AP calculus AB: ALL formulas and theorems to know

Sandwich Theorem

a function f that is sandwiched between two other functions, g and h. If g and h have the same limit as x approaches c, then f (the middle function) has that limit too

average rate of change

dividing distance by the time interval.

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

instantaneous rate of change

The rate of change, or slope of a tangent line, or value of the derivative, at a point of a graph.

h= a very small change in time, but it cant be 0 bc itll be undefined.

Intermediate Value Theorem

If a function is continuous between a and b, then it takes on every value between f(a) and f(b)

if f is a continuous function on the closed interval [a, b] and d is a number between f (a) and f (b), then the Intermediate Value Theorem guarantees that there is at least one number c between a and b, where f (c) = d.

Derivative at a point

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

Formal definition of derivative

IRC formula.

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

Product Rule

u'v + uv'

Quotient Rule

(u'v-uv')/v²

Chain Rule

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

derivative of inverse

average velocity

∆s/∆t

Instantaneous velocity

Derivative of position s(t) at a point.

ds/dt

speed

Absolute value of velocity.

|v(t)|

Acceleration

derivative of velocity. (units= m/s²)

first derivative test for local extrema

For a continuous function f:

1) If f' changes sign, f has a local maximum or minimum value at c

2) If f' does not change sign at a critical point c, then f has no local extreme values at c

Second Derivative Test for local extrema

1. if f'(c)=0 and f''(c)>0, then f has local minimum at c. Concave up.

2. if f'(c)=0 and f''(c)<0, then f has a local maximum at c. Concave down.

3. FAILS if f''(c) = 0 or if f''(c) DNE

HOW TO USE SECOND DERIVATIVE TEST

1. use first derivative to find critical points

2. sub critical points into second derivative. if f''(x)<0, it is concave down, and a MAX.

Intermediate Value Theorem for Derivatives

If f is differentiable on [a,b], then f'(x) takes on all values between f'(a) and f'(b).

mean value theorem

CHECK CONDITIONS FIRST

if f(x) is continuous on the closed interval and differentiable on the open interval,,,,,, the slope of tangent line equals the slope of the secant line (secant line goes through 2 points), at least once in the interval (a, b)

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

IRC=ARC : Mean Value Theorem

mean value theorem for derivatives

IRC = ARC

continuous on closed, differentiable on the open. ALWAYS CHECK CONDITIONS. ONLY WORKS ON CLOSED INTERVALS

Extreme Value Theorem

If f is continuous over a CLOSED interval, then f has at LEAST one maximum and minimum value over that interval

Rolle's Theorem

If f(x) is continuous on the closed interval [a,b], AND differentiable on the open (a,b), AND f(a)=f(b), then there is at least one number x=c in (a,b) where f'(c)=0

rectangle surface area formula

SA = 2lw + 2wh + 2lh

right cylinder surface area formula

SA= 2πrh + 2πr²

right cylinder with no top surface area formula

SA= 2πrh + πr²

open top and square base surface area formula

SA= x² +4xh

sphere surface area formula

SA= 4πr²

right cylinder volume formula

V= πr²h

right cylinder with no top volume formula

V= πr²h (the same)

sphere volume formula

V = 4/3πr³

cone volume formula

V = 1/3πr²h

distance formula

√(x₂-x₁)²+(y₂-y₁)²

left riemann sum

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

◦ UNDERESTIMATE OF AREA UNDER CURVE IF CURVE IS INCREASING

◦ OVERESTIMATE OF AREA UNDER CURVE IF CURVE IS DECREASING

◦ Interval *(adding up all y values on curve that the left side of rectangle touches. add all y values except last one)

right riemann sum

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

◦ UNDERESTIMATE OF AREA UNDER CURVE IF CURVE IS DECREASING

◦ OVERESTIMATE OF AREA UNDER CURVE IF CURVE IS INCREASING

◦ Interval *(adding up all y values on curve that the right side of rectangle touches. add all y values except the first)

Midpoint Riemann Sum

midpoint of rectangle touches the curve.

length of interval*(all y values on curve that the midpoint of the rectangle touches)

area of each trapezoid

interval/2 * (b₁+b₂)

trapezoidal rule

use trapezoids to evaluate integrals (estimate area).

average of LRAM and RRAM.

(interval/2) * (y0 +2y1 + 2y2 + 2y3 ... + yn)

overestimates the integral where the graph is concave up, and underestimates the integral where the graph is concave down.

definite integral

has upper and lower bounds a & b. find antiderivative, F(b) - F(a)

area under a curve

∫ f(x) dx

integrate over interval a to b

average value of f(x)

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

Mean Value Theorem for Definite Integrals

average value of integral = actual value. MUST check conditions (continuous on closed interval)

If g(x) = ∫ f(t) dt on the 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).

Fundamental Theorem of Calculus part 2

• variable in derivative (usually d/dx) matches the variable (upper bound) in the integral symbol

• lower limit is a constant, upper limit is a variable

• we have an integral and a derivative

Integration by parts formula

∫u dv= uv-∫vdu

u= use LIPET

dv= rest of integrand given

exponential growth equation

k= constant of proportionality.

used if y changes at a rate proportional to the amount present

exponential differential equation

dP/dt= kP

logistic growth equation

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

M= carrying capacity

k= growth constant

A= a constant

logistic differential equation

2 logistic differential equations:

dP/dt = kP(M - P)

AND

dP/dt = kP(1-P/M)

M = carrying capacity

k= proportionality constant

P= population

area between two curves (dx)

∫ f(x) - g(x) over interval a to b

f(x) is top function and

g(x) is bottom function

(dx: vertical strips)

area between two curves (dy)

dy= horizontal strips bc width of horizontal strip=dy

∫ f(x) - g(x) over interval a to b

f(x) is right function and

g(x) is left function

1. solve the equations to be x=, so you can have it in terms of dy

2. find y values of intersections (these will be bounds for integral)

3. subtract RIGHT - LEFT functions

volume of solid of revolution - disks

π ∫ r² dx over interval a to b

r = distance from curve to axis of revolution

volume of solid of revolution - washers

π ∫ R² - r² dx over interval a to b

R = farthest distance from outside curve to axis of revolution,

r = closest distance from inside curve to axis of revolution

Shells Method

V= 2π ∫R(x) * h(x) dx

length of curve for rectangular functions

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

L'Hopitals rule

used to find limits when substitution gives you 0/0 or ∞/∞

find derivative of numerator and denominator separately, then evaluate limit