AP Calculus BC Flashcards

Vocabulary flashcards for derivative and integral rules, limit rules, and theorems.

Product Rule

f(x)g(x) = f(x)g'(x) + g(x)f'(x)

Quotient Rule

d/dx [f(x)/g(x)] = [g(x)f'(x) - f(x)g'(x)] / (g(x))^2

Chain Rule

d/dx [f(g(x))] = f'(g(x)) * g'(x)

Derivative of sin(x)

cos(x)

Derivative of cos(x)

-sin(x)

Derivative of tan(x)

sec²(x)

Derivative of cot(x)

-csc²(x)

Derivative of sec(x)

sec(x)tan(x)

Derivative of csc(x)

-csc(x)cot(x)

Derivative of ln(x)

1/x

Derivative of e^x

e^x

Derivative of a^x

a^x * ln(a)

Integral of x^n

(x^(n+1))/(n+1) + C

Integral of e^x

e^x + C

Integral of cos(x)

sin(x) + C

Integral of sin(x)

-cos(x) + C

Integral of sec²(x)

tan(x) + C

Integral of csc²(x)

-cot(x) + C

Limit when solving for 0/0

Factor, use the conjugate method, clean up messy fractions, or use L'Hospital's Rule

L'Hospital's Rule

If lim (f(x)/g(x)) = 0/0 or ±∞/±∞, then lim (f(x)/g(x)) = lim (f'(x)/g'(x))

Definition of Continuity

f is continuous at x=c if 1) f(c) is defined; 2) lim x->c f(x) exists; 3) lim x->c f(x) = f(c).

Definition of the Derivative

f'(x) = lim h->0 [f(x+h) - f(x)] / h

Average Rate of Change

[f(b) - f(a)] / (b - a)

Intermediate Value Theorem

If f is continuous on [a, b] and d 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) = d.

Mean Value Theorem

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

Extreme Value Theorem

If a function is continuous on a closed interval, then the function is guaranteed to have an absolute maximum and an absolute minimum in the interval.

Mean Value Theorem (Integrals)

If f is continuous on [a, b], then there exists a value c in [a, b] such that f(c) = [1/(b-a)] * ∫[a to b] f(x) dx.

First Fundamental Theorem of Calculus

∫[a to b] f'(x) dx = f(b) - f(a)

Second Fundamental Theorem of Calculus

d/dx [∫[a to x] f(t) dt] = f(x)

Definition of a Critical Number

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.

First Derivative Test

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

Second Derivative Test

1) If f'(c)=0 and f''(c)>0, then f(c) is a relative minimum. 2) If f'(c)=0 and f''(c)<0, then f(c) is a relative maximum.

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.

Definition of an Inflection Point

A function f has an inflection point at (c, f(c)) if 1) 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.

Trapezoidal Rule

A= (b-a)/(2n) * [1st height + 2(middle heights) + last height]

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

fave = [1/(b-a)] * ∫[a to b] f(x) dx

Euler's Method

y(i+1) = y(i) + dy/dx * Δx

Integration by Parts

∫ u dv = uv - ∫ v du

Law of Exponential Change

dy/dx = ky

Disk Method (Horizontal Axis)

V = π ∫ [R(x)]^2 dx, where R(x) = upper bound - lower bound

Washer Method (Horizontal Axis)

V = π ∫ ([R(x)]^2 - [r(x)]^2) dx

Arc Length

AL = ∫[a to b] √[1 + (f'(x))^2] dx

Velocity

v(t) = s'(t)

Acceleration

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

Speed

|v(t)|

Parametric Arc Length

L = ∫[a to b] √[(dx/dt)^2 + (dy/dt)^2] dt

Area inside a polar curve

A = (1/2) ∫[a to b] r^2 dθ

Taylor Polynomial

P_n(x) = f(c) + f'(c)(x-c) + (f''(c)/2!)(x-c)^2 + … + (f^n(c)/n!)(x-c)^n

Maclaurin Series for 1/(1-x)

Σ x^n from n=0 to ∞

Maclaurin Series for cos(x)

Σ (-1)^n * (x^(2n))/(2n)! from n=0 to ∞

Maclaurin Series for e^x

Σ (x^n)/n! from n=0 to ∞

Maclaurin Series for sin(x)

Σ (-1)^n * (x^(2n+1))/(2n+1)! from n=0 to ∞