AP Calc AB Formulas

Basic Derivative (Power Rule)

f(x^n)= nX^(n-1)

Derivative of sin(x)

= cosx

Derivative of cos(x)

= -sinx

Derivative of tan(x)

= sec^2(x)

Derivative of cot(x)

= -csc^2(x)

Derivative of sec(x)

= secxtanx

Derivative of csc(x)

= -cscxcotx

Derivative of ln(u)

= 1/u u'

Derivative of e^x

= e^x

Chain Rule

d/dx f(g(x)) = f'(g(x)) g'(x)

product rule

f'(x)g(x)+f(x)g'(x)

Quotient Rule

g(x)f'(x)-f(x)g'(x)/g(x)^2

Intermediate Value Theorem

If f is continuous on [a,b] and y is a number between f(a) and f(b), then there exists at least one number x=c in the open interval (a,b) such that f(c)=y

Mean Value Theorem

If f(x) is continuous on [a,b] AND the first derivative exists on the interval (a, b), then there is at least one number x=c in (a,b) such that

f '(c) = [f(b) - f(a)]/(b - a)

average value of a function

1/b-a integral from a to b f(x)dx

Extreme Value Theorem

If f is continuous on [a,b] then f has an absolute maximum and an absolute minimum on [a,b].

Derivative of an Inverse Function

g'(x) = 1/ f'(g(x)) where g(x) is the inverse of f(x)

Average Rate of change

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

Critical Point

dy/dx=0 or undefined

Local Minimum

dy/dx goes (-,+) or second derivative > 0

Local Maximum

dy/dx goes (+,-) or second derivative < 0

Point of Inflection

second derivative goes from (+,-) or (-,+)

f'(x) < 0

f(x) is decreasing

f'(x) > 0

f(x) is increasing

f'(x) = 0 or DNE

Critical Values at x

Relative Maximum

f'(x) = 0 or DNE AND sign of f'(x) changes from + to -

Relative Minimum

f'(x) = 0 or DNE AND sign of f'(x) changes from - to +

f''(x) > 0

concave up

f''(x) < 0

concave down

f'(x) = 0 and sign change of f''(x) changes then there is a

point of inflection at x

Horizontal Asymptotes

1. If the largest exponent in the numerator is < largest exponent in the numerator in the denominator then the lim as x approaches positive and negative infinity of f(x)=0

2. If the largest exponent in the numerator is > largest exponent in the numerator in the denominator then the lim as x approaches positive and negative infinity of f(x)=DNE

3. If the largest exponent in the numerator is = largest exponent in the numerator in the denominator then the lim as x approaches positive and negative infinity of f(x)=a/b

x(t) - physics problem

= position function

v(t) - physics problem

= velocity function

a(t) - physics problem

= acceleration function

The derivative of position is

velocity

The derivative of velocity is

acceleration

The integral of acceleration is

velocity

The integral of velocity is

position

Speed is

absolute value of velocity

If acceleration and velocity have the SAME SIGN, then

the speed is INCREASING, particle is moving right

If acceleration and velocity have DIFFERENT SIGNS, then

the speed is DECREASING, particle is moving left

Displacement

∫ v(t)

Distance

∫ absolute value of position

Average Velocity

= final position - initial position / total time

ln e

= 1

ln 1

= 0

ln(MN)

= lnM + lnN

ln(M/N)

= lnM - lnN

ln(M)^P

= P ln(M)

The Fundamental Theorem of Calculus

from a to b ∫ f(x) dx = F(b) - F(a)

Corollary to FTC

from a to g(u) ∫ f(t) dt = f(g(u)) du/dx

sin(0)

= 0

sin(π/6)

= 1/2

sin(π/4)

= √(2)/2

sin(π/3)

= √(3)/2

sin(π/2)

= 1

sin(π)

= 0

cos(0)

= 1

cos(π/6)

= √(3)/2

cos(π/4)

= √(2)/2

cos(π/3)

= 1/2

cos(π/2)

= 0

cos(π)

= -1

tan(0)

= 0

tan(π/6)

= √(3)/3

tan(π/4)

= 1

tan(π/3)

= √(3)

tan(π/2)

undefined

tan(π)

= 0

Riemann Sum

Approximating Area under the curve using rectangles

sin^2x+cos^2x

= 1

tan(x)

= sinx/cosx

cot(x)

= cosx/sinx

csc(x)

= 1/sinx

sec(x)

= 1/cosx

∫ u^n du

= (u^(n+1))/(n+1) +C

∫ 1/u du

= ln |u| + C

∫ e^u du

= e^u + C

∫ sinu du

= -cosu + C

∫ cosu du

= sinu + C

∫ tanu du

= -ln |cosu + C|

∫ cotu du

= ln |sinu| + C

∫ secu du

= ln |secu + tanu| + C

∫ cscu du

= -ln |cscu + cotu| + C

∫ sec^2 du

= tanu + C

∫ csc^2 dy

= -cotu + C

∫ secu tanu du

= secu + C

∫ cscu cotu du

= -cscu + C

Derivative of inverse sin

Derivative of inverse tan

Derivative of inverse of sec

Integral of du/sqrt(a^2-u^2)

sin^-1(u/a) + C

integral of du/a^2+u^2

(1/a)arctan(u/a) + C

Integral of du/u √u^2-a^2

(1/a)arcsec(|u|/a) + C