AP Calculus AB Formula List Review

Flashcards to help memorize key definitions, theorems, and formulas for the AP Calculus AB exam.

What is the definition of e?

e = lim (1 + 1/x)^x as x approaches infinity

What is the definition of the derivative f'(x)?

f'(x) = lim (f(x+h) - f(x))/h as h approaches 0

What is the alternative form of the derivative f'(c)?

f'(c) = lim (f(x) - f(c))/(x - c) as x approaches c

What are the three conditions for a function f to be continuous at x=c?

1) f(c) is defined; 2) lim f(x) exists as x approaches c; 3) lim f(x) = f(c) as x approaches c

What is the formula for the average rate of change of f(x) on the interval [a, b]?

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

What is the definition of absolute value |x|?

x if x ≥ 0, and -x if x < 0

What does the Intermediate Value Theorem state?

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.

What does Rolle's Theorem state?

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 f'(c) = 0.

What does the Mean Value Theorem state?

If f is continuous on [a, b] and differentiable on (a, b), then there exists a number c on (a, b) such that f'(c) = (f(b) - f(a))/(b - a).

What is the derivative of sin(u)?

cos(u) * du/dx

What is the derivative of cos(u)?

-sin(u) * du/dx

What is the derivative of tan(u)?

sec^2(u) * du/dx

What is the derivative of cot(u)?

-csc^2(u) * du/dx

What is the derivative of sec(u)?

sec(u)tan(u) * du/dx

What is the derivative of csc(u)?

-csc(u)cot(u) * du/dx

What is the derivative of log_a(u)?

(1/(u*ln(a))) * du/dx

What is the derivative of a^u?

a^u * ln(a) * du/dx

What is the derivative of ln(u)?

(1/u) * du/dx

What is the derivative of e^u?

e^u * du/dx

What is the definition of a critical number c?

Let f be defined at c. If f'(c)=0 or if f' is undefined at c, then c is a critical number of f.

What does the First Derivative Test state regarding relative extrema?

If f'(x) changes from negative to positive at x = c, then (c, f(c)) is a relative minimum. If f'(x) changes from positive to negative at x = c, then (c, f(c)) is a relative maximum.

What does the Second Derivative Test state regarding relative extrema?

If f'(c) = 0 and f''(c) > 0, then (c, f(c)) is a relative minimum. If f'(c) = 0 and f''(c) < 0, then (c, f(c)) is a relative maximum.

What is the definition of concavity?

The graph of f is concave upward on I if f' is increasing on the interval and concave downward on I if f' is decreasing on the interval.

What is the Test for Concavity?

If f''(x) > 0 for all x in the interval I, then the graph of f is concave upward in I. If f''(x) < 0 for all x in the interval I, then the graph of f is concave downward in I.

What defines an inflection point at (c, f(c))?

1) if f''(c)=0 or f''(c) does not exist and 2) if f'' changes sign from positive to negative or negative to positive at x=c OR if f'(x) changes from increasing to decreasing or decreasing to increasing at x = c.

What is the definition of a definite integral of f(x) from a to b?

∫f(x)dx from a to b = lim ∑ f(x_k) * (Δx) as Δx approaches 0

What is the integral of x^n dx?

(x^(n+1))/(n+1) + C, n ≠ -1

What is the integral of cos(u) du?

sin(u) + C

What is the integral of sin(u) du?

-cos(u) + C

What is the integral of sec^2(u) du?

tan(u) + C

What is the integral of csc^2(u) du?

-cot(u) + C

What is the integral of sec(u)tan(u) du?

sec(u) + C

What is the integral of csc(u)cot(u) du?

-csc(u) + C

What is the integral of du/u?

ln|u| + C

What is the integral of tan(u) du?

-ln|cos(u)| + C

What is the integral of cot(u) du?

ln|sin(u)| + C

What is the integral of sec(u) du?

ln|sec(u) + tan(u)| + C

What is the integral of csc(u) du?

-ln|csc(u) + cot(u)| + C

What is the integral of e^u du?

e^u + C

What is the integral of a^u du?

(a^u)/ln(a) + C

What does the First Fundamental Theorem of Calculus state?

d/dx ∫f'(x) dx from a to b = f(b) - f(a)

What does the Second Fundamental Theorem of Calculus state?

d/dx ∫f(t) dt from a to x = f(x)

What is the Chain Rule Version of the Second Fundamental Theorem of Calculus?

d/dx ∫f(t) dt from a to g(x) = f(g(x)) * g'(x)

What is the average value of f(x) on [a, b]?

f_AVE = (1/(b-a)) ∫f(x)dx from a to b

What is the derivative of arcsin(u)?

(1/√(1-u^2)) * du/dx

What is the derivative of arccos(u)?

(-1/√(1-u^2)) * du/dx

What is the derivative of arctan(u)?

(1/(1+u^2)) * du/dx

What is the derivative of arccot(u)?

(-1/(1+u^2)) * du/dx

What is the derivative of arcsec(u)?

(1/(|u|√(u^2-1))) * du/dx

What is the derivative of arccsc(u)?

(-1/(|u|√(u^2-1))) * du/dx

What is the integral of du/√(a^2-u^2)?

arcsin(u/a) + C

What is the integral of du/(u√(u^2-a^2))?

(1/a) * arcsec(|u|/a) + C

What is the integral of du/(a^2+u^2)?

(1/a) * arctan(u/a) + C

How is the volume calculated by cross sections taken perpendicular to the x-axis?

V = ∫A(x)dx, where A(x) = area of each cross section

How is volume around a horizontal axis calculated by discs?

V = π∫[r(x)]^2 dx

How is volume around a horizontal axis calculated by washers?

V = π∫([R(x)]^2 - [r(x)]^2)dx

Given a position function s(t), what is the velocity v(t)?

v(t) = s'(t)

Given a position function s(t), what is the acceleration a(t)?

a(t) = v'(t) = s''(t)

How is displacement calculated from x=a to x=b?

Displacement = ∫v(t)dt from a to b

How is total distance traveled calculated from x=a to x=b?

Total Distance = ∫|v(t)|dt from a to b

When is the speed of an object increasing?

The speed of the object is increasing when its velocity and acceleration have the same sign.

When is the speed of an object decreasing?

The speed of the object is decreasing when its velocity and acceleration have opposite signs.