AP Calculus - Ultimate Justifications Guide

To justify that f has a relative maximum at x = a

• State/show that f' changes from positive to negative at x = a

• Show that f' is always positive at x = a

• Demonstrate that f'' is positive at x = a

• Prove that f is increasing at x = a

To justify that f has a critical point at x = a, which of the following must be true?

• f'(a) = 0 or undefined

• f(a) is a local maximum

• f(a) is a local minimum

• f'(a) > 0

To justify that f has an absolute minimum at x = a

• Show that f has a critical point at x = a and f(a) has the lowest value of all critical points and endpoints

• Demonstrate that f is continuous at x = a

• Prove that f is differentiable at x = a

• Establish that f has a local minimum at x = a

To justify that f has a relative minimum at x = a, which of the following must be shown?

• f' changes from positive to negative at x = a

• f' changes from negative to positive at x = a

• f'' is positive at x = a

• f' is zero at x = a

Mean Value Theorem

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

• If f is continuous on [a, b] and differentiable on (a, b), then f(c) = (f(a) + f(b)) / 2 for some c in (a, b).

• If f is continuous on [a, b] and differentiable on (a, b), then f'(c) = f(b) - f(a) for some c in (a, b).

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

To justify that f is concave down on the interval (a, b)

• State/show that f'' < 0 on the interval (a, b)

• State/show that f' < 0 on the interval (a, b)

• State/show that f'' > 0 on the interval (a, b)

• State/show that f is decreasing on the interval (a, b)

Right Riemann Sum Approximation

• A right Riemann sum is an underapproximation for the area under a curve if the function is decreasing on the interval.

• A right Riemann sum is an overapproximation for the area under a curve if the function is increasing on the interval.

• A right Riemann sum is an underapproximation or overapproximation for the area under a curve if the function is decreasing or increasing on the interval.

• A right Riemann sum is always an exact approximation for the area under a curve.

To justify that f is concave up on the interval (a, b), which of the following must be shown?

• f'' < 0 on the interval (a, b)

• f'' = 0 on the interval (a, b)

• f'' > 0 on the interval (a, b)

• f' > 0 on the interval (a, b)

Particle Changes Direction

• A particle changes direction at t = k if v changes signs at t = k.

• A particle changes direction at t = k if acceleration is constant.

• A particle changes direction at t = k if it travels in a circular path.

• A particle changes direction at t = k if its speed increases.

Intermediate Value Theorem

• If f is continuous on [a, b], then f(a) ≤ k ≤ f(b) for some x on the interval [a, b].

• If f is discontinuous on [a, b], then f(a) ≤ k ≤ f(b) for some x on the interval [a, b].

• If f is continuous on [a, b], then f(a) = k = f(b) for some x on the interval [a, b].

• If f is continuous on [a, b], then f(a) < k < f(b) for all x on the interval [a, b].