Exam 3: Applications of Differentiation

Find the absolute maximum of the function. on the interval

Given . (a) Find the intervals on which f is increasing or decreasing. (b) Find the relative maxima and relative minima of F.

For what values of does the curve have maximum and minimum points for the given function ?

A ballast is dropped from a stationary hot-air balloon that is at an altitude of 256 ft. Find (a) an expression for the altitude of the ballast after t seconds, (b) the time when it strikes the ground, and (c) its velocity when it strikes the ground. (Disregard air resistance and take .)

Find the horizontal and vertical asymptotes of the graph of the function .

The speed of traffic flow on a certain highway between 6 A.M. and 10 A.M. on a typical weekday is approximated by the functon where is measured in miles per hour and t is measured in hours, with corresponding to 6 A.M. Find the interval where f is increasing, the interval where f is decreasing, and the relative extrema of f. Interpret your results.

The graph below is the graph of function f on the interval Find the absolute maximum and absolute minimum values of f (if they exist) and where they are attained.

Find the minimum points of the function.

The function satisfies the hypotheses of Rolle's Theorem on the interval . Find all values of c that satisfy the conclusion of the theorem.

(a) Find the number c whose existence is guaranteed by the Mean Value Theorem for Integrals for the function f on [a, b], and (b) sketch the graph of f on [a, b] and the rectangle with base on [a, b] that has the same area as that of the region under the graph of f. f (x) = x² + x; [0, 1]

Evaluate , and tell whether its antiderivative F is increasing or decreasing at the point radians.

The function satisfies the hypotheses of Rolle's Theorem on the interval . Find all values of c that satisfy the conclusion of the theorem.

Use Newton's method with the specified initial approximation to find , the third approximation to the root of the given equation. (Give your answer to four decimal places.)

If of material is available to make a box with a square base and an open top, find the largest possible volume of the box.

Sketch the curve. ,

You are given the graph of f. Find the horizontal and vertical asymptotes of the graph of f.

At 4:00 P.M. a car's speedometer reads 25 . At 4:15 it reads 72 . At some time between 4:00 and 4:15 the acceleration is exactly x . Find x.

Find the limit.

Approximate the zero of in to within 0.00001.

Use the Second Derivative Test to find the relative extrema, if any, of the function .