area below x-axis is Formal definition of derivative Particle is moving to the right/up area above x-axis is

Formal definition of derivative

limit as x approaches a of [f(x)-f(a)]/(x-a) y = sin(x), y' = 6th degree Taylor Polynomial length of parametric curve Alternate definition of derivative

Alternate definition of derivative

increasing When f '(x) is positive, f(x) is given rate equation, R(t) and inital condition when t = a, R(t) = y₁ find final value when t = b Product rule Derivatives Average Rate of Change

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

decreasing When f '(x) is negative, f(x) is length of curve Linearization ratio test

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

relative minimum When f '(x) changes from negative to positive, f(x) has a y = a^x, y' = area between two curves volume of solid of revolution - no washer

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

relative maximum y = tan⁻¹(x), y' = When f '(x) changes from positive to negative, f(x) has a mean value theorem When f '(x) is increasing, f(x) is

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

concave up dP/dt = kP(M - P) methods of integration When f '(x) changes from increasing to decreasing or decreasing to increasing, f(x) has a When f '(x) is increasing, f(x) is

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

AP Calculus AB

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Calculus - Geometry

user_hodr Created by 6 mon ago

Cards in this deck(100)

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.

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

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

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

increasing

decreasing

relative minimum

relative maximum

concave up

concave down

point of inflection

corner, cusp, vertical tangent, discontinuity

uv' + vu'

(uv'-vu')/v²

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

product rule

quotient rule

chain rule

velocity is positive

velocity is negative

speed

y' = cos(x)

y' = -sin(x)

y' = sec²(x)

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

y' = sec(x)tan(x)

y' = -csc²(x)

y' = 1/√(1 - x²)

y' = -1/√(1 - x²)

y' = 1/(1 + x²)

y' = -1/(1 + x²)

y' = e^x

y' = a^x ln(a)

y' = 1/x

y' = 1/(x lna)

critical points and endpoints

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)

f(x) has a relative minimum

f(x) has a relative maximum

use tangent line to approximate values of the function

derivative

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

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

use trapezoids to evaluate integrals (estimate area)

area of trapezoid

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

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

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

positive

negative

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

g'(x) = f(x)

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

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

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

zero

undefined

substitution, parts, partial fractions

a function and it's derivative are in the integrand

two different types of functions are multiplied

uv - ∫ v du

integrand is a rational function with a factorable denominator

logistic differential equation, M = carrying capacity

logistic growth equation

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

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

∫ v(t) over interval a to b

∫ abs[v(t)] over interval a to b

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

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

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

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

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

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

0/0, ∞/∞, ∞*0, ∞ - ∞, 1^∞, 0⁰, ∞⁰

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

polynomial with infinite number of terms, includes general term

if terms grow without bound, series diverges

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

alternating series converges and general term converges with another test

alternating series converges and general term diverges with another test

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

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

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

if integral converges, series converges

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

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

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

dy/dx = dy/dt / dx/dt

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

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

√(dx/dt)² + (dy/dt)² not an integral!

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

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

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

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AP Calculus AB

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