## Description

1.

Given f '(x) = (x + 1)(6 + 3x), find the x-coordinate for the relative minimum on the graph of f(x). (5 points)

2.

Use the graph of f '(x) below to find the x values of the relative maximum on the graph of f(x):

(5 points)

3.

Given the position function s(t), s(t) = t^{3} + 4t, where s is measured in meters and t is in seconds, find the velocity and acceleration of the particle at t = 2 seconds. (5 points)

4.

Given the position function, , between t = 0 and t = 9, where s is given in feet and t is measured in seconds, find the interval in seconds where the particle is moving to the right. (5 points)

5.

Given the relationship x^{3} + 2y^{2} =10, with y > 0 and = 4 units/min., find the value of at the instant y = 1 unit. (5 points)

6.

The base of a triangle is decreasing at the rate of 1 ft/sec, while the height is increasing at the rate of 2 ft/sec. At what rate is the area of the triangle changing when the base is 10 ft and the height is 70 ft? Remember to use the product rule when you find the expression for . (5 points)

7.

Find . (5 points)

8.

Evaluate . (5 points)

9.

Which of the following statements is/are true? (5 points)

I. If f '(x) exists and is nonzero for all x, then f(1) ≠ f(0).

II. If f is differentiable for all x and f (-1) = f(1), then there is a number c, such that |c| < 1 and f '(c) = 0.

III. If f '(c) = 0, then f has a local maximum or minimum at x = c.

10.

The local linear approximation of a function f will always be less than or equal to the function's value if, for all x in an interval containing the point of tangency, (5 points)

1.

Find the x-coordinates of any relative extrema and inflection point(s) for the function f(x) = 6x^{(1/3)} + 3x^{(4/3)}. You must justify your answer using an analysis of f '(x) and f "(x).

2.

What is the maximum volume in cubic inches of an open box to be made from a 16-inch by 30-inch piece of cardboard by cutting out squares of equal sides from the four corners and bending up the sides? Your work must include a statement of the function and its derivative. Give one decimal place in your final answer.

3.

The position function of a particle in rectilinear motion is given by s(t) = t^{3} - 12t^{2} + 45t + 4 for t ≥ 0 with t measured in seconds and s(t) measured in feet. Find the position and acceleration of the particle at the instant when the particle reverses direction. Include units in your answer.

4.

A circle is growing so that the radius is increasing at the rate of 3 cm/min. How fast is the area of the circle changing at the instant the radius is 12 cm? Include units in your answer.

5.

The side of a square is measured to be 12 ft with a possible error of ±0.1 ft. Use linear approximation or differentials to estimate the error in the calculated area. Include units in your answer.

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