AP Calc AB

Quotient ID’s

(tan x = sin x/cos x) OR (cot x = cos x/sin x)

Reciprocal ID’s

(sec x = 1/cos x) OR (csc x = 1/sin x)

Pythagorean ID’s

(sin²x + cos²x = 1) OR (sec²x - tan²x = 1)

2sinxcosx = sin2x

Double Angle id #1

cos²x-sin²x = cos2x

Double Angle id #2

sin (-x) = - sin x

Even-odd ID (sin)

cos (-x) = cos x

Even-odd ID (cos)

tan(-x) = - tanx

Even-odd ID (tan)

Cos Personality

Mean + selfish (doesn’t like sin, goes against + and -)

Sin (A+B)

sinA cosB + cosA sinB

Cos (A+B)

cosA cosB - sinA sinB

Sin (A-B)

sinA cosB- cosA sinB

Cos (A-B)

cosA cosB + sinA sinB

Distance b/w 2 points

(x₂-x₁)² + (y₂-y₁)² (square root of)

Midpoint Formula

(x₁+x₂)/2 , (y₁+y₂)/2

ln (0)

undefined

ln (1)

0

ln (e)

1

Law of Logarithms = ln(ab)

lna + lnb

Law of Logarithms = ln(a/b)

lna - lnb

Law of Logarithms = ln(aⁿ)

n*lna

Law of Logarithms = ln(1/a)

-lna

Average Rate of Change is also called

slope of the secant line

Average Rate of Change Formula

f(b) - f(a) / b - a

Squeeze Theorem

If f(x) </= g(x) </= h(x), then as x —→ a, f(x) —> L and h(x) —> L then g(x) —→ L

lime^x (- infinity)

0

limex (infinity)

infinity

limlnx —> 0+

- infinity

limlnx —> infinity

infinity

lim1/x

0

lim sinx/x

1

lim1-cosx/x

0

limarctanx (—> infinity)

pi/2

limarctanx (—> - infinity)

- pi/2

lim1/x (—> 0)

- infinity

lim1/x (—> 0+)

infinity

Definition of a vertical asymptote

x=a iff lim (—> a+) = ±infinity OR lim (—> a-)

Definition of a horizontal asymptote

y=a iff lim (—> infinity) = a OR lim (—> - infinity) = a

Continuity

1) f(a) exists, 2) lim (x—>a) f(x) exists, 3) lim (x—>a) f(x) = f(a)

Intermediate Value Theorem (IVT)

1) f is cont. on the closed interval [a,b] 2) f(a) not equal to f(b) 3) k is between f(a) and f(b)

If IVT meets requirements,

Then there exists a number c between a and b for which f(c) = k

Definition of the derivative/limit of the difference quotient

f’(x) = lim (h—>0) f(x+h) - f(x) / h

Definition of the derivative/alternate form

f’(x) = lim (x—>b) f(x) - f(b) / x-b 

Tangent Line

y-f(a) = f’(a) (x-a) 

Normal Line

The line perpendicular to the tangent line at the point of tangency

Three Reasons for a function, f, will not be differentiable at a point x=a

1) f not cont. at x=a 2) the graph of f has a corner/cusp at x=a 3) the graph of f has a vertical tangent at x=a

Differentiation Rules

f and g are functions of x

Product Differentiation Rule

d/dx (f g) = f’ g + f * g’ 

Quotient Differentiation Rule

d/dx (f/g) = f’*g - f*g’/g²

Chain Rule:

if h(x) = f(g(x)), then h’(x) = f’(g(x)) * g’(x)

Chain Rule Words

the derivative of the outside evaluated at the inside times the derivative of the inside

d/dx ( sinx) =

cosx

d/dx (cscx) =

-cscx * cotx

d/dx (e^x) =

e^x

d/dx (lnx) =

-1/x

d/dx (cosx) =

-sinx

d/dx (secx) =

secx * tanx

d/dx (logax) =

1/xlna

d/dx (tanx) =

sec²x

d/dx (cotx)

-csc²x

lim (f+-g) =

limf +- limg

lim (c*f) =

c*limf

lim c =

c

lim (fg) =

limf * limg

lim f/g = 

limf / limg for limg does not equal 0

lim f(g(x)) =

f(limg(x))

lim (f(x))^n =

(limf(x))^n

lim f(g(x)) =

f(limg(x))

Extreme Value Theorem (EVT)

1) If f is cont. on a closed interval [a,b], 2) Then f has an absolute max and an absolute min on the interval [a,b] 

Mean Value Theorem

If: 1) f is cont. on [a,b], 2) differentiable on (a,b) Then: there exists a number c between a and b such that f’(c ) = f(b) - f(a) / b-a

Minimum or maximum refers to

y-value

Position, Velocity, Acceleration Application

1. Particle at rest when v(t) = 0

2. Speed is increasing if v and a have same sign

3. Speed is decreasing if v and a have opposite signs

Definition of concavity

1. f is concave up if f’ is increasing

2. f is concave down if f’ is decreasing

Definition of a point of inflection (POI)

A point on the graph of f where the concavity of f changes.

2nd derivative test for relative extrema:

Given f’(c ) = 0, then

1) If f’’(c ) < 0, then f(c ) is a max

2) If f’’(c ) > 0, then f(c ) is a min

3) If f’’(c ) = 0, then the test is inconclusive

xⁿdu =

xⁿ⁺¹/n+1 + C

a^udu =

a^u/lna + C

sinu du =

-cosu + C

cscu cotu du =

-cscu + C

sin^-1u + C

sin^-1 u/a + C

cosu du =

sinu + C

secu tanu du =

secu + C

tan^-1u + C

1/a tan^-1 u/a + C

e^u du =

e^u + C

sec² u du =

tanu + C

csc² u du =

-cotu + C

1/a arc sec IuI/a + C

arc sec IuI + C

0

1/u du =

ln IuI + C