1. lim_{x→∞} \frac{\sqrt{x^{4} + x^{2} + 1} + cos(x^{2} + π)}{x^{2}} =

(a) 1  (b) ∞  (c) 3  (d) 2  (e) 0

2. If the second derivative of a function f is given by

f′′(x) = \frac{x(x − 2)(x − 4)(x − 6)}{(x^{2} + 1)}

then the number of inflection points of the graph of f is 3. The shortest distance from the origin to the line 2x + y = 4 is

(a) 2  (b) 16/5  (c) 4  (d) 1  (e) \frac{4}{5}\sqrt{5}

4. If f′(x) = x^{−2}, f(1) = −1 then f(2) =?

(a) −2  (b) −1/2  (c) 0  (d) 1/2  (e) 2

5. If f(x) = x^{2} , 0 ≤ x ≤ 6 , find the Riemann sum with n = 3 taking the sample points to be \underline{Midpoints}.

(a) 35  (b) 18  (c) 74  (d) 9  (e) 70

7. Find all functions f such that f′′(x) = sin x + cos x ( A, B are constants)

(a) − sin x − cos x + A + B (b) sin x − cos x + A + B (c) sin x + cos x + Ax + B
(d) − sin x − cos x + Ax + B  (e) sin x + cos x + A + B

8. Find the area under the graph of the function

f(x) = \sqrt{x}

from x = 0 to x = 4 .

(a) 1  (b) \frac{16}{3}  (c) \frac{14}{3}  (d) \frac{1}{4}  (e) \frac{11}{2}

9. Let f(x) = 8x^{3} − x^{4} . Notice that

f′(x) = 4x^{2}(6 − x)
f′′(x) = 12x(4 − x)

Which of the following describes the graph of f(x) ?

(a) local minimum at x = 0; local maximum at x = 6; inflection points only at x = 0 and x = 4

(b) local maximum at x = 6; inflection points only at x = 0 and x = 4

(c) local minimum at x = 0 and x = 6; inflection point only at x = 4

(d) local minimum at x = 0; local maximum at x = 6; no inflection points

(e) local maximum at x = 6; inflection point only at x = 4 and x = 2

10. Find all the horizontal and vertical asymptotes of y = \frac{x + 1}{\sqrt{x − 1}} .

(a) y = −1; no vertical asymptote  (b) y = −1; x = −1 (c) No horizontal asymptote; x = 1
(d) y = 0; x = −1  (e) y = 1; x = 1

11. Evaluate the integral ∫_{−1}^{1}(1 − |x|) dx

(a) −\frac{1}{2}  (b) \frac{1}{2}  (c) 0  (d) −1  (e) 1

12. Evaluate the integral ∫_{0}^{1}(x + 1)^{3} dx

(a) \frac{1}{4}  (b) \frac{15}{16}  (c) \frac{7}{3}  (d) 7  (e) \frac{15}{4}

13. Let f(x) = \frac{x}{x^{2} − 1} . Then f′(x) = −\frac{x^{2} + 1}{(x^{2} − 1)^{2}} and f′′(x) = \frac{2x(x^{2} + 3)}{(x^{2} − 1)^{3}} .

(a) Find all horizontal and vertical asymptotes.

(b) Find all critical points.

(c) Find the intervals on which the function is increasing or decreasing.

(d) Sketch the graph of f . Indicate the intervals on which the function is concave up or down.

14. Evaluate the following integrals: (a) ∫ 3x^{2}\sqrt{x^{3} + 1} dx =

(b) ∫ \frac{sin x}{cos^{3} x} dx =

15. A box with a square base and open top must have a volume of 32, 000 cm^{3} . Find the dimensions of the box that minimize the amount of material used.

16. Use Part 1 of the Fundamental Theorem of Calculus to find the derivative g′(x) of the function g(x) = ∫_{0}^{x} \sqrt{1 + 2t} dt

Version 1 color:

Math 119 Calculus, Exam 3 November 28, 2000

This Examination contains 16 problems, worth a total of 100 points, on 11 sheets of paper including the front cover. The first 12 problems are multiple choice with no partial credit, and each is worth 5 points. Please cross × the correct answers for the multiple choice questions 1–12. The last 4 problems are partial credit problems worth 10 points each. For these problems, show your computations and clearly mark your answers on the page. Books and notes are not allowed. You may use your calculator.

\underline{Name:} \underline{Prof:}

1. •  b  c  d  e

2. a  •  c  d  e

3. a  b  c  d  •

4. a  •  c  d  e

5. a  b  c  d  •

6. a  •  c  d  e

7. a  b  c  •  e

8. a  •  c  d  e

9. a  •  c  d  e

10. a  b  •  d  e

11. a  b  c  d  •

12. a  b  c  d  •

Mult. Choice

13

14

15

16

Total

Math 119 Calculus, Exam 3 November 28, 2000

This Examination contains 16 problems, worth a total of 100 points, on 11 sheets of paper including the front cover. The first 12 problems are multiple choice with no partial credit, and each is worth 5 points. Please cross × the correct answers for the multiple choice questions 1–12. The last 4 problems are partial credit problems worth 10 points each. For these problems, show your computations and clearly mark your answers on the page. Books and notes are not allowed. You may use your calculator.

\underline{Name:} \underline{Prof:}

1. a  b  c  d  e

2. a  b  c  d  e

3. a  b  c  d  e

4. a  b  c  d  e

5. a  b  c  d  e

6. a  b  c  d  e

7. a  b  c  d  e

8. a  b  c  d  e

9. a  b  c  d  e

10. a  b  c  d  e

11. a  b  c  d  e

12. a  b  c  d  e

Mult. Choice

13

14

15

16

Total

Sign your name:

1. lim_{x→∞} \frac{\sqrt{x^{4} + x^{2} + 1} + cos(x^{2} + π)}{x^{2}} =

(a) 2  (b) 3  (c) 0  (d) 1  (e) ∞

2. If the second derivative of a function f is given by

f′′(x) = \frac{x(x − 2)(x − 4)(x − 6)}{(x^{2} + 1)}

then the number of inflection points of the graph of f is 3. The shortest distance from the origin to the line 2x + y = 4 is

(a) 2  (b) 16/5  (c) 1  (d) 4  (e) \frac{4}{5}\sqrt{5}

4. If f′(x) = x^{−2}, f(1) = −1 then f(2) =?

(a) 1/2  (b) 2  (c) −1/2  (d) 0  (e) −2

5. If f(x) = x^{2} , 0 ≤ x ≤ 6 , find the Riemann sum with n = 3 taking the sample points to be \underline{Midpoints}.

(a) 35  (b) 9  (c) 70  (d) 74  (e) 18

7. Find all functions f such that f′′(x) = sin x + cos x ( A, B are constants)

(a) sin x + cos x + Ax + B (b) − sin x − cos x + A + B (c) sin x − cos x + A + B
(d) − sin x − cos x + Ax + B  (e) sin x + cos x + A + B

8. Find the area under the graph of the function

f(x) = \sqrt{x}

from x = 0 to x = 4 .

(a) 1  (b) \frac{14}{3}  (c) \frac{16}{3}  (d) \frac{11}{2}  (e) \frac{1}{4}

9. Let f(x) = 8x^{3} − x^{4} . Notice that

f′(x) = 4x^{2}(6 − x)
f′′(x) = 12x(4 − x)

Which of the following describes the graph of f(x) ?

(a) local maximum at x = 6; inflection point only at x = 4 and x = 2

(b) local minimum at x = 0; local maximum at x = 6; inflection points only at x = 0 and x = 4

(c) local minimum at x = 0; local maximum at x = 6; no inflection points

(d) local minimum at x = 0 and x = 6; inflection point only at x = 4

(e) local maximum at x = 6; inflection points only at x = 0 and x = 4

10. Find all the horizontal and vertical asymptotes of y = \frac{x + 1}{\sqrt{x − 1}} .

(a) y = 0; x = −1 (b) y = 1; x = 1 (c) No horizontal asymptote; x = 1 (d) y = −1; x = −1

(e) y = −1; no vertical asymptote 11. Evaluate the integral ∫_{−1}^{1}(1 − |x|) dx

(a) −\frac{1}{2}  (b) 1  (c) −1  (d) \frac{1}{2}  (e) 0

12. Evaluate the integral ∫_{0}^{1}(x + 1)^{3} dx

(a) \frac{7}{3}  (b) 7  (c) \frac{15}{4}  (d) \frac{1}{4}  (e) \frac{15}{16}

13. Let f(x) = \frac{x}{x^{2} − 1} . Then f′(x) = −\frac{x^{2} + 1}{(x^{2} − 1)^{2}} and f′′(x) = \frac{2x(x^{2} + 3)}{(x^{2} − 1)^{3}} .

(a) Find all horizontal and vertical asymptotes.

(b) Find all critical points.

(c) Find the intervals on which the function is increasing or decreasing.

(d) Sketch the graph of f . Indicate the intervals on which the function is concave up or down.

14. Evaluate the following integrals: (a) ∫ 3x^{2}\sqrt{x^{3} + 1} dx =

(b) ∫ \frac{sin x}{cos^{3} x} dx =

15. A box with a square base and open top must have a volume of 32, 000 cm^{3} . Find the dimensions of the box that minimize the amount of material used.

16. Use Part 1 of the Fundamental Theorem of Calculus to find the derivative g′(x) of the function g(x) = ∫_{0}^{x} \sqrt{1 + 2t} dt

Math 119 Calculus, Exam 3 November 28, 2000

This Examination contains 16 problems, worth a total of 100 points, on 11 sheets of paper including the front cover. The first 12 problems are multiple choice with no partial credit, and each is worth 5 points. Please cross × the correct answers for the multiple choice questions 1–12. The last 4 problems are partial credit problems worth 10 points each. For these problems, show your computations and clearly mark your answers on the page. Books and notes are not allowed. You may use your calculator.

\underline{Name:} \underline{Prof:}

1. a  b  c  •  e

2. a  b  c  d  •

3. a  b  c  d  •

4. a  b  •  d  e

5. a  b  •  d  e

6. •  b  c  d  e

7. a  b  c  •  e

8. a  b  •  d  e

9. a  b  c  d  •

10. a  b  •  d  e

11. a  •  c  d  e

12. a  b  •  d  e

Mult. Choice

13

14

15

16

Total