AP Calc BC: Formulas (Final)

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

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)

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

increasing

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

decreasing

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

relative minimum

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

relative maximum

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

concave up

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

concave down

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

point of inflection

When is a function not differentiable

corner, cusp, vertical tangent, discontinuity

Product Rule

uv' + vu'

Quotient Rule

(uv'-vu')/v²

Chain Rule

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

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

product rule

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

quotient rule

y = cos²(3x)

chain rule

Particle is moving to the right/up

velocity is positive

Particle is moving to the left/down

velocity is negative

absolute value of velocity

speed

y = sin(x), y' =

y' = cos(x)

y = cos(x), y' =

y' = -sin(x)

y = tan(x), y' =

y' = sec²(x)

y = csc(x), y' =

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

y = sec(x), y' =

y' = sec(x)tan(x)

y = cot(x), y' =

y' = -csc²(x)

y = sin⁻¹(x), y' =

y' = 1/√(1 - x²)

y = cos⁻¹(x), y' =

y' = -1/√(1 - x²)

y = tan⁻¹(x), y' =

y' = 1/(1 + x²)

y = cot⁻¹(x), y' =

y' = -1/(1 + x²)

y = e^x, y' =

y' = e^x

y = a^x, y' =

y' = a^x ln(a)

y = ln(x), y' =

y' = 1/x

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

y' = 1/(x lna)

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

critical points and endpoints

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)

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

f(x) has a relative minimum

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

f(x) has a relative maximum

Linearization

use tangent line to approximate values of the function

rate

derivative

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)

trapezoidal rule

use trapezoids to evaluate integrals (estimate area)

[(h1 - h2)/2]*base

area of trapezoid

definite integral

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

indefinite integral

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

area under a curve

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

area above x-axis is

positive

area below x-axis is

negative

average value of f(x)

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

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

g'(x) = f(x)

Fundamental Theorem of Calculus

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

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

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

To draw a slope field,

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

slope of horizontal line

zero

slope of vertical line

undefined

methods of integration

substitution, parts, partial fractions

use substitution to integrate when

a function and it's derivative are in the integrand

use integration by parts when

two different types of functions are multiplied

∫ u dv =

uv - ∫ v du

use partial fractions to integrate when

integrand is a rational function with a factorable denominator

dP/dt = kP(M - P)

logistic differential equation, M = carrying capacity

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

logistic growth equation

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

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

given v(t) find displacement

∫ v(t) over interval a to b

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

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

volume of solid of revolution - no washer

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

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

length of curve

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

L'Hopitals rule

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

indeterminate forms

0/0, ∞/∞, ∞*0, ∞ - ∞, 1^∞, 0⁰, ∞⁰

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

nth term test

if terms grow without bound, series diverges

alternating series test

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

converges absolutely

alternating series converges and general term converges with another test

converges conditionally

alternating series converges and general term diverges with another test

ratio test

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

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

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

geometric series test

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

p-series test

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

derivative of parametrically defined curve

x(t) and y(t)

dy/dx = dy/dt / dx/dt

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

length of parametric curve

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

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

√(dx/dt)² + (dy/dt)² not an integral!

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

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

area inside polar curve

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

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.