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Calculus I Sample Final Exam

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CALCULUS I (MATH 156) SUMMER 2013 — FINAL EXAM JULY 26, 2013

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This exam consists of 8 questions and 1 bonus question. Show all your work. No work, no credit. Good Luck!

Question Points Out of 1 18 2 20 3 18 4 15 5 8 6 8 7 14 8 7 9(Bonus) 7 Total 115

18 points

1. For the function f (x) = x3 − 6x2 + 9x − 3 (a) find f (x).

(b) determine all the critical points of f.

(c) find the intervals where f is increasing and where it is decreasing.

(d) classify each critical point as relative maximum or minimum.

(e) Find f (x).

(f) Find the intervals where the graph of f is concave up and concave down.

(g) Determine the inflection points.

Page 2

20 points

2. Evaluate the following limits: (a) lim x2 − 4x + 4 x→2 x3 + 5x2 − 14x

(b) lim

x2 x→0 cos 8x − 1

(c) lim

x − 8x2 x→∞ 12x2 + 5x

(d) lim

e3x − 1 x→0 ex − x

(e) lim x2 e−x x→∞ Page 3

18 points

3. Find the following indefinite integrals: (a) 3 cos 5x − √ + 6e3x dx x

(b)



4x dx x2 + 1

(c)

x2 +

√ x

x−5

dx

Page 4

15 points

4. Evaluate the following definite integrals:
1

(a)
0

x4 + 3x3 + 1dx

e2

(b)
1

(ln x)2 dx x

π 4

(c)
0

(1 + etan x ) sec2 xdx

Page 5

8 points

5. Sketch and find the area of the region that lies under y = ex and above the x axis over the interval 0 ≤ x ≤ 7.

8 points

6. A rectangular plot of farmland will be bounded on one side by a river and on the other three sides by a single-strand electric fence. With 800 m of wire at your disposal, what is the largest area you can enclose and what are its dimensions?

Page 6

14 points

7. For the function f (x) = e−x (a) find f (x)

2

(b) find an equation of the tangent line to the graph of f at x = 0.

(c) find the absolute maximum and absolute minimum of f on the interval [−2, 1].

Page 7

7 points

8. Sketch the graph of a function f that has all of the following properties: i. f (x) > 0 when x < −4, and 0 < x < 3. ii. f (x) < 0 when −4 < x < 0 and x > 3. iii. f ”(x) > 0 when −1 < x < 1 iv. f ”(x) < 0 when x < −1 and x > 1. v. f (0) = −4, f (−1) = 2 = f (1).

Page 8

7 points

9. [Bonus] Solve the initial value problem ds = cos t + sin t dt s(π) = 1.

Page 9

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