ap calc ab exam

derivative

slope of the graph at any particular point

trapezoid formula

A=(a/2+b+c+d+e/2)Δx

vertical asymptotes

occur where the denominator approaches 0 and the numerator does not

horizontal asymptotes

found by comparing the degree of the numerator and the degree of the denominator (or the lead coefficients)

non-horizontal asymptotes

found by doing long division

LIMIT RULES

LIMIT RULES

limit of a sum is equal to the sum of the limits

\lim_{x\rightarrow a}\left(f\left(x\right)+g\left(x\right)\right)=\lim_{x\rightarrow a}f\left(x\right)+\lim_{x\rightarrow a}g\left(x\right)

limit of a constant times a function equals the constant times the limit of the function

\lim_{x\rightarrow a}\left(c\cdot f\left(x\right)\right)=c\cdot\lim_{x\rightarrow a}\left(f\left(x\right)\right)

the limit of a product equals the product of the limits

\lim_{x\rightarrow a}\left(f\left(x\right)\cdot g\left(x\right)\right)=\lim_{x\rightarrow a}\left(f\left(x\right)\right)\cdot\lim_{x\to a}\left(g\left(x\right)\right)

the limit of a quotient equals the quotient of the limits

\lim_{x\to a}\frac{f\left(x\right)}{g\left(x\right)}=\frac{\lim_{x\to a}\left(f\left(x\right)\right)}{\lim_{x\to a}\left(g\left(x\right)\right)}

the limit of a quotient DNE if the denominator is zero while the numerator is not zero

the value of a limit as x➡a is the same as the value of the polynomial at x=a

\lim_{x\to a}p\left(x\right)=p\left(a\right)

cusp

a point at which the function is continuous but the derivative of the function is discontinuous

intermediate value theorem

if f is continuous over [a,b] and y is a number between f(a) and f(b), then there is some number c in the interval [a,b] such that f(c)=y

difference quotient

\frac{f\left(x\right)-f\left(c\right)}{x-c}

calculating a derivative

take the limit of the difference quotient

mathematical definition of a derivative

\lim_{\Delta x\rightarrow0}\frac{f\left(x+\Delta x\right)-f\left(x\right)}{\Delta x}

derivative of a power function

if f(x)=xⁿ then f'(x)= nx^n-1

derivative of a sum

the sum of the derivatives (same as limits rule)

derivative of constant times a function

\frac{d}{\differentialD x}kf=kf^{\prime}

derivative of a constant

0

linear approximation

derivative chain rule

derivative of the outer function with respect to the inner times the derivative of the inner function

derivative product rule

first times the derivative of the second plus second times the derivative of the first: uv'+u'v

quotient rule

low d high - high d low over low times low

derivative of sin(x)

cos x

derivative of cos(x)

-sin x

derivative of tan(x)

(sec x)(sec x)

derivative of csc(x)

-csc x cot x

derivative of sec(x)

sec x tan x

derivative of cot(x)

-csc² x

rate of change practical consequence

the slope of f at any point (x,y) is the reciprocal of the slope of f' at the corresponding point (y,x)

extreme value theorem

if f(x) is continuous on [a,b], then f(x) has both an absolute maximum and minimum on [a,b]

rolle's theorem

between 2 zeros of a function, there is a place where the slope is zero
requirements
1. a function is differentiable on the open interval (a,b)
2. the function is continuous at x=a and x=b
3. f(a)=f(b)

mean value theorem

at some point in the interval [a,b] there is a place where the slope of the curve equals the slope of the secant line

if f is rising...

f' is positive

if f is rising at an increasing rate...

the value of f' gets larger

if f' has a positive slope

f'' has a positive value

if f is concave up

f' is increasing, and f'' is positive

average velocity

Δx/Δt

instantaneous velocity

dx/dt

speed

how fast something is moving (absolute value of velocity)

velocity

how fast something is moving and in what direction

constant of integration

"+ c"

definite integral

the area under a graph

integral multiplication rule

integral power rule

the integral of the sum of functions...

is equal to the sum of their integrals

fundamental theorem of calculus

\int_{a}^{b}\!f^{\prime}x\,dx=f\left(b\right)-f\left(a\right)

integrals from a higher to lower number

the sum of two adjacent areas

even function

symmetrical about the y-axis

integral properties of even function

upper and lower limit are equal and opposite. definite integral includes the same area on the left of the axis as it does on the right

odd function

symmetrical about the origin

integral properties of odd function

upper and lower limit are equal and opposite. definite integral includes the same area on one side above the axis as it does on the other side belwo the axis

integrand

expression between integral and "dx"

average value

derivative of ln(x)

1/x

integral of ln(x)

x ln x - x + c

when differentiating b^x

multiply by ln b

when integrating b^x

divide by ln b

logarithmic differentiation

take the logarithm of both sides
use the properties of logarithms to change the exponent to multiplication
differentiate both sides with respect to x. don’t forget that the chain rule applies

∫sin(x) dx

-cos x + c

∫cos(x) dx

sin x + c

∫tan(x) dx

-ln |cos x| + c = ln |sec x| + c

∫cot(x) dx

- ln |csc x| + c = ln |sin x| + c

∫sec(x) dx

ln |sec x + tan x| + c

∫csc(x) dx

ln |csc x - cot x| + c = - ln |csc x + cot x| + c

l'hospital's rule

volume of a paraboloid

dV = (circumference)(dA)

plane slicing a paraboloid

r² is some function of x. h is “dx”

cylindrical shells

dV=2\pi x\left(h-y\right)\differentialD x or vice versa