AP Calculus BC Flashcards

Flashcards for AP Calculus BC review, covering pre-calculus identities, limits, continuity, derivatives, and integration techniques.

Quotient Identities

sin(x)/cos(x) = tan(x), cos(x)/sin(x) = cot(x)

Reciprocal Identities

sec(x) = 1/cos(x), csc(x) = 1/sin(x)

Pythagorean Identities

sin^2(x) + cos^2(x) = 1, sec^2(x) - tan^2(x) = 1

Double Angle Identities

sin(2x) = 2sin(x)cos(x), cos(2x) = cos^2(x) - sin^2(x)

Even-Odd Identities

sin(-x) = -sin(x), cos(-x) = cos(x), tan(-x) = -tan(x)

Sum & Difference Identities

sin(A + B) = sin(A)cos(B) + cos(A)sin(B), cos(A + B) = cos(A)cos(B) - sin(A)sin(B)

Distance Between Two Points

sqrt((x2 - x1)^2 + (y2 - y1)^2)

Midpoint Formula

((x1 + x2)/2, (y1 + y2)/2)

Laws of Logarithms

ln(ab) = ln(a) + ln(b), ln(a/b) = ln(a) - ln(b), ln(a^n) = n*ln(a)

One-sided Limits

lim (x->a-) f(x) = L (from left), lim (x->a+) f(x) = L (from right)

Definition of a Limit

lim (x->a) f(x) = L iff lim (x->a-) f(x) = lim (x->a+) f(x) = L

Definition of Vertical Asymptote

The line x = a is a vertical asymptote iff lim (x->a) f(x) = ±∞

Definition of Horizontal Asymptote

The line y = a is a horizontal asymptote iff lim (x->±∞) f(x) = a

Definition of Continuity

A function f is continuous at x = a iff 1. f(a) exists, 2. lim (x->a) f(x) exists, 3. lim (x->a) f(x) = f(a)

Intermediate Value Theorem (IVT)

If 1. f is continuous on [a, b], 2. f(a) ≠ f(b), 3. k is between f(a) and f(b), then there exists a number c between a and b for which f(c) = k

Squeeze Theorem

If f(x) ≤ g(x) ≤ h(x) and lim (x->a) f(x) = L and lim (x->a) h(x) = L, then lim (x->a) g(x) = L

Definition of the Derivative

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

Normal Line

The line perpendicular to the tangent line at the point of tangency.

Average Rate of Change

Δy/Δx = (f(b) - f(a))/(b - a)

Chain Rule

If h(x) = f(g(x)), then h'(x) = f'(g(x)) * g'(x)

Product Rule

(fg)' = f'g + fg'

Quotient Rule

(f/g)' = (f'g - fg')/g^2

L'Hopital's Rule

If lim (x->a) f(x)/g(x) is of the form 0/0 or ∞/∞, then lim (x->a) f(x)/g(x) = lim (x->a) f'(x)/g'(x)

Extreme Value Theorem (EVT)

If f is continuous on a closed interval [a, b], then f has an absolute max and an absolute min on the interval [a, b].

Definition of Critical Points

A critical point of f is a number c such that either f'(c) = 0 or f'(c) does not exist.

Mean Value Theorem

If 1. f is continuous on [a, b] and 2. differentiable on (a, b), then there exists a number c between a and b such that f'(c) = (f(b) - f(a))/(b - a)

Rolle's Theorem

If 1. f is continuous on [a, b], 2. Differentiable on (a, b), and 3. f(a) = f(b), then there is at least one number c on (a, b) such that f'(c) = 0

First Derivative Test for Relative Extrema

If f' changes from positive to negative at c, then f(c) is a relative max. If f' changes from negative to positive at c, then f(c) is a relative min.

Test for Concavity

1. f'' > 0 means f is concave up. 2. f'' < 0 means f is concave down.

Definition of a Point of Inflection (POI)

A point on the graph of f where the concavity of f changes.

Second Derivative Test for Relative Extrema

Given f'(c) = 0, then if f''(c) < 0, then f(c) is a max; if f''(c) > 0, then f(c) is a min.

Fundamental Theorem of Calculus (Part 1)

∫[a, b] f(x) dx = F(b) - F(a), where F(x) is an anti-derivative of f(x).

Average Value of a Function

f_avg = (1/(b-a)) ∫[a, b] f(x) dx

Fundamental Theorem of Calculus (Part 2)

d/dx ∫[c, x] f(t) dt = f(x)

Area Between Two Curves

A = ∫a, b - g(x)) dx, where f(x) > g(x)

Volumes of Solids (Known Cross Sections)

V = ∫[a, b] A(x) dx

Solids of Revolution (Discs)

V = π ∫[a, b] R(x)^2 dx

Arc Length

L = ∫[a, b] √(1 + (f'(x))^2) dx

Integration by Parts

∫ u dv = uv - ∫ v du

Euler's Method

ynew = yold + m * Δx, where m = dy/dx

Logistic Growth

dP/dt = kP(L - P), where L is the carrying capacity.

Speed of a 2-D Vector

speed = √((dx/dt)^2 + (dy/dt)^2)

Polar Area

Area = (1/2) ∫[α, β] r^2 dθ

Divergence Test (nth term test)

If lim (n->∞) a_n ≠ 0, then the series diverges.

Geometric Series

∑ ar^n converges if |r| < 1 and diverges if |r| ≥ 1. Sum = a/(1-r)

P-series

∑ 1/n^p converges if p > 1 and diverges if p ≤ 1

Integral Test

If f(n) = an, f is continuous, positive, and decreasing, then ∑ an and ∫ f(x) dx both converge or both diverge.

Direct Comparison Test

If 0 < an ≤ bn and ∑ bn converges, then ∑ an converges. If 0 < bn ≤ an and ∑ bn diverges, then ∑ an diverges.

Limit Comparison Test

If lim (n->∞) an/bn = L > 0, then ∑ an and ∑ bn both converge or both diverge.

Alternating Series Test

∑ (-1)^n an converges if an is decreasing and lim (n->∞) a_n = 0

Ratio Test

If lim (n->∞) |a(n+1)/an| < 1, then the series converges. If lim (n->∞) |a(n+1)/an| > 1, then the series diverges.

Root Test

If lim (n->∞) (an)^(1/n) < 1, then the series converges. If lim (n->∞) (an)^(1/n) > 1, then the series diverges.

Maclaurin Series for e^x

e^x = ∑ (x^n)/n! from n=0 to ∞. Interval of convergence: (-∞, ∞)

Maclaurin Series for sin(x)

sin(x) = ∑ ((-1)^n * x^(2n+1))/(2n+1)! from n=0 to ∞. Interval of convergence: (-∞, ∞)

Maclaurin Series for cos(x)

cos(x) = ∑ ((-1)^n * x^(2n))/(2n)! from n=0 to ∞. Interval of convergence: (-∞, ∞)