{a}{( s - c ) ^{2} - a ^{2}}

L {e^{c t} sin (a t)} L {e^{c t} cos (a t)} L {e^{c t} sinh (a t)} L {e^{c t} cosh (a t)} = \frac{a}{( s - c ) ^{2} + a ^{2}} = \frac{s - c}{( s - c ) ^{2} + a ^{2}} = \frac{a}{( s - c ) ^{2} - a ^{2}} = \frac{s - c}{( s - c ) ^{2} - a ^{2}}

Type: Basic Math (and reversed card)::Card 2::Back

Deck: Math::Calculus::Identities::Laplace Transforms

Tags: Math Calculus Identity Laplace-Transform

\frac{a}{( s - c ) ^{2} - a ^{2}}

L {e^{c t} sinh (a t)}

L {e^{c t} sin (a t)} L {e^{c t} cos (a t)} L {e^{c t} sinh (a t)} L {e^{c t} cosh (a t)} = \frac{a}{( s - c ) ^{2} + a ^{2}} = \frac{s - c}{( s - c ) ^{2} + a ^{2}} = \frac{a}{( s - c ) ^{2} - a ^{2}} = \frac{s - c}{( s - c ) ^{2} - a ^{2}}

Type: Basic Math (and reversed card)::Card 1::Back

Deck: Math::Calculus::Identities::Laplace Transforms

Tags: Math Calculus Identity Laplace-Transform

L {e^{c t} cosh (a t)}

\frac{s - c}{( s - c ) ^{2} - a ^{2}}

L {e^{c t} sin (a t)} L {e^{c t} cos (a t)} L {e^{c t} sinh (a t)} L {e^{c t} cosh (a t)} = \frac{a}{( s - c ) ^{2} + a ^{2}} = \frac{s - c}{( s - c ) ^{2} + a ^{2}} = \frac{a}{( s - c ) ^{2} - a ^{2}} = \frac{s - c}{( s - c ) ^{2} - a ^{2}}

Type: Basic Math (and reversed card)::Card 2::Back

Deck: Math::Calculus::Identities::Laplace Transforms

Tags: Math Calculus Identity Laplace-Transform

\frac{s - c}{( s - c ) ^{2} - a ^{2}}

L {e^{c t} cosh (a t)}

L {e^{c t} sin (a t)} L {e^{c t} cos (a t)} L {e^{c t} sinh (a t)} L {e^{c t} cosh (a t)} = \frac{a}{( s - c ) ^{2} + a ^{2}} = \frac{s - c}{( s - c ) ^{2} + a ^{2}} = \frac{a}{( s - c ) ^{2} - a ^{2}} = \frac{s - c}{( s - c ) ^{2} - a ^{2}}

Type: Basic Math (and reversed card)::Card 1::Back

Deck: Math::Calculus::Identities::Laplace Transforms

Tags: Math Calculus Identity Laplace-Transform

L {e^{c t} f (t)}

F (s - c)

Type: Basic Math (and reversed card)::Card 2::Back

Deck: Math::Calculus::Identities::Laplace Transforms

Tags: Math Calculus Identity Laplace-Transform

F (s - c)

L {e^{c t} f (t)}

Type: Basic Math (and reversed card)::Card 1::Back

Deck: Math::Calculus::Identities::Laplace Transforms

Tags: Math Calculus Identity Laplace-Transform

L {δ (t - c)}

e^{- cs}

Type: Basic Math (and reversed card)::Card 2::Back

Deck: Math::Calculus::Identities::Laplace Transforms

Tags: Math Calculus Identity Laplace-Transform

e^{- cs}

L {δ (t - c)}

Type: Basic Math (and reversed card)::Card 1::Back

Deck: Math::Calculus::Identities::Laplace Transforms

Tags: Math Calculus Identity Laplace-Transform

L {u (t - c)}

e^{- cs} \frac{1}{s}

Type: Basic Math (and reversed card)::Card 2::Back

Deck: Math::Calculus::Identities::Laplace Transforms

Tags: Math Calculus Identity Laplace-Transform

e^{- cs} \frac{1}{s}

L {u (t - c)}

Type: Basic Math (and reversed card)::Card 1::Back

Deck: Math::Calculus::Identities::Laplace Transforms

Tags: Math Calculus Identity Laplace-Transform

L {u (t - c) f (t - c)}

e^{- cs} F (s)

Type: Basic Math (and reversed card)::Card 2::Back

Deck: Math::Calculus::Identities::Laplace Transforms

Tags: Math Calculus Identity Laplace-Transform

e^{- cs} F (s)

L {u (t - c) f (t - c)}

Type: Basic Math (and reversed card)::Card 1::Back

Deck: Math::Calculus::Identities::Laplace Transforms

Tags: Math Calculus Identity Laplace-Transform

L {\frac{f ( t )}{t}}

\int_{s}^{\infty} F (u) d u

Type: Basic Math (and reversed card)::Card 2::Back

Deck: Math::Calculus::Identities::Laplace Transforms

Tags: Math Calculus Identity Laplace-Transform

\int_{s}^{\infty} F (u) d u

L {\frac{f ( t )}{t}}

Type: Basic Math (and reversed card)::Card 1::Back

Deck: Math::Calculus::Identities::Laplace Transforms

Tags: Math Calculus Identity Laplace-Transform

L {\int_{0}^{t} f (u) d u}

\frac{F ( s )}{s}

Type: Basic Math (and reversed card)::Card 2::Back

Deck: Math::Calculus::Identities::Laplace Transforms

Tags: Math Calculus Identity Laplace-Transform

\frac{F ( s )}{s}

L {\int_{0}^{t} f (u) d u}

Type: Basic Math (and reversed card)::Card 1::Back

Deck: Math::Calculus::Identities::Laplace Transforms

Tags: Math Calculus Identity Laplace-Transform

L {f^{'} (t)}

s F (s) - f (0)

Type: Basic Math (and reversed card)::Card 2::Back

Deck: Math::Calculus::Identities::Laplace Transforms

Tags: Math Calculus Identity Laplace-Transform

s F (s) - f (0)

L {f^{'} (t)}

Type: Basic Math (and reversed card)::Card 1::Back

Deck: Math::Calculus::Identities::Laplace Transforms

Tags: Math Calculus Identity Laplace-Transform

L {f^{''} (t)}

s^{2} F (s) - s f (0) - f^{'} (0)

Type: Basic Math (and reversed card)::Card 2::Back

Deck: Math::Calculus::Identities::Laplace Transforms

Tags: Math Calculus Identity Laplace-Transform

s^{2} F (s) - s f (0) - f^{'} (0)

L {f^{''} (t)}

Type: Basic Math::Card 1::Back

Deck: Math::Calculus::Limits

Tags: Math Calculus Limits Theory

What is a continuous function?

A function $f (x)$ is continuous at a point $x = c$ if:

$f (c)$ exists: The function is defined at the point $c$ .
$lim_{x \to c} f (x)$ exists: The limit of the function as $x$ approaches $c$ from both the left and right sides is the same.
$lim_{x \to c} f (x) = f (c)$ : The limit of the function as $x$ approaches $c$ is equal to the actual value of the function at $c$ .

If a function is continuous at every point in its domain, it is called a continuous function.

Type: Basic Math::Card 1::Back

Deck: Math::Calculus::Differentiation

Tags: Math Calculus Differentiation Theory

The definition of the a critical point.

A critical point of a function $f (x)$ is a point in the domain of $f$ where either:

The derivative $f^{'} (x)$ is zero, or
The derivative $f^{'} (x)$ does not exist.

Type: Basic Math::Card 1::Back

Deck: Math::Calculus::Differentiation

Tags: Math Calculus Differentiation Theory

The definition of the Extreme Value Theorem (EVT).

The Extreme Value Theorem states that if a real-valued function $f$ is continuous on the closed and bounded interval $[a, b]$ , then $f$ must attain a maximum and a minimum, each at least once.

That is, there exist numbers $c$ and $d$ in $[a, b]$ such that: $f (c) \leq f (x) \leq f (d) \forall x \in [a, b]$ .

In simpler terms, a continuous function on a closed interval will always have a highest point and a lowest point.

Type: Basic Math::Card 1::Back

Deck: Math::Calculus::Differentiation

Tags: Math Calculus Differentiation Theory

The definition of the Rolle's Theorem.

The Rolle's Theorem is a special case of the Mean Value Theorem.

The Rolle's Theorem states that if a function $f (x)$ is:

Continuous on a closed interval $[a, b]$
Differentiable on the open interval $(a, b)$
$f (a) = f (b)$ (the function has the same value at the endpoints)

Then there exists at least one point $c$ in the open interval $(a, b)$ such that $f^{'} (c) = 0$ .

Geometrically, this means if a continuous and differentiable curve starts and ends at the same height, there must be at least one point in between where the tangent line is horizontal (i.e., the slope is zero).

Type: Basic Math::Card 1::Back

Deck: Math::Calculus::Differentiation

Tags: Math Calculus Differentiation Theory

The definition of the Mean Value Theorem (MVT).

The Mean Value Theorem is a generalization of Rolle's Theorem.

The Mean Value Theorem states that if a function $f (x)$ is:

Continuous on a closed interval $[a, b]$
Differentiable on the open interval $(a, b)$

Then there exists at least one point $c$ in the open interval $(a, b)$ such that the instantaneous rate of change (the derivative $f^{'} (c)$ ) is equal to the average rate of change over the interval.

Mathematically, this is expressed as:

f^{'} (c) = \frac{f ( b ) - f ( a )}{b - a}

Or, alternatively:

f (c) = \frac{1}{b - a} \int_{a}^{b} f (x) d x

The Rolle's Theorem can be derived from the Mean Value Theorem: if $f (a) = f (b)$ , then the right side of the Mean Value Theorem equation becomes

f^{'} (c) = \frac{f ( a ) - f ( a )}{b - a} = \frac{0}{b - a}

which means $f^{'} (c) = 0$ .

Type: Basic Math::Card 1::Back

Deck: Math::Calculus::Differentiation

Tags: Math Calculus Differentiation Theory

The definition of the Intermediate Value Theorem (IVT).

The Intermediate Value Theorem states that if a function $f (x)$ is continuous on a closed interval $[a, b]$ , and $L$ is any value between $f (a)$ and $f (b)$ , then there exists at least one number $c$ in the interval $[a, b]$ such that $f (c) = L$ .

Intuitively, if you can draw the graph of a function without lifting your pencil, and it goes from one y-value to another, it must pass through all the y-values in between.

Type: Basic Math::Card 1::Back

Deck: Math::Calculus::Differentiation

Tags: Math Calculus Differentiation Theory

Prove that $\frac{d x}{d y} \cdot \frac{d y}{d x} = 1$

y x f^{- 1} (f (x)) = f (x) = f^{- 1} (y) = x

Differentiate both sides with respect to $x$ using the chain rule:

(f^{- 1})^{'} (f (x)) \cdot f^{'} (x) = 1

which is the same as

\frac{d x}{d y} \cdot \frac{d y}{d x} = 1

because

(f^{- 1})^{'} (f (x)) = (f^{- 1})^{'} (y) = \frac{d x}{d y}

and

f^{'} (x) = \frac{d y}{d x}

Type: Basic Math::Card 1::Back

Deck: Math::Calculus::Differentiation

Tags: Math Calculus Differentiation Theory

Derive $\frac{d}{d x} ln (x)$

y x = ln (x) = e^{y}

Differentiate both sides implicitly with respect to $x$ using the chain rule:

\frac{d}{d x} [x] 1 \frac{d y}{d x} = \frac{d}{d x} [e^{y}] = e^{y} \cdot \frac{d y}{d x} = \frac{1}{e ^{y}}

Express $e^{y}$ in terms of $x$ :

e^{y} = e^{l n x} = x

Substitute back:

\frac{d y}{d x} = \frac{1}{x}

Type: Basic Math::Card 1::Back

Deck: Math::Calculus::Differentiation

Tags: Math Calculus Differentiation Theory

Derive $\frac{d}{d x} arcsin (x)$

y x = arcsin (x) = sin (y)

Differentiate both sides implicitly with respect to $x$ using the chain rule:

\frac{d}{d x} [x] 1 \frac{d y}{d x} = \frac{d}{d x} [sin (y)] = cos (y) \cdot \frac{d y}{d x} = \frac{1}{cos ( y )}

Express $cos (y)$ in terms of $x$ :

cos (y) = 1 - sin^{2} (y) = 1 - x^{2}

Substitute back:

\frac{d y}{d x} = \frac{1}{1 - x ^{2}}

Type: Basic Math::Card 1::Back

Deck: Math::Calculus::Differentiation

Tags: Math Calculus Differentiation Theory

Derive $\frac{d}{d x} arccos (x)$

y x = arccos (x) = cos (y)

Differentiate both sides implicitly with respect to $x$ using the chain rule:

\frac{d}{d x} [x] 1 \frac{d y}{d x} = \frac{d}{d x} [cos (y)] = - sin (y) \cdot \frac{d y}{d x} = - \frac{1}{sin ( y )}

Express $sin (y)$ in terms of $x$ :

sin (y) = 1 - cos^{2} (y) = 1 - x^{2}

Substitute back:

\frac{d y}{d x} = - \frac{1}{1 - x ^{2}}

Type: Basic Math::Card 1::Back

Deck: Math::Calculus::Differentiation

Tags: Math Calculus Differentiation Theory

Derive $\frac{d}{d x} arctan (x)$

y x = arctan (x) = tan (y)

Differentiate both sides implicitly with respect to $x$ using the chain rule:

\frac{d}{d x} [x] 1 \frac{d y}{d x} = \frac{d}{d x} [tan (y)] = sec^{2} (y) \cdot \frac{d y}{d x} = \frac{1}{sec ^{2} ( y )}

Express $sec^{2} (y)$ in terms of $x$ :

sec^{2} (y) = 1 + tan^{2} (y) = 1 + x^{2}

Substitute back:

\frac{d y}{d x} = \frac{1}{1 + x ^{2}}

Type: Basic Math::Card 1::Back

Deck: Math::Calculus::Integration

Tags: Math Calculus Integration Theory

What is the fundamental theorem of calculus (FTC)?

Consider a function $f$ which is continuous on $[a, b]$ .

If $F (x) = \int_{a}^{x} (t) d t$ , then $F^{'} (x) = f (x)$ on $(a, b)$ .
If $F$ is any antiderivative of $f$ on $[a, b]$ (i.e., $F^{'} (x) = f (x)$ ), then $\int_{a}^{b} f (x) d x = F (b) - F (a)$ .

The fundamental theorem of calculus is arguably the most important theorem in elementary calculus, as it establishes the profound connection between differentiation and integration.

Type: Basic Math::Card 1::Back

Deck: Math::Calculus::Sequences and Series

Tags: Math Calculus Sequences-and-Series Theory

What is an infinite sequence?

An infinite sequence is an ordered and indexed list of numbers of the form

a_{1}, a_{2}, a_{3}, \dots, a_{n}, \dots

denoted as

{a_{n}}_{n = 1}^{\infty}

or simply

{a_{n}} .

Each number $a_{n}$ is a term of the sequence.

The subscript $n$ is the index variable of the sequence.

Type: Basic Math::Card 1::Back

Deck: Math::Calculus::Sequences and Series

Tags: Math Calculus Sequences-and-Series Theory

What are the different ways to define a sequence?

An infinite sequence can be defined in various ways:

By listing terms ${2, 4, 6, 8, \dots}$ .
By an explicit formula that defines the $n$ -th term directly in terms of $n$ , $a_{n} = f (n)$ .
By a recurrence relation, a rule that defines a term based on previous terms in the sequence, $a_{n} = f (a_{n - 1}, a_{n - 2}, \dots)$ .

Type: Basic Math::Card 1::Back

Deck: Math::Calculus::Sequences and Series

Tags: Math Calculus Sequences-and-Series Theory

What is the limit of a sequence?

A sequence ${a_{n}}$ converges to a real number $L$ if for all $ϵ > 0$ , there exists an integer $N$ such that $∣ a_{n} - L ∣ < ϵ$ if $n \geq N$ . The number $L$ is the limit of the sequence and we write

n \to \infty lim a_{n} = L or a_{n} \to L .

In this case, we say the sequence ${a_{n}}$ is a convergent sequence. If a sequence does not converge, it is a divergent sequence, and we say the limit does not exist.

Type: Basic Math::Card 1::Back

Deck: Math::Calculus::Sequences and Series

Tags: Math Calculus Sequences-and-Series Theory

What is the squeeze theorem for sequences?

Consider sequences ${a_{n}}$ , ${b_{n}}$ , and ${c_{n}}$ . Suppose there exists an integer $N$ such that $a_{n} \leq b_{n} \leq c_{n}$ for all $n \geq N$ . If there exists a real number $L$ such that

n \to \infty lim a_{n} = L = n \to \infty lim c_{n},

then ${b_{n}}$ converges and

n \to \infty lim b_{n} = L

Type: Basic Math::Card 1::Back

Deck: Math::Calculus::Sequences and Series

Tags: Math Calculus Sequences-and-Series Theory

What is a bounded sequence?

A sequence ${a_{n}}$ is bounded above if there exists a real number $M$ such that

a_{n} \leq M

for all positive integers $n$ .

A sequence ${a_{n}}$ is bounded below if there exists a real number $M$ such that

M \leq a_{n}

for all positive integers $n$ .

A sequence ${a_{n}}$ is a bounded sequence if it is bounded above and bounded below.

If a sequence is not bounded, it is an unbounded sequence.

Type: Basic Math::Card 1::Back

Deck: Math::Calculus::Sequences and Series

Tags: Math Calculus Sequences-and-Series Theory

What is the relationship between convergent and bounded sequences?

If a sequence ${a_{n}}$ converges, then it is bounded.

Type: Basic Math::Card 1::Back

Deck: Math::Calculus::Sequences and Series

Tags: Math Calculus Sequences-and-Series Theory

What is a monotone sequence?

A sequence ${a_{n}}$ is increasing for all $n \geq n_{0}$ if

a_{n} \leq a_{n + 1} for all n \geq n_{0} .

A sequence ${a_{n}}$ is decreasing for all $n \geq n_{0}$ if

a_{n} \geq a_{n + 1} for all n \geq n_{0} .

A sequence ${a_{n}}$ is a monotone sequence for all $n \geq n_{0}$ if it is increasing for all $n \geq n_{0}$ or decreasing for all $n \geq n_{0}$ .

Type: Basic Math::Card 1::Back

Deck: Math::Calculus::Sequences and Series

Tags: Math Calculus Sequences-and-Series Theory

What is the monotone convergence theorem?

If ${a_{n}}$ is a bounded sequence and there exists a positive integer $n_{0}$ such that ${a_{n}}$ is monotone for all $n \geq n_{0}$ , then ${a_{n}}$ converges.

Type: Basic Math::Card 1::Back

Deck: Math::Calculus::Sequences and Series

Tags: Math Calculus Sequences-and-Series Theory

What is an infinite series?

An infinite series is a sum of the terms of an infinite sequnce $a_{n}$ :

n = 1 \sum \infty a_{n} = a_{1} + a_{2} + a_{3} + \dots

Type: Basic Math::Card 1::Back

Deck: Math::Calculus::Sequences and Series

Tags: Math Calculus Sequences-and-Series Theory

What is the harmonic series?

The harmonic series is an infinite series of the following form

n = 1 \sum \infty \frac{1}{n}

The harmonic series is a special case of the p-series with $p = 1$ .

Type: Basic Math::Card 1::Back

Deck: Math::Calculus::Sequences and Series

Tags: Math Calculus Sequences-and-Series Theory

What are the convergence properties of the harmonic series?

The harmonic series $\sum_{n = 1}^{\infty} \frac{1}{n}$ diverges to infinity.

The alternate harmonic series $\sum_{n = 1}^{\infty} (- 1)^{n + 1} \frac{1}{n}$ converges to $ln 2$ .

Type: Basic Math::Card 1::Back

Deck: Math::Calculus::Sequences and Series

Tags: Math Calculus Sequences-and-Series Theory

What is the alternating harmonic series?

The alternating harmonic series is an infinite series of the following form

n = 1 \sum \infty (- 1)^{n + 1} \frac{1}{n} = 1 - \frac{1}{2} + \frac{1}{3} - \frac{1}{4} + \dots

Type: Basic Math::Card 1::Back

Deck: Math::Calculus::Sequences and Series

Tags: Math Calculus Sequences-and-Series Theory

What are the convergence properties of the alternate harmonic series?

The alternate harmonic series $\sum_{n = 1}^{\infty} (- 1)^{n + 1} \frac{1}{n}$ converges to $ln 2$ .

The harmonic series $\sum_{n = 1}^{\infty} \frac{1}{n}$ diverges to infinity.

Type: Basic Math::Card 1::Back

Deck: Math::Calculus::Sequences and Series

Tags: Math Calculus Sequences-and-Series Theory

What is the geometric series?

The geometric series is an infinite series of the following form

n = 0 \sum \infty a r^{n} = a + a r + a r^{2} + a r^{3} + \dots

Type: Basic Math::Card 1::Back

Deck: Math::Calculus::Sequences and Series

Tags: Math Calculus Sequences-and-Series Theory

What are the convergence properties of the geometric series?

The geometric series $\sum_{n = 0}^{\infty} a r^{n}$ converges to $\frac{a}{1 - r}$ if $∣ r ∣ < 1$ , otherwise it diverges.

Type: Basic Math::Card 1::Back

Deck: Math::Calculus::Sequences and Series

Tags: Math Calculus Sequences-and-Series Theory

What is the p-series?

The p-series series is an infinite series of the following form

n = 1 \sum \infty \frac{1}{n ^{p}} = \frac{1}{1 ^{p}} + \frac{1}{2 ^{p}} + \frac{1}{3 ^{p}} + \dots

Type: Basic Math::Card 1::Back

Deck: Math::Calculus::Sequences and Series

Tags: Math Calculus Sequences-and-Series Theory

What are the convergence properties of the p-series?

The p-series $\sum_{n = 1}^{\infty} \frac{1}{n ^{p}}$ converges if $p > 1$ , otherwise it diverges.

Type: Basic Math::Card 1::Back

Deck: Math::Calculus::Sequences and Series

Tags: Math Calculus Sequences-and-Series Theory

What is the telescoping series?

The telescoping series is an infinite series of the following form

n = 1 \sum \infty a_{n} - a_{n + 1} .

In other words, the telescoping series is a series in which most of the terms cancel in each of the partial sums, leaving only some of the first terms and some of the last terms.

Type: Basic Math::Card 1::Back

Deck: Math::Calculus::Sequences and Series

Tags: Math Calculus Sequences-and-Series Theory

What are the algebraic properties of convergent series?

Let $\sum_{n = 1}^{\infty} a_{n}$ and $\sum_{n = 1}^{\infty} b_{n}$ be convergent series. Then the following algebraic properties hold.

Sum Rule: The series $\sum_{n = 1}^{\infty} (a_{n} + b_{n})$ converges and $\sum_{n = 1}^{\infty} (a_{n} + b_{n}) = \sum_{n = 1}^{\infty} a_{n} + \sum_{n = 1}^{\infty} b_{n}$ .
Difference Rule: The series $\sum_{n = 1}^{\infty} (a_{n} - b_{n})$ converges and $\sum_{n = 1}^{\infty} (a_{n} - b_{n}) = \sum_{n = 1}^{\infty} a_{n} - \sum_{n = 1}^{\infty} b_{n}$ .
Constant Multiple Rule: For any real number $c$ , the series $\sum_{n = 1}^{\infty} c a_{n}$ converges and $\sum_{n = 1}^{\infty} c a_{n} = c \sum_{n = 1}^{\infty} a_{n}$ .

Type: Basic Math::Card 1::Back

Deck: Math::Calculus::Sequences and Series

Tags: Math Calculus Sequences-and-Series Theory

What is the divergence test for a series?

Consider the sequence $a_{n}$ . If $lim_{n \to \infty} a_{n} \neq = 0$ or $lim_{n \to \infty} a_{n}$ does not exist, then the series $\sum_{n = 1}^{\infty} a_{n}$ diverges.

The converse is not true. That is, if $lim_{n \to \infty} a_{n} = 0$ , we cannot make any conclusion about the convergence of $\sum_{n = 1}^{\infty} a_{n}$ .

Type: Basic Math::Card 1::Back

Deck: Math::Calculus::Sequences and Series

Tags: Math Calculus Sequences-and-Series Theory

What is the integral test?

Suppose $\sum_{n = 0}^{\infty} a_{n}$ is a series with positive terms $a_{n}$ . Suppose there exists a function $f$ and a positive integer $N$ such that the following three conditions are satisfied:

$f$ is continuous,
$f$ is decreasing, and
$f (n) = a_{n}$ for all integers $n \geq N$ .

Then $\sum_{n = 1}^{\infty} a_{n}$ and $\int_{N}^{\infty} f (x) d x$ both converge or both diverge.

Although convergence of the integral implies convergence of the related series, it does not imply that the value of the integral and the series are the same.

Type: Basic Math::Card 1::Back

Deck: Math::Calculus::Sequences and Series

Tags: Math Calculus Sequences-and-Series Theory

How to estimate the value of a series?

TODO

Type: Basic Math::Card 1::Back

Deck: Math::Calculus::Sequences and Series

Tags: Math Calculus Sequences-and-Series Theory

How to estimate the remainder of a series?

TODO

Type: Basic Math::Card 1::Back

Deck: Math::Calculus::Sequences and Series

Tags: Math Calculus Sequences-and-Series Theory

What is the direct comparison test?

Suppose there exists an integer $N$ such that $0 \leq a_{n} \leq b_{n}$ for all $n \geq N$ . If $\sum_{n = 1}^{\infty} b_{n}$ converges, then $\sum_{n = 1}^{\infty} a_{n}$ converges.

Suppose there exists an integer $N$ such that $a_{n} \geq b_{n} \geq 0$ for all $n \geq N$ . If $\sum_{n = 1}^{\infty} b_{n}$ diverges, then $\sum_{n = 1}^{\infty} a_{n}$ diverges.

Type: Basic Math::Card 1::Back

Deck: Math::Calculus::Sequences and Series

Tags: Math Calculus Sequences-and-Series Theory

What is the limit comparison test?

Let $a_{n}$ , $b_{n} \geq 0$ for all $n \geq 1$ .

If $lim_{n \to \infty} a_{n} / b_{n} = L \neq = 0$ , then $\sum_{n = 1}^{\infty} a_{n}$ and $\sum_{n = 1}^{\infty} b_{n}$ both converge or both diverge.

If $lim_{n \to \infty} a_{n} / b_{n} = 0$ and $\sum_{n = 1}^{\infty} b_{n}$ converges, then $\sum_{n = 1}^{\infty} a_{n}$ converges.

If $lim_{n \to \infty} a_{n} / b_{n} = \infty$ and $\sum_{n = 1}^{\infty} b_{n}$ diverges, then $\sum_{n = 1}^{\infty} a_{n}$ diverges.

Type: Basic Math::Card 1::Back

Deck: Math::Calculus::Sequences and Series

Tags: Math Calculus Sequences-and-Series Theory

What is the alternating series?

The alternating series is a sum of the terms of an infinite sequnce $a_{n}$ whose terms alternate between positive and negative values:

n = 1 \sum \infty (- 1)^{n + 1} a_{n} = a_{1} - a_{2} + a_{3} - \dots

n = 1 \sum \infty (- 1)^{n} a_{n} = - a_{1} + a_{2} - a_{3} + \dots

Type: Basic Math::Card 1::Back

Deck: Math::Calculus::Sequences and Series

Tags: Math Calculus Sequences-and-Series Theory

What is the alternating series test?

An alternating series of the form $\sum_{n = 1}^{\infty} (- 1)^{n + 1} a_{n}$ or $\sum_{n = 1}^{\infty} (- 1)^{n} a_{n}$ converges if $0 < a_{n + 1} \leq a_{n}$ for all $n \geq 1$ and $lim_{n \to \infty} a_{n} = 0$ .

Type: Basic Math::Card 1::Back

Deck: Math::Calculus::Sequences and Series

Tags: Math Calculus Sequences-and-Series Theory

What is the remainder of an alternating series?

TODO

Type: Basic Math::Card 1::Back

Deck: Math::Calculus::Sequences and Series

Tags: Math Calculus Sequences-and-Series Theory

What is the relationship between absolute and conditional convergence?

A series $\sum_{n = 1}^{\infty} a_{n}$ exhibits absolute convergence if $\sum_{n = 1}^{\infty} ∣ a_{n} ∣$ converges.

A series $\sum_{n = 1}^{\infty} a_{n}$ exhibits conditional convergence if $\sum_{n = 1}^{\infty} a_{n}$ converges but $\sum_{n = 1}^{\infty} ∣ a_{n} ∣$ diverges.

If $\sum_{n = 1}^{\infty} ∣ a_{n} ∣$ converges, then $\sum_{n = 1}^{\infty} a_{n}$ converges.

Type: Basic Math::Card 1::Back

Deck: Math::Calculus::Sequences and Series

Tags: Math Calculus Sequences-and-Series Theory

What is the ratio test?

Consider the series $\sum_{n = 1}^{\infty} a_{n}$ with nonzero terms. Let

ρ = n \to \infty lim \frac{a _{n + 1}}{a _{n}}

If $0 \leq ρ < 1$ , then $\sum_{n = 1}^{\infty} a_{n}$ converges absolutely.
If $ρ > 1$ or $ρ = \infty$ , then $\sum_{n = 1}^{\infty} a_{n}$ diverges.
If $ρ = 1$ , the test does not provide any information.

Type: Basic Math::Card 1::Back

Deck: Math::Calculus::Sequences and Series

Tags: Math Calculus Sequences-and-Series Theory

What is the root test?

Consider the series $\sum_{n = 1}^{\infty} a_{n}$ . Let

ρ = n \to \infty lim n ∣ a_{n} ∣

If $0 \leq ρ < 1$ , then $\sum_{n = 1}^{\infty} a_{n}$ converges absolutely.
If $ρ > 1$ or $ρ = \infty$ , then $\sum_{n = 1}^{\infty} a_{n}$ diverges.
If $ρ = 1$ , the test does not provide any information.

Type: Basic Math::Card 1::Back

Deck: Math::Calculus::Sequences and Series

Tags: Math Calculus Sequences-and-Series Theory

What is the power series?

The power series centered at $x = 0$ is an infinite series of the following form

n = 0 \sum \infty c_{n} x^{n} = c_{0} + c_{1} x + c_{2} x^{2} + \dots

The power series centered at $x = a$ is an infinite series of the following form

n = 0 \sum \infty c_{n} (x - a)^{n} = c_{0} + c_{1} (x - a) + c_{2} (x - a)^{2} + \dots

where $x$ is a variable and the coefficients $c_{n}$ are constants.

Type: Basic Math::Card 1::Back

Deck: Math::Calculus::Sequences and Series

Tags: Math Calculus Sequences-and-Series Theory

What is the center of a power series?

The center of the power series $\sum_{n = 0}^{\infty} c_{n} (x - a)^{n}$ is the point $x = a$ , the value around which the series is expanded.

Type: Basic Math::Card 1::Back

Deck: Math::Calculus::Sequences and Series

Tags: Math Calculus Sequences-and-Series Theory

What are the convergence properties of a power series?

Consider the power series $\sum_{n = 0}^{\infty} c_{n} (x - a)^{n}$ . The series satisfies exactly one of the following properties:

The series converges at $x = a$ and diverges for all $x \neq = a$ .
The series converges for all real numbers $x$ .
There exists a real number $R > 0$ such that the series converges if $∣ x - a ∣ < R$ and diverges if $∣ x - a ∣ > R$ . At the values $x$ where $∣ x - a ∣ = R$ , the series may converge or diverge.

Type: Basic Math::Card 1::Back

Deck: Math::Calculus::Sequences and Series

Tags: Math Calculus Sequences-and-Series Theory

What is the interval of convergence of a power series?

The interval of convergence $I$ of the power series $\sum_{n = 0}^{\infty} c_{n} (x - a)^{n}$ is the set of all real numbers $x$ such that the series converges. It's the "domain" of the power series.

Type: Basic Math::Card 1::Back

Deck: Math::Calculus::Vector Calculus

Tags: Math Calculus Vectors

Vector equation of the line.

r = p + t d

where:

$r$ is the position vector of any point on the line,
$p$ is the position vector of a specific point on the line,
$d$ is the direction vector of the line, and
$t$ is a scalar parameter.

Type: Basic Math::Card 1::Back

Deck: Math::Calculus::Vector Calculus

Tags: Math Calculus Vectors

Parametric equation of the line.

x y z = p_{x} + t d_{x} = p_{y} + t d_{y} = p_{z} + t d_{z}

where:

$(p_{x}, p_{y}, p_{z})$ are the coordinates of the specific point on the line,
$(d_{x}, d_{y}, d_{z})$ are the components of the direction vector, and
$t$ is a scalar parameter.

Type: Basic Math::Card 1::Back

Deck: Math::Calculus::Vector Calculus

Tags: Math Calculus Vectors

Symmetric equation of the line.

\frac{x - p _{x}}{d _{x}} = \frac{y - p _{y}}{d _{y}} = \frac{z - p _{z}}{d _{z}}

where:

$(p_{x}, p_{y}, p_{z})$ are the coordinates of the specific point on the line,
$(d_{x}, d_{y}, d_{z})$ are the components of the direction vector.

Type: Basic Math::Card 1::Back

Deck: Math::Calculus::Vector Calculus

Tags: Math Calculus Vectors

Vector equation of the plane.

(r - p) \cdot n = 0

or equivalently:

(r - p) \cdot n r \cdot n - p \cdot n r \cdot n r \cdot n = 0 = 0 = p \cdot n = c

where:

$r$ is the position vector of any point on the plane,
$p$ is the position vector of a specific point on the plane,
$n$ is a normal vector to the plane, and
$c = r \cdot n$ is a scalar constant.

Type: Basic Math::Card 1::Back

Deck: Math::Calculus::Vector Calculus

Tags: Math Calculus Vectors

Sclar equation of the plane.

a x + b y + cz = d

where:

$(a, b, c)$ are the components of the normal vector $n$ ,
$(x, y, z)$ are the coordinates of any point on the plane, and
$d$ is a scalar constant.

Type: Basic Math::Card 1::Back

Deck: Math::Calculus::Vector Calculus

Tags: Math Calculus Vectors

Parametric equation of the plane.

r = p + s b + t c

where:

$r$ is the position vector of any point on the plane,
$p$ is the position vector of a specific point on the plane,
$b$ is the first vector parallel to the plane,
$c$ is the second vector parallel to the plane,
$s$ and $t$ are scalar parameters.

Type: Basic Math::Card 1::Back

Deck: Math::Calculus::Vector Calculus

Tags: Math Calculus Vectors

Whatis a Quadric Surface?

A Quadric Surface is a surface in three-dimensional space whose equation is a second-degree polynomial in the variables $x$ , $y$ and $z$ .

The most general Quadric Surface has the form

A x^{2} + B y^{2} + C z^{2} + D x y + E yz + F x z + G x + Hy + I z + J = 0

where $A$ , $B$ , $C$ ,... are all constants.

Type: Basic Math::Card 1::Back

Deck: Math::Calculus::Vector Calculus

Tags: Math Calculus Vectors

Ellipsoid centered at the origin.

\frac{x ^{2}}{a ^{2}} + \frac{y ^{2}}{b ^{2}} + \frac{z ^{2}}{c ^{2}} = 1

Type: Basic Math::Card 1::Back

Deck: Math::Calculus::Vector Calculus

Tags: Math Calculus Vectors

Elliptic Paraboloid centered at the origin.

z = \frac{x ^{2}}{a ^{2}} + \frac{y ^{2}}{b ^{2}}

Type: Basic Math::Card 1::Back

Deck: Math::Calculus::Vector Calculus

Tags: Math Calculus Vectors

Hyperbolic Paraboloid centered at the origin.

z = \frac{x ^{2}}{a ^{2}} - \frac{y ^{2}}{b ^{2}}

Type: Basic Math::Card 1::Back

Deck: Math::Calculus::Vector Calculus

Tags: Math Calculus Vectors

Hyperboloid of One Sheet centered at the origin.

\frac{x ^{2}}{a ^{2}} + \frac{y ^{2}}{b ^{2}} - \frac{z ^{2}}{c ^{2}} = 1

Type: Basic Math::Card 1::Back

Deck: Math::Calculus::Vector Calculus

Tags: Math Calculus Vectors

Hyperboloid of Two Sheets centered at the origin.

\frac{x ^{2}}{a ^{2}} + \frac{y ^{2}}{b ^{2}} - \frac{z ^{2}}{c ^{2}} = - 1

Type: Basic Math::Card 1::Back

Deck: Math::Calculus::Vector Calculus

Tags: Math Calculus Vectors

Elliptic Cone centered at the origin.

\frac{z ^{2}}{c ^{2}} = \frac{x ^{2}}{a ^{2}} + \frac{y ^{2}}{b ^{2}}

Type: Basic Math::Card 1::Back

Deck: Math::Calculus::Vector Calculus

Tags: Math Calculus Vectors

What is the algebraic definition the Dot Product?

Consider two vectors $a$ and $b$ in an $n$ -dimensional space $a = ⟨ a_{1}, a_{2}, \dots, a_{n} ⟩$ , $b = ⟨ b_{1}, b_{2}, \dots, b_{n} ⟩$ .

The Dot Product of $a \cdot b$ is computed as the sum of the products of the corresponding components:

a \cdot b = a_{1} b_{1} + a_{2} b_{2} + \dots + a_{n} b_{n} = i = 1 \sum n a_{i} b_{i}

Type: Basic Math::Card 1::Back

Deck: Math::Calculus::Vector Calculus

Tags: Math Calculus Vectors

What is the geometric definition the Dot Product?

Consider two vectors $a$ and $b$ in an $n$ -dimensional space $a = ⟨ a_{1}, a_{2}, \dots, a_{n} ⟩$ , $b = ⟨ b_{1}, b_{2}, \dots, b_{n} ⟩$ .

The Dot Product of $a \cdot b$ is defined in terms of the magnitudes (lengths) of the vectors and the angle between them:

a \cdot b = ∥ a ∥ ∥ b ∥ cos θ

where

$∥ a ∥$ is the magnitude (length) of vector $a$ ,
$∥ a ∥$ is the magnitude (length) of vector $b$ , and
$θ$ is the angle between the two vectors.

Type: Basic Math::Card 1::Back

Deck: Math::Calculus::Vector Calculus

Tags: Math Calculus Vectors

What is the algebraic definition the Cross Product?

Consider two vectors $a$ and $b$ in the 3-dimensional space $a = ⟨ a_{1}, a_{2}, a_{3} ⟩$ , $b = ⟨ b_{1}, b_{2}, b_{3} ⟩$ .

The Cross Product of $a \times b$ is the determinant

a \times b = i a_{1} b_{1} j a_{2} b_{2} k a_{3} b_{3}

which expands to

a \times b = (a_{2} b_{3} - a_{3} b_{2}) i - (a_{1} b_{3} - a_{3} b_{1}) j + (a_{1} b_{2} - a_{2} b_{1}) k

Type: Basic Math::Card 1::Back

Deck: Math::Calculus::Vector Calculus

Tags: Math Calculus Vectors

What is the geometric definition the Cross Product?

Consider two vectors $a$ and $b$ in the 3-dimensional space $a = ⟨ a_{1}, a_{2}, a_{3} ⟩$ , $b = ⟨ b_{1}, b_{2}, b_{3} ⟩$ .

The Cross Product $a \times b$ is a vector that:

is perpendicular to both $a$ and $b$ (so orthogonal to the plane containing them),
has magnitude equal to the area of the parallelogram spanned by $a$ and $b$ : $∥ a \times b ∥ = ∥ a ∥∥ b ∥ sin θ$ where $θ$ is the angle between $a$ and $b$ , and
has direction given by the right-hand rule.

Type: Basic Math::Card 1::Back

Deck: Math::Calculus::Vector Calculus

Tags: Math Calculus Vectors

Unit Tangent Vector $T$ to a parameterized curve.

The Unit Tangent Vector is the normalized velocity vector.

T (t) = \frac{r ^{'} ( t )}{∥ r ^{'} ( t ) ∥}

where:

$r (t) = ⟨ x (t), y (t), z (t)⟩$ is a vector-valued function, and
$r^{'} (t) = ⟨ x^{'} (t), y^{'} (t), z^{'} (t)⟩$ is the velocity vector.

Type: Basic Math::Card 1::Back

Deck: Math::Calculus::Vector Calculus

Tags: Math Calculus Vectors

Principal Normal Vector $N$ to a parameterized curve.

The Principal Normal Vector to a curve $C$ is a unit vector that

is perpendicular to the Unit Tangent Vector, and
points in the direction that the curve is bending.

N (t) = \frac{T ^{'} ( t )}{∥ T ^{'} ( t ) ∥}

where:

$T (t)$ is the Unit Tangeng Vector.

Type: Basic Math::Card 1::Back

Deck: Math::Calculus::Vector Calculus

Tags: Math Calculus Vectors

Binormal Vector $B$ to a parameterized curve.

B (t) = T (t) \times N (t)

where:

$T (t)$ is the Unit Tangeng Vector, and
$N (t)$ is the Principal Normal Vector.

Type: Basic Math::Card 1::Back

Deck: Math::Calculus::Vector Calculus

Tags: Math Calculus Vectors

What is a Vector Valued Function (VVF)?

A Vector Valued Function (VVF)is a function where the input is a scalar and the output is a vector.

r : D \subseteq R \to R^{n}

r (t) = (f_{1} (t), f_{2} (t), \dots, f_{n} (t))

Examples:

r (t) = (cos (t), sin (t))

r (t) = (t, t^{2}, t^{3})

Function of Several Variables: $r : D \subseteq R^{n} \to R$

Vector-Valued Function of Several Variables: $F : D \subseteq R^{m} \to R^{n}$

Type: Basic Math::Card 1::Back

Deck: Math::Calculus::Vector Calculus

Tags: Math Calculus Vectors

What is a Function of Several Variables (FSV)?

A Function of Several Variables (FSV) is a function where the input is a vector and the output is a scalar.

f : D \subseteq R^{n} \to R

f (x_{1}, x_{2}, \dots, x_{n})

Examples:

f (x, y) = x^{2} + y^{2}

f (x) = ∥ x ∥^{2}

Function of Several Variables: $r : D \subseteq R^{n} \to R$

Vector-Valued Function of Several Variables: $F : D \subseteq R^{m} \to R^{n}$

Type: Basic Math::Card 1::Back

Deck: Math::Calculus::Vector Calculus

Tags: Math Calculus Vectors

What kind of a field does a Function of Several Variables define?

A Function of Several Variables $r : D \subseteq R^{n} \to R$ defines a Scalar Field.

Function of Several Variables: $r : D \subseteq R^{n} \to R$

Vector-Valued Function of Several Variables: $F : D \subseteq R^{m} \to R^{n}$

Type: Basic Math::Card 1::Back

Deck: Math::Calculus::Vector Calculus

Tags: Math Calculus Vectors

What is a Vector-Valued Function of Several Variables (VVFSV)?

A Vector-Valued Function of Several Variables (VVFSV)is a function where the input is a vector $x \in R^{m}$ and the output is a vector $y \in R^{n}$ .

F : D \subseteq R^{m} \to R^{n}

F (x) = (f_{1} (x), f_{2} (x), \dots, f_{n} (x))

x = (x_{1}, x_{2}, \dots, x_{m})

Examples:

F (x, y) = (x^{2} - y, x + y^{2})

F (x, y, z) = (x y, yz, x z)

Type: Basic Math::Card 1::Back

Deck: Math::Calculus::Vector Calculus

Tags: Math Calculus Vectors

What kind of a field does a Vector-Valued Function of Several Variables define?

A Vector-Valued Function of Several Variables $F : D \subseteq R^{m} \to R^{n}$ defines a Vector Field.

Function of Several Variables: $r : D \subseteq R^{n} \to R$

Vector-Valued Function of Several Variables: $F : D \subseteq R^{m} \to R^{n}$

Type: Basic Math::Card 1::Back

Deck: Math::Calculus::ODE

Tags: Math Calculus Differential-Equation ODE Theory

What is an ordinary differential equation (ODE)?

An ordinary differential equation is a mathematical equation that relates an unknown function of a single independent variable to its derivatives with respect to that variable.

A general $n$ -th order ordinary differential equation can be expressed as

F (x, y, y^{'}, y^{''}, \dots, y^{(n)}) = 0

where

$x$ is the independent variable.
$y$ is the unknown function of $x$ , i.e., $y = y (x)$ .
$y^{'}, y^{''}, \dots, y^{(n)}$ are the first, second, ..., $n$ -th derivatives of $y$ with respect to $x$ .
$F$ is a given function relating all these variables and derivatives.

The solution for a differential equation is a function which satisfies the above expression.

Type: Basic Math::Card 1::Back

Deck: Math::Calculus::ODE

Tags: Math Calculus Differential-Equation ODE Theory

What is the order of an ordinary differential equation?

The order of an ordinary differential equation is the order of the highest derivative in the equation.

Type: Basic Math::Card 1::Back

Deck: Math::Calculus::ODE

Tags: Math Calculus Differential-Equation ODE Theory

What is an autonomous ordinary differential equation?

An autonomous ordinary differential equation is an equation where the independent variable does not explicitly appear in the equation.

An autonomous ordinary differential equation can be expressed in the form:

\frac{d y}{d x} = f (y)

where $f (y)$ is a function that depends only on the dependent variable $y$ .

The equation can be solved by separating the variables and integrating both sides:

\frac{1}{f ( y )} \cdot \frac{d y}{d x} \int \frac{1}{f ( y )} \cdot \frac{d y}{d x} d x \int \frac{d y}{f ( y )} = 1 = \int 1 d x = x + C

where $C$ is the constant of integration.

Type: Basic Math::Card 1::Back

Deck: Math::Calculus::ODE

Tags: Math Calculus Differential-Equation ODE Theory

What is a separable ordinary differential equation?

A first-order ordinary differential equation is separable if it can be written in the form

\frac{d y}{d x} = h (x) g (y)

where $h (x)$ is a function of $x$ alone and $g (y)$ is a function of $y$ alone.

Once the variables are separated, we can integrate both sides of the equation with respect to $x$ as follows:

\int \frac{1}{g ( y )} \cdot \frac{d y}{d x} d x \int \frac{d y}{g ( y )} G (y) = \int h (x) d x = \int h (x) d x = H (x) + C

Type: Basic Math::Card 1::Back

Deck: Math::Calculus::Laplace Transform

Tags: Math Calculus Laplace-Transform Theory

What is the definition of the Laplace transform?

The Laplace transform of a function $f (t)$ is a function $F (s)$ defined as

F (s) = L {f} = \int_{0}^{\infty} e^{- s t} f (t) d t

where $f (t)$ is a function of exponential order defined for all $t \geq 0$ .

Type: Basic Math::Card 1::Back

Deck: Math::Calculus::Laplace Transform

Tags: Math Calculus Laplace-Transform Theory

What is the first translation or shifting property of the Laplace transform?

If $f (t)$ has the transform $F (s)$ then

L {e^{c t} f (t)} = F (s - c)

L^{- 1} {F (s - c)} = e^{c t} f (t)

Example:

Since

L {cos 2 t} = \frac{s}{s ^{2} + 4}

we have

L {e^{3 t} cos 2 t} = F (s - 3) = \frac{( s - 3 )}{( s - 3 ) ^{2} + 4}

Type: Basic Math::Card 1::Back

Deck: Math::Calculus::Laplace Transform

Tags: Math Calculus Laplace-Transform Theory

What is the second translation or shifting property of the Laplace transform?

If $f (t)$ has the transform $F (s)$ then

L {f (t - c) u (t - c)} = e^{- cs} F (s)

L^{- 1} {e^{- cs} F (s)} = f (t - c) u (t - c)

where $u (t - c)$ is the unit step function, which is defined as

u (t - c) = {10 t \geq c t < c

Example:

Since

L {t^{3}} = \frac{6}{s ^{4}}

we have

L {f (t - 2) u (t - 2)} = e^{- 2 s} F (s) = e^{- 2 s} \frac{6}{s ^{4}}

Type: Basic Math::Card 1::Back

Deck: Math::Calculus::Laplace Transform

Tags: Math Calculus Laplace-Transform Theory

What is the change of scale property of the Laplace transform?

If $f (t)$ has the transform $F (s)$ then

L {f (c t)} = \frac{1}{c} F (\frac{s}{c})

L^{- 1} {\frac{1}{c} F (\frac{s}{c})} = f (c t)

Example:

Since

L {sin t} = \frac{1}{s ^{2} + 1}

we have

L {sin 3 t} = \frac{1}{3} F (\frac{s}{3}) = \frac{1}{3} \frac{1}{( s /3 ) ^{2} + 1} = \frac{3}{s ^{2} + 9}

Type: Basic Math::Card 1::Back

Deck: Math::Calculus::Laplace Transform

Tags: Math Calculus Laplace-Transform Theory

What is the unit step function $u (t - c)$ ?

The unit step function (Heaviside function) $u (t - c)$ is defined as

u (t - c) = {10 t \geq c t < c

Type: Basic Math::Card 1::Back

Deck: Math::Calculus::Laplace Transform

Tags: Math Calculus Laplace-Transform Theory

What is the effect of multiplying by the unit step function $u (t - c)$ ?

Let $f (t) = 0$ for all $t < 0$ .

$f (t) u (t - c)$ with $c > 0$ is $f (t)$ switched on at the point $c$ .
$f (t - c) u (t - c)$ with $c > 0$ is $f (t)$ shifted (translated) to the right by the amount $c$ .

Type: Basic Math::Card 1::Back

Deck: Math::Calculus::Laplace Transform

Tags: Math Calculus Laplace-Transform Theory

What is the unit impulse function $δ (t - c)$ ?

The unit impulse function (Dirac delta function) $δ (t - c)$ is a function that has the following properties.

Infinite at the origin $c$ , zero everywhere else:

δ (t - c) = {\infty 0 t = c t \neq = c

Area of unity:

\int_{0}^{\infty} δ (t - c) d t = 1

Shifting or sampling property:

\int_{- \infty}^{\infty} f (t) δ (t - c) d t = f (c)

The unit impulse function is technically not a function but a generalized function or distribution.

Type: Basic Math::Card 1::Back

Deck: Math::Calculus::Laplace Transform

Tags: Math Calculus Laplace-Transform Theory

What is the derivative of the unit step function?

The derivative of the unit step function (Heaviside function) is the unit impulse function (Dirac delta function);

\frac{d}{d t} u (t) = δ (t)

Type: Basic Math::Card 1::Back

Deck: Math::Linear Algebra

Tags: Math Linear-Algebra

What is a linear equation?

A linear equation is an algebraic equation in which each term is either a constant or the product of a constant and a single variable. Linear equations can have one or more variables.

The general form of a linear equation in two variables $x$ and $y$ is:

a x + b y = c

where $a$ , $b$ , and $c$ are constants, and $a$ and $b$ are not both zero.

Type: Basic Math::Card 1::Back

Deck: Math::Linear Algebra

Tags: Math Linear-Algebra

What is a system of linear equations?

A system of linear equations is a collection of one or more linear equations involving the same set of variables.

For example, the following is a system of two linear equations with two variables $x$ and $y$ :

2 x + 3 y 4 x - y = 6 = 5

The solution to a system of linear equations is the set of values for the variables that satisfy all the equations simultaneously.

Type: Basic Math::Card 1::Back

Deck: Math::Linear Algebra

Tags: Math Linear-Algebra

What is a homogeneous system of linear equations?

A homogeneous system of linear equations is a system in which all of the constant terms are zero.

For example, the following is a homogeneous system of two linear equations with two variables $x$ and $y$ :

2 x + 3 y 4 x - y = 0 = 0

A homogeneous system always has at least one solution, known as the trivial solution, where all variables are equal to zero (e.g., $x = 0$ and $y = 0$ ).

Type: Basic Math::Card 1::Back

Deck: Math::Linear Algebra

Tags: Math Linear-Algebra

What is the difference between a inconsistent and an consistent system of linear equations?

A system of linear equations is consistent if there is at least one set of values for the variables that satisfies all the equations simultaneously. In other words, a consistent system has at least one solution.

A system of linear equations is inconsistent if there are no solutions. This occurs when the equations represent parallel lines (in two dimensions) or planes (in three dimensions) that do not intersect.

Type: Basic Math::Card 1::Back

Deck: Math::Linear Algebra

Tags: Math Linear-Algebra

What is the difference between a dependent and an independent system of linear equations?

A system of linear equations is independent if the equations are not multiples of each other, meaning they provide unique information about the variables. An independent system has a unique solution.

A system of linear equations is dependent if at least one equation can be derived from the others, meaning they do not provide unique information about the variables. A dependent system has either infinitely many solutions or no solution.

Type: Basic Math::Card 1::Back

Deck: Math::Linear Algebra

Tags: Math Linear-Algebra Matrix

What is a diagonal matrix?

A diagonal matrix is a square matrix in which all entries outside the main diagonal are zero. The main diagonal runs from the top left to the bottom right of the matrix.

For example, the following is a $3 \times 3$ diagonal matrix:

a_{11} 00 0 a_{22} 0 00 a_{33}

where $a_{11}$ , $a_{22}$ , and $a_{33}$ are the diagonal elements.

Type: Basic Math::Card 1::Back

Deck: Math::Linear Algebra

Tags: Math Linear-Algebra Matrix

What is a upper-trianguar matrix?

An upper-triangular matrix is a square matrix in which all entries below the main diagonal are zero. The main diagonal runs from the top left to the bottom right of the matrix.

For example, the following is a $3 \times 3$ upper-triangular matrix:

a_{11} 00 a_{12} a_{22} 0 a_{13} a_{23} a_{33}

Type: Basic Math::Card 1::Back

Deck: Math::Linear Algebra

Tags: Math Linear-Algebra Matrix

What is a lower-trianguar matrix?

An lower-triangular matrix is a square matrix in which all entries above the main diagonal are zero. The main diagonal runs from the top left to the bottom right of the matrix.

For example, the following is a $3 \times 3$ lower-triangular matrix:

a_{11} a_{21} a_{31} 0 a_{22} a_{32} 00 a_{33}

Type: Basic Math::Card 1::Back

Deck: Math::Linear Algebra

Tags: Math Linear-Algebra Matrix

What is the identity matrix $I$ ?

The identity matrix $I$ is a square matrix in which all the elements of the main diagonal are ones, and all other elements are zeros. The main diagonal runs from the top left to the bottom right of the matrix.

For example, the following is a $3 \times 3$ identity matrix:

100010001

The identity matrix acts as the multiplicative identity in matrix multiplication, meaning that for any matrix $A$ of compatible size:

A I = I A = A

Type: Basic Math::Card 1::Back

Deck: Math::Linear Algebra

Tags: Math Linear-Algebra Matrix

What is a symmetric matrix?

A symmetric matrix is a square matrix that is equal to its transpose. In other words, a matrix $A$ is symmetric if:

A = A^{T}

This means that the elements across the main diagonal are mirror images of each other, i.e., $a_{ij} = a_{ji}$ for all $i$ and $j$ .

For example, the following is a $3 \times 3$ symmetric matrix:

123245356

Type: Basic Math::Card 1::Back

Deck: Math::Linear Algebra

Tags: Math Linear-Algebra Matrix

What are the multiplication properties of matrices?

Matrix multiplication follows the distributive law:

A (B \pm C) (A \pm B) C = A B \pm A C = A C \pm BC

Matrix multiplication follows the associative law:

A (BC) = (A B) C

In general, matrix multiplication is not commutative:

A B \neq = B A

Except when multiplying by the identity matrix $I$ :

A I = I A

Type: Basic Math::Card 1::Back

Deck: Math::Linear Algebra

Tags: Math Linear-Algebra Matrix

Is matrix multiplication distributive?

Yes, matrix multiplication is distributive. For matrices $A$ , $B$ , and $C$ (of compatible sizes), the following properties hold:

A (B \pm C) (A \pm B) C = A B \pm A C = A C \pm BC

Type: Basic Math::Card 1::Back

Deck: Math::Linear Algebra

Tags: Math Linear-Algebra Matrix

Is matrix multiplication associative?

Yes, matrix multiplication is associative. For matrices $A$ , $B$ , and $C$ (of compatible sizes), the following properties hold:

A (BC) = (A B) C

Type: Basic Math::Card 1::Back

Deck: Math::Linear Algebra

Tags: Math Linear-Algebra Matrix

Is matrix multiplication commutative?

No, matrix multiplication is not commutative. For any matrices $A$ and $B$ :

A B \neq = B A

Except when multiplying by the identity matrix $I$ :

A I = I A

Type: Basic Math::Card 1::Back

Deck: Math::Linear Algebra

Tags: Math Linear-Algebra Matrix

What is the inverse of a square matrix?

The inverse of a square matrix $A$ is a square matrix $A^{- 1}$ such that:

A \cdot A^{- 1} = A^{- 1} \cdot A = I

where $I$ is the identity matrix of the same size as $A$ .

Type: Basic Math::Card 1::Back

Deck: Math::Linear Algebra

Tags: Math Linear-Algebra Matrix

What are the properties of symmetric matrices?

Suppose that is $A$ symmetric matrix. Then, we have the following properties:

$- A$ is symmetric.
$c A$ is symmetric for any $c \neq = 0$ .
$A^{m}$ is symmetric for any positive integer $m$ .
$A^{- 1}$ is symmetric for nonsingular $A$ .

If $A$ and $B$ symmetric matrices, we have the following properties:

The sum $A + B$ is symmetric.
The product $A \cdot B$ is not always symmetric.

Finally:

$A + A^{T}$ is symmetric for any square matrix $A$ .
$A \cdot A^{T}$ and $A^{T} \cdot A$ is symmetric for any $m \times n$ matrix $A$ .

Any diagonal matrix is symmetric.

Type: Basic Math::Card 1::Back

Deck: Math::Linear Algebra

Tags: Math Linear-Algebra Matrix

If $A$ is a symmetric matrix, is $A^{m}$ symmetric?:

Yes, $A^{m}$ is symmetric for a symmetric matrix $A$ and a positive integer $m$ .

Type: Basic Math::Card 1::Back

Deck: Math::Linear Algebra

Tags: Math Linear-Algebra Matrix

If $A$ is a symmetric matrix, is $A^{- 1}$ symmetric?:

Yes, $A^{- 1}$ is symmetric for a nonsingular symmetric matrix $A$ .

Type: Basic Math::Card 1::Back

Deck: Math::Linear Algebra

Tags: Math Linear-Algebra Matrix

If $A$ and $B$ are symmetric matrices, is their sum $A + B$ symmetric?:

Yes, the sum $A + B$ is symmetric for two symmetric matrices $A$ and $B$ .

Type: Basic Math::Card 1::Back

Deck: Math::Linear Algebra

Tags: Math Linear-Algebra Matrix

If $A$ and $B$ are symmetric matrices, is their product $A \cdot B$ symmetric?:

No, the product $A \cdot B$ is not always symmetric.

Type: Basic Math::Card 1::Back

Deck: Math::Linear Algebra

Tags: Math Linear-Algebra Matrix

For which matrix $A$ the sum $A + A^{T}$ is symmetric?

The sum $A + A^{T}$ is symmetric for any square $n \times n$ matrix $A$ .

Type: Basic Math::Card 1::Back

Deck: Math::Linear Algebra

Tags: Math Linear-Algebra Matrix

For which matrix $A$ the product $A \cdot A^{T}$ is symmetric?

The product $A \cdot A^{T}$ is symmetric for any rectangular $m \times n$ matrix $A$ .

Type: Basic Math::Card 1::Back

Deck: Math::Linear Algebra

Tags: Math Linear-Algebra Matrix

When a diagonal matrix is symmetric?

Any diagonal matrix is symmetric.

Type: Basic Math::Card 1::Back

Deck: Math::Linear Algebra

Tags: Math Linear-Algebra Determinant

What is a singular square matrix?

A square matrix $A$ is singular if its determinant is zero:

det (A) = 0

A singular square matrix $A$ cannot be inverted, meaning $A^{- 1}$ does not exist.

Type: Basic Math::Card 1::Back

Deck: Math::Linear Algebra

Tags: Math Linear-Algebra Determinant

When a matrix does not have an inverse?

A matrix $A$ does not have an inverse if:

It is not square.
It is singular, meaning its determinant is zero $det (A) = 0$ .

Type: Basic Math::Card 1::Back

Deck: Math::Linear Algebra

Tags: Math Linear-Algebra Determinant

When the determinant of a matrix is zero?

A square matrix has zero determinant when its rows or columns are linearly dependent - meaning one row (or column) can be written as a linear combination of the others.

Type: Basic Math::Card 1::Back

Deck: Math::Linear Algebra

Tags: Math Linear-Algebra Determinant

What is the geometric interpretation of the absolute determinant value of a matrix?

For a square matrix $A$ , the absolute value of the determinant $∣ det (A) ∣$ is the factor by which the matrix scales $n$ -dimensional volume:

$∣ det (A) ∣ = 0$ : the transformation flattens space into a lower dimension (e.g., collapsing area or volume to zero).
$∣ det (A) ∣ = 1$ : the transformation preserves area/volume (it may rotate, reflect, or shear, but doesn't scale).
$∣ det (A) ∣ > 1$ : volumes are expanded.
$∣ det (A) ∣ < 1$ : volumes are contracted.

Type: Basic Math::Card 1::Back

Deck: Math::Linear Algebra

Tags: Math Linear-Algebra Determinant

What is the geometric interpretation of the zero determinant of a matrix?

A determinant is zero whenever the transformation collapses space into a lower dimension.

Type: Basic Math::Card 1::Back

Deck: Math::Linear Algebra

Tags: Math Linear-Algebra Determinant

What is the determinant of a triangular matrix?

The determinant of a triangular matrix is the product of its diagonal entries.

For a diagonal matrix:

det (A) = a_{11} 0 ⋮ 0 0 a_{22} ⋮ 0 \dots \dots ⋱ \dots 00 ⋮ a_{nn} = i = 1 \prod n a_{ii} = a_{11} \cdot a_{22} \cdot a_{33} \dots a_{nn}

For an upper-triangular matrix:

det (A) = a_{11} 0 ⋮ 0 a_{12} a_{22} ⋮ 0 \dots \dots ⋱ \dots a_{nn} a_{2 n} ⋮ a_{nn} = i = 1 \prod n a_{ii} = a_{11} \cdot a_{22} \cdot a_{33} \dots a_{nn}

For a lower-triangular matrix:

det (A) = a_{11} a_{21} ⋮ a_{n 1} 0 a_{22} ⋮ a_{n 2} \dots \dots ⋱ \dots 00 ⋮ a_{nn} = i = 1 \prod n a_{ii} = a_{11} \cdot a_{22} \cdot a_{33} \dots a_{nn}

Type: Basic Math::Card 1::Back

Deck: Math::Linear Algebra

Tags: Math Linear-Algebra Determinant

When the determinant of a triangular matrix is zero?

The determinant of a triangular matrix is zero whenever one of its diagonal entries is zero.

Type: Basic Math::Card 1::Back

Deck: Math::Linear Algebra

Tags: Math Linear-Algebra Determinant

What is the determinant of a $2 \times 2$ matrix $A$ ?

Given a $2 \times 2$ matrix $A = [a c b d]$ its determinant is computed as:

det (A) = a c b d = a d - b c

Type: Basic Math::Card 1::Back

Deck: Math::Linear Algebra

Tags: Math Linear-Algebra Determinant

What is the determinant of a $3 \times 3$ matrix $A$ ?

Given a $3 \times 3$ matrix $A = a d g b e h c f i$ its determinant is computed as:

det (A) = a d g b e h c f i = a e h f i - b d g f i + c d g e h = a (e i - f h) - b (d i - f g) + c (d h - e g)

It is simply a cofactor expansion along the first row.

Type: Basic Math::Card 1::Back

Deck: Math::Linear Algebra

Tags: Math Linear-Algebra Determinant

What is the minor $M_{ij}$ of the entry $a_{ij}$ in a matrix $A$ ?

The minor $M_{ij}$ of the entry $a_{ij}$ in a matrix $A$ is the determinant of the remaining entries when the row and column containing $a_{ij}$ are removed.

Example:

For the matrix $A = 147258369$ , the minor $M_{22}$ of the entry $a_{22} = 5$ is computed by removing the second row and second column, resulting in the submatrix $[1739]$ :

M_{22} = 1739

Type: Basic Math::Card 1::Back

Deck: Math::Linear Algebra

Tags: Math Linear-Algebra Determinant

What is the cofactor $C_{ij}$ of the entry $a_{ij}$ in a matrix $A$ ?

The cofactor $C_{ij}$ of the entry $a_{ij}$ is in a matrix $A$ is given by:

C_{ij} = (- 1)^{i + j} \cdot M_{ij}

where $M_{ij}$ is the minor of the entry $a_{ij}$ .

Example:

For the matrix $A = 147258369$ , the cofactor $C_{21}$ of the entry $a_{21} = 4$ is

C_{ij} = (- 1)^{2 + 1} \cdot M_{21} = - 2839

Cofactors are a useful tool in helping us to compute determinants of larger matrices.

Type: Basic Math::Card 1::Back

Deck: Math::Linear Algebra

Tags: Math Linear-Algebra Determinant

How to use the cofactor expansion to compute the determinant of a square matrix $A$ ?

The determinant of a square matrix $A$ can be computed using cofactors by expanding along any row or column.

For example, the cofactor expansion across the first row $i = 1$ is:

det (A) = j = 1 \sum n a_{1 j} C_{1 j} = a_{11} C_{11} + a_{12} C_{12} + \dots + a_{1 n} C_{1 n}

Or, the cofactor expansion across the second column $j = 2$ is:

det (A) = i = 1 \sum m a_{i 2} C_{i 2} = a_{12} C_{12} + a_{22} C_{22} + \dots + a_{m 2} C_{m 2}

Type: Basic Math::Card 1::Back

Deck: Math::Linear Algebra

Tags: Math Linear-Algebra Determinant

What is the Laplace expansion to compute the determinant of a square matrix $A$ ?

The Laplace expansion to compute the determinant is the cofactor expansion along any row or column.

Type: Basic Math::Card 1::Back

Deck: Math::Linear Algebra

Tags: Math Linear-Algebra Determinant

What is the inverse of a $2 \times 2$ matrix $A$ ?

Given a $2 \times 2$ matrix $A = [a c b d]$ , its inverse $A^{- 1}$ is computed as:

A^{- 1} = \frac{1}{det ( A )} [d - c - b a] = \frac{1}{a d - b c} [d - c - b a]

Type: Basic Math::Card 1::Back

Deck: Math::Linear Algebra

Tags: Math Linear-Algebra Matrix

What is the affine transformation?

An affine transformation is a function between affine spaces that preserves points, straight lines, and planes, but not necessarily distances or angles. Examples include translation, scaling, rotation, and shearing.

An affine transformation $T : R^{2} \to R^{2}$ is defined as

x = A u + b

where $x$ , $u$ and $b$ are $2 \times 1$ column vectors and $A$ is a $2 \times 2$ matrix.

Type: Basic Math::Card 1::Back

Deck: Math::Linear Algebra

Tags: Math Linear-Algebra Matrix

What does it mean for an affine transformation to preserve orientation?

Preserving orientation means that the transformation does not flip the space, maintaining the clockwise or counterclockwise arrangement of points.

Type: Basic Math::Card 1::Back

Deck: Math::Linear Algebra

Tags: Math Linear-Algebra Matrix

When an affine transformation preserves orientation?

An affine transformation $T$ preserves orientation if

det (A) > 0

and does not preserve orientation if the determinant is negative:

det (A) < 0

where the matrix $A$ that represents the linear transformation associated with $T$ .

Type: Basic Math::Card 1::Back

Deck: Math::Linear Algebra

Tags: Math Linear-Algebra Matrix

When an affine transformation is invertible?

An affine transformation $T$ is invertible if and only if the matrix $A$ that represents the linear transformation associated with $T$ is invertible, that is $det (A) \neq = 0$ .

Type: Basic Math::Card 1::Back

Deck: Math::Linear Algebra

Tags: Math Linear-Algebra Matrix

What is an eigenvector $v$ of a square matrix $A$ ?

Given a matrix $A$ an eigenvector of $A$ is a non-zero vector $v$ such that

A v = λ v

for some scalar $λ \in R$ , $λ \neq = 0$ . The scalar $λ$ is called the eigenvalue corresponding to the eigenvector $v$ .

Type: Basic Math::Card 1::Back

Deck: Math::Extra

Tags: Math Calculus Center-of-Mass

What is the formula for the moment of a system of particles about the origin?

The moment $M_{O}$ of a system of particles about the origin is given by the formula:

M_{O} = i = 1 \sum n m_{i} r_{i}

$m_{i}$ is the mass of the $i$ -th particle, and
$r_{i}$ is the position vector of the $i$ -th particle.

Type: Basic Math::Card 1::Back

Deck: Math::Extra

Tags: Math Calculus Center-of-Mass

What is the formula for the center of mass of a system of particles?

The center of mass $R$ of a system of particles is given by the formula:

R = \frac{1}{m} i = 1 \sum n m_{i} r_{i}

where

$m = \sum_{i = 1}^{n} m_{i}$ is the total mass of the system,
$m_{i}$ is the mass of the $i$ -th particle, and
$r_{i}$ is the position vector of the $i$ -th particle.

Type: Basic Math::Card 1::Back

Deck: Math::Extra

Tags: Math Algebra Mean

The definition of the Arithmetic Mean (AM).

The Arithmetic Mean (AM) of two numbers $a$ and $b$ is given by:

A M = \frac{a + b}{2}

The general formula for $n$ numbers $x_{1}, x_{2}, \dots, x_{n}$ is:

A M = \frac{x _{1} + x _{2} + \dots + x _{n}}{n} = \frac{1}{n} i = 1 \sum n x_{i}

Type: Basic Math::Card 1::Back

Deck: Math::Extra

Tags: Math Algebra Mean

The definition of the Geometric Mean (GM).

The Geometric Mean (GM) of two numbers $a$ and $b$ is given by:

GM = ab

The general formula for $n$ numbers $x_{1}, x_{2}, \dots, x_{n}$ is:

GM = (x_{1} \cdot x_{2} \cdot \dots \cdot x_{n})^{1/ n} = (i = 1 \prod n x_{i})^{1/ n}

Type: Basic Math::Card 1::Back

Deck: Math::Extra

Tags: Math Algebra Mean

The definition of the Quadratic Mean (QM).

The Quadratic Mean (QM) of two numbers $a$ and $b$ is given by:

QM = \frac{a ^{2} + b ^{2}}{2}

This is also known as the Root Mean Square (RMS).

The general formula for $n$ numbers $x_{1}, x_{2}, \dots, x_{n}$ is:

QM = \frac{x _{1}^{2} + x _{2}^{2} + \dots + x _{n}^{2}}{n} = \frac{1}{n} i = 1 \sum n x_{i}^{2}

Type: Basic Math::Card 1::Back

Deck: Math::Extra

Tags: Math Algebra Mean

The definition of the Harmonic Mean (HM).

The Harmonic Mean (HM) of two numbers $a$ and $b$ is given by:

H M = \frac{2}{1/ a + 1/ b} = \frac{2 ab}{a + b}

The general formula for $n$ numbers $x_{1}, x_{2}, \dots, x_{n}$ is:

H M = \frac{n}{\frac{1}{x _{1}} + \frac{1}{x _{2}} + \dots + \frac{1}{x _{n}}} = \frac{n}{\sum _{i = 1}^{n} \frac{1}{x _{i}}}

Type: Basic Math::Card 1::Back

Deck: Math::Extra

Tags: Math Algebra Mean

AM-GM Inequality

\frac{a + b}{2} \geq ab

where AM is the Arithmetic Mean and GM is the Geometric Mean of two non-negative numbers $a$ and $b$ .

Extra

More generally, it is known that the following inequality holds for any set of positive numbers:

QM \geq A M \geq GM \geq H M

where

QM is the Quadratic Mean
AM is the Arithmetic Mean
GM is the Geometric Mean
HM is the Harmonic Mean

Type: Basic Math::Card 1::Back

Deck: Math::Extra

Tags: Math Algebra Mean

What is the general formula for means?

The general formula is the Kolmogorov–Nagumo (Quasi-Arithmetic) Mean:

M_{f} (x_{1}, x_{2}, \dots, x_{n}) = f^{- 1} (\frac{1}{n} i = 1 \sum n f (x_{i})) = f^{- 1} (\frac{f ( x _{1} ) + f ( x _{2} ) + \dots + f ( x _{n} )}{n})

where $f$ is any continuous strictly monotone function.

If $f (x) = x$ , we get the Arithmetic Mean.
If $f (x) = ln x$ , we get the Geometric Mean (requires $x_{i} > 0$ ).
If $f (x) = 1/ x$ , we get the Harmonic Mean (requires $x_{i} \neq = 0$ ).
If $f (x) = x^{p}$ (with $p \neq = 0$ ), we get the Power Mean, with $p = 2$ being the Quadratic Mean.

So the Quasi-Arithmetic Mean is the most general form that unites all the classical means.