AP Calculus AB Review

Flashcards covering limits, derivatives, theorems, and applications.

What condition must be met for the limit of a function f(x) as x approaches a to exist?

The limit from the left (x→a-) must equal the limit from the right (x→a+), and both must equal L.

State L'Hopital's Rule

If lim(x→c) f(x) = 0 and lim(x→c) g(x) = 0, or if lim(x→c) f(x) = ±∞ and lim(x→c) g(x) = ±∞, then lim(x→c) f(x)/g(x) = lim(x→c) f'(x)/g'(x) = L.

Give the formula of the derivative

f'(x) = lim (h→0) [f(x+h) - f(x)] / h or f'(x) = lim (x→a) [f(x) - f(a)] / (x-a)

Under which conditions does a derivative NOT exist?

Corner, cusp, discontinuity, or vertical tangent.

What is the power rule?

d/dx [x^n] = nx^(n-1)

What is the quotient rule?

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

What is the chain rule?

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

What is the derivative of sin(u)?

cos(u) * du

What is the derivative of e^g(x)?

e^g(x) * g'(x)

State the Mean Value Theorem.

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

State Rolle's Theorem.

If f is continuous on [a, b], differentiable on (a, b), and f(a) = f(b), then there exists a c on (a, b) such that f'(c) = 0.

State the Intermediate Value Theorem.

If f is continuous on [a, b] and there is a value k between f(a) and f(b), then there exists at least one value c on (a, b) such that f(c) = k.

State the Extreme Value Theorem.

If f is continuous on [a, b], then f has an absolute maximum and an absolute minimum value at x = a, x = b, or when f'(x) = 0 or f'(x) is undefined.

If f''(x) is positive, then f(x) is…

Concave up

What is speed?

Speed = |v(t)|

When is speed increasing?

When velocity and acceleration have the same sign/direction

Net Distance

The distance from where the object begins and where it ends.

Total Distance

The sum of all distances moved in any direction.

State the Fundamental Theorem of Calculus Part 1

If f(x) is continuous on [a, b] and F(x) is the anti-derivative of f(x), then ∫[a to b] f(x) dx = F(b) - F(a).

State the Fundamental Theorem of Calculus Part 2

If F(x) = ∫[a to g(x)] f(t) dt, then F'(x) = f(g(x)) * g'(x).