1. Let f (x) be a polynomial with critical points at x = −3 and x = 4, and let k be a nonzero
constant.
What are the critical points of h(x) = f (kx)?
(a) x = −3 and x = 4
4
3
(b) x = − and x =
k
k
(c) x = 3k and x = 4k
(d) Cannot be determined
Z
5
f (x) dx
2. If f (x) is a continuous function on [−2, 5], then
−2
(a) = F (x) + C
(b) is a number.
(c) may not exist.
(d) is an antiderivative of f (x).
3. A car braked with a constant deceleration of 16 f t/s2 and traveled 200f t before coming to a
stop. How fast was the car traveling when the brakes were first applied?
(a) 130 ft/s
(b) 80 ft/s
(c) 70 ft/s
(d) 40 ft/s
(e) Cannot be determined
4. A rocket lifts off the surface of Earth from rest at a constant acceleration of 20 m/s2 . How far
will the rocket have traveled after 5 seconds?
(a) 250 m
(b) 500 m
(c) 2,500 m
(d) 50,000 m
(e) Cannot be determined
5. An rock dropped from a 18-meter cliff has acceleration −9.8 m/sec2 (due to the influence of
gravity). What is the function s(t) that models the rock’s position above the ground at time t?
Z 2
Z 2
6. Given that
H(t) dt = −5, determine the value of
(H(t) − t + 3) dt.
1
1
7. True/False: Indicate whether each of the following statements is True or False. If the
statement is true, explain how you know it’s true. If it is false, give a counterexample. (A
counterexample is an example that shows the statement is false.)
(a) If f 00 (c) = 0, then x = c is an inflection point.
(b) If f 0 (x) > 0 for x < 1 and f 0 (x) < 0 for x > 1, then f (x) has a local maximum at x = 1.
8. Antiderivatives and Initial Value Problems: See section 4.7 problems 17-46, 57-66, 69-92
9. Use Geometry to evaluate definite integrals: See section 5.3 problems 15-28
10. Find two positive numbers that are reciprocals of each other and their sum is a minimum.
11. Find two positive numbers whose sum is 300 and whose product is a maximum.
12. Maximize the area of a rectangle inscribed under the parabola 4 − x2 , where the base of the
rectangle is on the x-axis and the top corners are on the parabola.
√
13. Using calculus, determine the point on the curve f (x) = x that is closest to the point (3, 0).
14. The shape of a Norman window can be approximated by a rectangle with a semicircle on top.
What dimensions will admit the maximum amount of light if the perimeter of the window is P
inches?
15. Use the graph of h0 (x) below to answer the following questions.
(a) In the axes below, sketch a graph of a possible antiderivative of h0 (x) and label this graph
as h(x).
(b) Write a story involving real-world phenomena that can be described by h(x).
16. To the right is a graph showing the velocity of
an object with respect to time.
To the right is a graph showing the velocity of
an object with respect to time.
For each of the graphs above, answer the following questions:
(a) Where does the object have the largest position? On the graph, mark that point with a
capital P.
(b) Where does the object have the largest velocity? On the graph, mark that point with a
capital V.
(c) Where does the object have the largest speed? On the graph, mark that point with a
capital S.
(d) Write a story to match the graph.
17. Riemann Sums! See lab 2, homework, and section 5.1 in textbook. There are several practice
problems in the textbook.
18. Suppose that the function f (t) is continuous and always increasing and always negative. If G(t)
is an antiderivative of f (t), then what can we say about G(t)? Is it continuous? Increasing?
Decreasing? Concave up? Concave down? Explain.
19. Sketch the graph of a function that satisfies the following:
(a) lim f (x) = ∞
(g) f 0 (4) does not exist
(b) lim f (x) = −∞
(h) f 0 (x) > 0 for −4 < x < −2 and 2 < x < 4
(c) lim f (x) = ∞
(i) f 0 (x) < 0 for x < −4 and −2 < x < 2 and
x>4
x→−∞
x→∞
x→−2
(d) lim− f (x) = −∞
(e) lim+ f (x) = 2
(j) f 00 (x) > 0 for x < −2 and −2 < x < 0 and
2<x<4
(f) lim f (x) = 4 = f (4)
(k) f 00 (x) < 0 for 0 < x < 2 and x > 4
x→2
x→2
x→4
20. Sketch the graph of a function, G(x), that has the following properties:
• G(0) = 0
• G0 (−2) = G0 (1) = G0 (9) = 0
• lim G(x) = 0
• G0 (x) < 0 on (−∞, −2) ∪ (1, 6) ∪ (9, ∞)
• lim G(x) = −∞
• G00 (x) < 0 on (0, 6) ∪ (6, 12)
• G0 (x) > 0 on (−2, 1) ∪ (6, 9)
• G00 (x) > 0 on (−∞, 0) ∪ (12, ∞)
x→∞
x→6
Based on the given information, identify the critical point(s) of G(x): x =
Based on the given information, identify the inflection point(s) of G(x): x =
Based on the given information, identify the local maxima of G(x):
Based on the given information, identify the local minima of G(x):
Is there an absolute maximum? YES / NO (circle one) Explain why or why not in 1-2 sentences.
Is there an absolute minimum? YES / NO (circle one) Explain why or why not in 1-2 sentences.
21. Find f (x) that satisfies the given conditions:
√
(a) f 00 (x) = 3 x, f 0 (1) = 0, f (1) = 17
(b) f 000 (x) = − cos(2x), f 00 (0) = 1, f 0 (0) = 1, f (0) = −1
(c) The acceleration caused by gravity is −9.81 m/s2 . A hiker throws a pebble into a canyon
that is 350 meters deep with downward initial velocity of 10 m/s. For how many seconds is
the pebble in the air and what is the speed of impact?
22. Let f (x) be a differentiable function on a closed interval with x = a being one of the endpoints
of the interval. If f 0 (a) > 0 then,
(a) f could have either an absolute maximum or an absolute minimum at x = a.
(b) f cannot have an absolute maximum at x = a.
(c) f must have an absolute minimum at x = a.
23. True or False. If F 00 (5) = 0, then F (x) has an inflection point at x = 5.
24. True or False. If f 0 (x) = g 0 (x), then f (x) must be the same function as g(x).
25. Write a sum that represents the area of the rectangles drawn in the picture below, where the
curve is the graph of a function y = f (x) (be sure to use summation notation).