Mathematics

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MATH 140 Test 3 Review

1. Suppose that f is a continuous, differentiable function such that f(2) = 3 and f′(x) ≤ 4 for all x. Using the Mean Value Theorem, what can you say about f(7)?

2. Suppose that f is a continuous, differentiable function such that f(1) = 12 and f(8) = 3. What does the Mean Value Theorem tell you?

3. Let f(x) = x4 − 8×3 + 16×2.

(a) Find the critical points of f.

(b) Describe how you would find the absolute maximum and minimum of f on the interval [1, 6].

4. Find the absolute maximum and absolute minimum of f(x) = 1

x2 −x + 1 on the interval [0, 5].

5. You want to make a wooden box of width w, depth d and height h, where the width and depth are equal. The wood for the top and bottom costs 2 dollars per square foot, and the wood for the sides costs 1 dollar per square foot. Then the cost is

C = 4w2 + 4wh

and the volume is V = w2h.

If you want the box to have a volume of 8 cubic feet, then what should you pick for w in order to minimize the cost?

6. Suppose that the point (x, y) is on the curve y = √ x. If we want to make this point as close as

possible to the point (3, 0), then we can minimize the function

(x− 3)2 + y2.

Find the coordinates of the the point (x, y) on y = √ x that minimize this function.

7. For each part, draw a graph of a function that satisfies the given information.

(a) f′(x) < 0 and f′′(x) > 0 on the interval [1, 3]

(b) f′(x) < 0 and f′′(x) < 0 on the interval [1, 3] (c) f′(x) > 0 and f′′(x) > 0 on the interval [1, 3]

(d) f′(x) > 0 and f′′(x) < 0 on the interval [1, 3] (e) f′(1) = 0 and f′′(x) < 0 on the interval [0, 2] (f) f′(x) > 0 and f′′(x) = 0 on the interval [2, 4]

8. Let f(x) = 9×2 + cos x + 1

x +

1

x2 + 2.

(a) Find an antiderivative F (x) of f(x).

(b) Find the antiderivative F (x) of f(x) such that F (1) = sin 1.

9. Let f(x) = 4x + 5ex + 6 √ x + 7.

(a) Find an antiderivative F (x) of f(x).

(b) Find the antiderivative F (x) of f(x) such that F (0) = 3.

10. Estimate the area under the graph of y = x2

x2 + 1 between x = −4 and x = 4 by subdividing

[−4, 4] into four intervals of equal width, and using the midpoint of each interval as your sample point.

11. Estimate the area under the graph of y = 10− (x− 6)2

4 between x = 0 and x = 12 by subdividing

[0, 12] into three intervals of equal width, and using the right endpoint of each interval as your sample point.

12. Let f(x) = 4×2 + 1

x2 − 1

(a) Find the equation(s) of any vertical asymptotes.

(b) Find the equation(s) of any horizontal asymptotes.

(c) The derivative of f is f′(x) = −10x

(x2 − 1)2 . Determine where f is increasing and decreasing.

(d) Sketch a graph of f(x), using the information from the previous parts.

13. Let f(x) = x2 + 5

x3 − 8

(a) Find the equation(s) of any vertical asymptotes.

(b) Find the equation(s) of any horizontal asymptotes.

(c) The derivative of f is f′(x) = −x(x + 1)(x2 −x + 16)

(x3 − 8)2 . Determine where f is increasing and

decreasing. (Hint: x2 −x + 16 is never 0.) (d) Sketch a graph of f(x), using the information from the previous parts.