AP Calcuous - Ultimate Justifications Guide

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

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

To justify that f is differentiable at x = a

• State/show that f is continuous at x = a and @@\lim_{x \to a^-} f'(x) = \lim_{x \to a^+} f'(x)@@

• Demonstrate that f is continuous at x = a and @@f(a) = f'(a)@@

• Prove that f is continuous at x = a and @@f(a) = 0@@

• Show that f is differentiable at all points in the neighborhood of x = a

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

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

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

• Prove that f'' is positive at x = a

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

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 and f(a) is less than f(b) for all b

• Prove that the derivative of f at x = a is zero and f(a) is greater than f(b) for all b

• Establish that f is decreasing on the interval around x = a and f(a) is the lowest value

Tangent Line Approximation

• A tangent line approximation for f(a) is always an exact value of f(a).

• A tangent line approximation for f(a) is an underestimate or overestimate for the true value of f(a) if f is concave up or concave down near x = a.

• A tangent line approximation for f(a) is only valid when f is linear.

• A tangent line approximation for f(a) is a good estimate when f is concave up but not when it is concave down.

To justify that f is continuous at x = a, which of the following statements is correct?

• @@ ext{To show that } \\ \\ ext{lim}_{x o a^-} f(x) = ext{lim}_{x o a^+} f(x) = L ext{ and } ext{lim}_{x o a} f(x) = f(a) = L@@

• @@ ext{To show that } \\ \\ ext{lim}_{x o a^-} f(x) eq ext{lim}_{x o a^+} f(x)@@

• @@ ext{To show that } \\ \\ f(a) eq ext{lim}_{x o a} f(x)@@

• @@ ext{To show that } \\ \\ f(x) ext{ is not defined at } x = a@@

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 reaches maximum speed.

• A particle changes direction at t = k if it is acted upon by an external force.

Trapezoidal Approximation

• A trapezoidal approximation is an underapproximation or overapproximation for the area under a curve f between x = a and x = b if f is concave down or concave up on the interval (a, b).

• A trapezoidal approximation is always an exact representation of the area under a curve regardless of the function's concavity.

• A trapezoidal approximation can only be used for linear functions and cannot be applied to nonlinear functions.

• A trapezoidal approximation is a method used to calculate the derivative of a function.

Particle at Rest

• A particle is at rest if its velocity is greater than zero.

• A particle is at rest if its velocity is less than zero.

• A particle is at rest at t = k if v(k) = 0, where v is the velocity function.

• A particle is at rest if it has no mass.

Left Riemann Sum Approximation

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

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

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

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