AP Calculus BC Terms

Quotient Rule

(uv'-vu')/v²

indeterminate forms

0/0, ∞/∞, ∞*0, ∞ - ∞, 1^∞, 0⁰, ∞⁰

area inside one polar curve and outside another polar curve

1/2 ∫ R² - r² over interval from a to b, find a & b by setting equations equal, solve for theta.

area inside polar curve

1/2 ∫ r² over interval from a to b, find a & b by setting r = 0, solve for theta

average value of f(x)

= 1/(b-a) ∫ f(x) dx on interval a to b

Intermediate Value Theorem

If f(1)=-4 and f(6)=9, then there must be a x-value between 1 and 6 where f crosses the x-axis.

Average Rate of Change

Slope of secant line between two points, use to estimate instantanous rate of change at a point.

Instantenous Rate of Change

Slope of tangent line at a point, value of derivative at a point

use substitution to integrate when

a function and it's derivative are in the integrand

converges absolutely

alternating series converges and general term converges with another test

converges conditionally

alternating series converges and general term diverges with another test

[(h1 - h2)/2]*base

area of trapezoid

When f '(x) is decreasing, f(x) is

concave down

When f '(x) is increasing, f(x) is

concave up

When is a function not differentiable

corner, cusp, vertical tangent, discontinuity

To find absolute maximum on closed interval [a, b], you must consider...

critical points and endpoints

When f '(x) is negative, f(x) is

decreasing

rate

derivative

derivative of parametrically defined curve x(t) and y(t)

dy/dx = dy/dt / dx/dt

Chain Rule

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

If f '(x) = 0 and f"(x) < 0,

f(x) has a relative maximum

If f '(x) = 0 and f"(x) > 0,

f(x) has a relative minimum

second derivative of parametrically defined curve

find first derivative, dy/dx = dy/dt / dx/dt, then find derivative of first derivative, then divide by dx/dt

If g(x) = ∫ f(t) dt on interval 2 to x, then g'(x) =

g'(x) = f(x)

p-series test

general term = 1/n^p, converges if p > 1

geometric series test

general term = a₁r^n, converges if -1 < r < 1

definite integral

has limits a & b, find antiderivative, F(b) - F(a)

mean value theorem

if f(x) is continuous and differentiable, slope of tangent line equals slope of secant line at least once in the interval (a, b) f '(c) = [f(b) - f(a)]/(b - a)

integral test

if integral converges, series converges

limit comparison test

if lim as n approaches ∞ of ratio of comparison series/general term is positive and finite, then series behaves like comparison series

nth term test

if terms grow without bound, series diverges

When f '(x) is positive, f(x) is

increasing

use partial fractions to integrate when

integrand is a rational function with a factorable denominator

alternating series test

lim as n approaches zero of general term = 0 and terms decrease, series converges

ratio test

lim as n approaches ∞ of ratio of (n+1) term/nth term > 1, series converges

Formal definition of derivative

limit as h approaches 0 of [f(a+h)-f(a)]/h

Alternate definition of derivative

limit as x approaches a of [f(x)-f(a)]/(x-a)

dP/dt = kP(M - P)

logistic differential equation, M = carrying capacity

P = M / (1 + Ae^(-Mkt))

logistic growth equation

area below x-axis is

negative

indefinite integral

no limits, find antiderivative + C, use inital value to find C

To draw a slope field,

plug (x,y) coordinates into differential equation, draw short segments representing slope at each point

When f '(x) changes from increasing to decreasing or decreasing to increasing, f(x) has a

point of inflection

6th degree Taylor Polynomial

polynomial with finite number of terms, largest exponent is 6, find all derivatives up to the 6th derivative

Taylor series

polynomial with infinite number of terms, includes general term

area above x-axis is

positive

y = x cos(x), state rule used to find derivative

product rule

y = ln(x)/x², state rule used to find derivative

quotient rule

When f '(x) changes fro positive to negative, f(x) has a

relative maximum

When f '(x) changes from negative to positive, f(x) has a

relative minimum

To find particular solution to differential equation, dy/dx = x/y

separate variables, integrate + C, use initial condition to find C, solve for y

absolute value of velocity

speed

methods of integration

substitution, parts, partial fractions

given v(t) and initial position t = a, find final position when t = b

s₁+ Δs = s Δs = ∫ v(t) over interval a to b

use integration by parts when

two different types of functions are multiplied

slope of vertical line

undefined

find interval of convergence

use ratio test, set > 1 and solve absolute value equations, check endpoints

find radius of convergence

use ratio test, set > 1 and solve absolute value equations, radius = center - endpoint

left riemann sum

use rectangles with left-endpoints to evaluate integral (estimate area)

right riemann sum

use rectangles with right-endpoints to evaluate integrals (estimate area)

Linearization

use tangent line to approximate values of the function

L'Hopitals rule

use to find indeterminate limits, find derivative of numerator and denominator separately then evaluate limit

trapezoidal rule

use trapezoids to evaluate integrals (estimate area)

∫ u dv =

uv - ∫ v du

Product Rule

uv' + vu'

Particle is moving to the left/down

velocity is negative

Particle is moving to the right/up

velocity is positive

y = cot⁻¹(x), y' =

y' = -1/(1 + x²)

y = cos⁻¹(x), y' =

y' = -1/√(1 - x²)

y = csc(x), y' =

y' = -csc(x)cot(x)

y = cot(x), y' =

y' = -csc²(x)

y = cos(x), y' =

y' = -sin(x)

y = tan⁻¹(x), y' =

y' = 1/(1 + x²)

y = log (base a) x, y' =

y' = 1/(x lna)

y = ln(x), y' =

y' = 1/x

y = sin⁻¹(x), y' =

y' = 1/√(1 - x²)

y = a^x, y' =

y' = a^x ln(a)

y = sin(x), y' =

y' = cos(x)

y = e^x, y' =

y' = e^x

y = sec(x), y' =

y' = sec(x)tan(x)

y = tan(x), y' =

y' = sec²(x)

given rate equation, R(t) and inital condition when t = a, R(t) = y₁ find final value when t = b

y₁ + Δy = y Δy = ∫ R(t) over interval a to b

slope of horizontal line

zero

volume of solid of revolution - washer

π ∫ R² - r² dx over interval a to b, where R = distance from outside curve to axis of revolution, r = distance from inside curve to axis of revolution

volume of solid of revolution - no washer

π ∫ r² dx over interval a to b, where r = distance from curve to axis of revolution

given velocity vectors dx/dt and dy/dt, find speed

√(dx/dt)² + (dy/dt)² not an integral!

volume of solid with base in the plane and given cross-section

∫ A(x) dx over interval a to b, where A(x) is the area of the given cross-section in terms of x

given v(t) find total distance travelled

∫ abs[v(t)] over interval a to b

area between two curves

∫ f(x) - g(x) over interval a to b, where f(x) is top function and g(x) is bottom function

area under a curve

∫ f(x) dx integrate over interval a to b

Fundamental Theorem of Calculus

∫ f(x) dx on interval a to b = F(b) - F(a)

given v(t) find displacement

∫ v(t) over interval a to b

given velocity vectors dx/dt and dy/dt, find total distance travelled

∫ √ (dx/dt)² + (dy/dt)² over interval from a to b

length of parametric curve

∫ √ (dx/dt)² + (dy/dt)² over interval from a to b

length of curve

∫ √(1 + (dy/dx)²) dx over interval a to b