AP CALCULUSFORMULA LIST

Definition of e:

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Absolute value:

______________________________________________________________________________Definition of the derivative:

(original form: gives a function)

(alternative form: gives slope at a particular point)

____________________________________________________________________________Definition of continuity: f is continuous at c iff

1) f (c) is defined; 2)

3) _____________________________________________________________________________

Average rate of change of (slope)

Instantaneous rate of change implies use of derivative._____________________________________________________________________________Rolle's Theorem: If f is continuous on [a, b] and differentiable on (a, b) and if f (a) = f (b), then there is at least one number c on (a, b) such that _____________________________________________________________________________Mean Value Theorem: If f is continuous on [a, b] and differentiable on (a, b), then there

exists a number c on (a, b) such that

______________________________________________________________________________Intermediate Value Theorem: If f is continuous on [a, b] and k is any number between f (a) and f (b), then there is at least one number c between a and b

such that f (c) = k._____________________________________________________________________________

_____________________________________________________________________________Linear Approximation: to approximate y-value of a function using a tangent line with

point of tangency at

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Definition of a definite integral:

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Domains of “arc” trig functions:

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Limits of Trig Functions:

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Derivative Rules:

(power rule)

(product rule) (quotient rule)

(chain rule)

(derivative of an inverse)

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Integral Rules:

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Definition of a Critical Number:Let f be defined at c. If is undefined at c, then c is a critical number of f.______________________________________________________________________________

First Derivative Test: Let c be a critical number of a function f that is continuous on an open interval I containing c. If f is differentiable on the interval, except possibly at c, then can be classified as follows.

1) If changes from negative to positive at c, then is a relative minimum of f.

2) If changes from positive to negative at c, then is a relative maximum of f.______________________________________________________________________________

Second Derivative Test: Let f be a function such that the second derivative of f exists on an open interval containing c.1) If and , then is a relative minimum.

2) If and , then is a relative maximum.______________________________________________________________________________

Definition of Concavity:Let f be differentiable on an open interval I. The graph of f is concave upward on I if is increasing on the interval and concave downward on I if is decreasing on the interval.______________________________________________________________________________

Test for Concavity: Let f be a function whose second derivative exists on an open interval I.1) If for all x in I, then the graph of f is concave upward in I.

2) If for all x in I, then the graph of f is concave downward in I.______________________________________________________________________________

Definition of an Inflection Point:A function f has an inflection point at

1) if does not exist and2) if changes sign from positive to negative or negative to positive at OR if changes from increasing to decreasing or decreasing to increasing at x = c.______________________________________________________________________________

First Fundamental Theorem of Calculus:

Second Fundamental Theorem of Calculus:

Chain Rule Version:

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Average value of f (x) on [a, b]:

Volume around a horizontal axis by discs:

Volume around a horizontal axis by washers:

Volume by cross sections taken perpendicular to the x-axis:

______________________________________________________________________________If an object moves along a straight line with position function , then its

Velocity is

Speed =

Acceleration is

Displacement (change in position) from is Displacement =

Total Distance traveled from is Total Distance =

or Total Distance = , where changes sign at

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CALCULUS BC ONLY

Summation Using LimitsDefinition of the Area of a Region: Let f be continuous and non-negative on . The area of the region bounded by f, the x-axis, , and is:

Area =

where n = # of rectangles / intervalsi = particular rectangles / intervals

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Trig Integration:Wallis’ Formulas (definite integrals involving sine and cosine with and ):

1) If is odd ( ), then

2) If is even ( ), then

Integrals involving sine-cosine products with different angles:

Integrals involving sine-cosine products with different powers:-if the power of one function is odd, save one factor and convert the remaining factors to the other function using the Pythagorean identity -if the powers of both/all functions are even, make repeated use of the half-angle formulas

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Differential equation for logistic growth:

Integration by parts:

Arc length for functions:

Arc length for parametrics:

Arc length for polar: s = r2 +drdθ

⎛⎝⎜

⎞⎠⎟

2

dθa

b

∫_____________________________________________________________________________Parametric:If an object moves along a curve, its

Position vector =

Velocity vector =

Acceleration vector =

Speed (or magnitude of velocity) vector =

Derivative: dydx

=dy/ dtdx / dt

Second derivative: d2ydx2 =

ddt

dydx

⎡⎣⎢

⎤⎦⎥

dx / dtPolar:

tanθ =yx

and r2 =x2 + y2

Derivative (slope of a polar curve): dydx

=dy/ dθdx / dθ

=rcosθ +r'sinθ−rsinθ +r'cosθ

Tangents at the pole: where does r =0 and r ' ≠0

Area inside a polar curve:

Definition of a Taylor polynomial:If f has n derivatives, centered at c, then the polynomial

is called the nth Taylor polynomial for f at c.

Lagrange Error Bound for a Taylor Polynomial (or Taylor’s Theorem Remainder):If f is differentiable through order in an interval I containing c, then for each x in I, there exists z between x and c such that

The remainder represents the difference between the function and the polynomial. That is, .

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Alternating Series Remainder:If a series has terms that alternate, decrease in absolute value, and have a limit of 0 (so that the series converges by the Alternating Series Test), then the absolute value of the remainder involved in approximating the sum S by is less than the first neglected term. That is,

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Maclaurin series that you must know:

Euler’s Method: a numerical approach to approximating the particular solution of a differential equation that passes through a point, and has a known slope.

where y-value previous y-value

change in x-value from previous pointslope at previous point