AP Calc BC General

average rate of change

instantaneous rate of change

limit definition of a derivative

Mean Value Theorem

if f(x) is continuous on [a, b] and differentiable on (a, b), then x = c exists in that interval so that IROC = AROC at that x-value

average value of a function

Intermediate Value Theorem

if a f(x) is continuous on [a, b], then f(a) and f(b) passes through all y-values between them

Extreme Value Theorem

a function such as f(x) that is continuous on [a, b] has both an absolute min and an absolute max on that interval

found by using critical points + endpoints of f’(x) and then plugging into f(x)

definition of continuity

a function such as f(x)) is continuous at x = c if and only if:
the limit as x approaches c from the left = the limit as x approaches c from the right = f(c)

Squeeze Theorem

arc length for a function

arc length for a parametric equations / vectors

speed formula for parametric

total distance formula for parametric

polar area formula

second derivative for parametric

polar conversions

nth Term Test

Geometric Series Test

p-Series Test

Alternating Series Test

if writing justification, must write:

alternating terms

absolute value of terms decrease to 0

Integral Test

if the integral converges, the series converges

if the integral diverges, the series diverges

Ratio Test

Direct Comparison Test

if big converge, small converge

if small diverge, big diverge

Limit Comparison Test

product rule

Fo[DeSe] + Se[DeFo]

quotient rule

Lo[DeHi] - Hi[DeLo] / (Lo)²

chain rule

d/dx f(u) = f’(u) [u]

d/dx f(g(x)) = f’(g(x)) [g’(x)]

derivative of sin(u)

cos(u) [u’]

derivative of cos(u)

-sin(u) [u’]

derivative of a^x

a^x * ln(a)

derivative of tan(u)

sec²(u) [u’]

derivative of cot(u)

-csc²(u) [u’]

derivative of sec(u)

sec(u)tan(u) [u’]

derivative of csc(u)

-csc(u)cot(u) [u’]

derivative of e^u

e^u [u’]

derivative of arcsin(u)

derivative of arccos(u)

derivative of arctan(u)

integral of:

integral of:

integral of:

integral of:

integral of:

integral of:

integration by parts

ultraviolet voodoo

u v - ∫ v du

integral of:

integral of:

integral of:

integral of:

integral of:

integral of:

integral of:

integral of:

integral of:

Second Fundamental Theorem of Calculus

Start-Plus Accumulation / Fundamental Theorem of Calculus

where final value is f(b) and initial value is f(a)

Disc Volume Formula

Washer Volume Formula

Logistic Differential Equation

When is logistic differential growth the fastest?

P = ½ * M

Area of a Trapezoid

e^x as an elementary series expansion

sin(x) as an elementary series expansion

cos(x) as an elementary series expansion

ln(x) as an elementary series expansion

Alternating Series Error Bound

error <= | the next term or first unused term |

Euler’s Method set-up

Taylor Series set-up

MacLaurin Series set-up

LaGrange Error Bound

Candidate Test

use the critical numbers (where f’(x) is 0 and undefined) and the endpoints to check where relative mins / maxs exist

velocity function / vector

• the derivative of the position function / vector

• the integral of the acceleration function / vector

• the rate of change of the position

• positive for upward / rightward motion

• negative for downward / leftward motion

position function / vector

• the integral of the velocity function / vector

• gives the location of an object at a given time (t)

acceleration function / vector

• the derivative of the velocity function / vector

• the rate of change of velocity

total distance traveled for functions

• the total distance traveled by the object in the time interval

• found using a table of s(t) with t-values consisting of the endpoints and the critical numbers of v(t) = 0

displacement

• final position - initial position

• s(b) - s(a)

speed of a function

absolute value of velocity

properties of logarithms

• ln(a * b) = ln(a) + ln(b)

• ln(a / b) = ln(a) - ln(b)

• ln(aⁿ) = n * ln(a)

change of base definition (Exponential to Logarithmic)

x = e^y ←→ y = ln(x)

Procedure for Logarithmic Differentiation

1. take the natural log (ln) of both sides of the equation

2. simplify the right hand side of the equation

3. differentiate both sides

4. solve for y’

5. substitute for y if necessary

Procedure for Implicit Differentiation

1. differentiate both sides with respect to x (use dy/dx for y’)

2. collect all (dy/dx) onto one side

3. factor out (dy/dx)

4. simplify

derivatives of inverse functions

• if (a, b) is a point on f(x), then (b, a) is a point on f^-1(x)

• (f^-1)’(b) = 1 / f’(a)

• or the inverse derivative at the y-value is the reciprocal of the original derivative at the x-value

• if f(x) and g(x) are inverse functions, then: g’(x) = 1 / f’(g(x))

L’Hopital’s Rule

must clarify that it’s being used when f(x) and g(x) assume 0 / 0 form at the limit

point of inflection

• a point where the graph has a tangent line and where the graph changes concavity at that point

• found as the “critical numbers” or “potential points of inflection” of the second derivative

reading a f’(x) or f’’(x) graph

• f(x) is increasing where f’(x) is positive

• f(x) is decreasing where f’(x) is negative

• f(x) is concave up where f’’(x) is positive

• f(x) is concave down where f’’(x) is negative

• f(x) has a relative min where the f’(x) graph goes from negative to positive

• f(x) has a relative max where the f’(x) graph goes from positive to negative

• f(x) has a point of inflection where the f’(x) graph has relative extreme

Second Derivative Test for Relative Extrema

• use a f’’(x) number line using f’(x)’s critical numbers

• done to find concavity (if negative, then relative max; if positive, then relative min)

• only do if specifically asked to

Curve Sketching Recipe

• write down domain

• reduce f(x)

• find vertical asymptotes (domain restrictions of f_red(x)) and holes (domain restrictions of f(x) and NOT f_red(x))

• give x and y-intercepts

• find the end behavior / horizontal asymptotes (limit behavior as x approaches infinity)

• increasing / decreasing intervals and relative extrema points (showing a f’(x) number line)

• find concavity and points of inflection (showing a f’’(x) number line)