1. Let f(x) be a differentiable function of x . Which of the following expressions is the derivative of the function g(x) = \frac{f(x)}{x^{2}} ?

(a) \frac{2xf(x) − x^{2}f′(x)}{x^{4}} (b) 2xf(x) − \frac{f′(x)}{x^{3}} (c) −2x^{−3}f′(x) + x^{−2}f(x) (d) \frac{x^{2}f′(x) − 2xf(x)}{x^{4}} (e) 2xf(x) + \frac{f′(x)}{x^{3}}

2. Assume x ∈ [0, \frac{π}{2} ] . If sin x = \frac{3}{5} then tan x =

(a) \frac{4}{3}  (b) \frac{5}{4}  (c) \frac{4}{5}  (d) \frac{5}{3}  (e) \frac{3}{4}

3. Suppose that f = g ∘ h + h ∘ g , g(0) = 1 , h(0) = 2 , g′(0) = 3 , g′(2) = 5 , h′(0) = 6 , h′(1) = 7 then f′(0) =

(a) 51  (b) 41  (c) 31  (d) 11  (e) 21

4. Which of the following quantities is equal to

lim_{h→0} \frac{(2 + h)^{3} − 8}{h}
(a) 0  (b) 3h^{2}  (c) 1  (d) f′(2), f(x) = x^{3}  (e) f′(h), f(x) = x^{2}

5. The volume V of a cube is increasing at a rate of 300cm^{3} per minute. How fast is the length L of each side increasing where L = 10 cm ?

(a) 1cm/min  (b) 100cm/min  (c) 30cm/min  (d) 3cm/min  (e) 10cm/min

6. Compute the indefinite integral

∫ \frac{2x}{(x^{2} + 1)^{5}} dx =

(a) \frac{(x^{2} + 1)^{4}}{4} + C  (b) \frac{1}{x^{4}} + C  (c) −\frac{(x^{2} + 1)^{4}}{4} + C  (d) \frac{1}{4(x^{2} + 1)^{4}} + C (e) −\frac{1}{4(x^{2} + 1)^{4}} + C

7. The function f(x) = \frac{x^{2}}{x^{2} − 1} is increasing on the interval(s):

(a) (−1, 0) ∪ (0, 1) (b) (−∞, 0) (c) (−1, 1) (d) (−∞, −1) ∪ (−1, 0) (e) (0, 1) ∪ (1, ∞)

8. Which of the following is \frac{f(x + h) − f(x)}{h} if f(x) = x^{2} + 7x ? Note that you are not being asked to compute the limit as h → 0 .

(a) 2x + 7  (b) 2x + 7x + h (c) 2x + 7 + h  (d) 2x + 7h  (e) 2xh + h^{2}

9. Let

f(x) = (sin(2x))^{3}.

Which of the following is f′(x) ?

(a) −6(sin(2x))^{2} cos(2x) (b) 6(sin(2x))^{2} cos(2x) (c) 3(sin(2x))^{2} cos(2x) (d) (sin(2x))^{2}

(e) 3 sin(x) cos(x)

10. Which of the following equals

∫ (cos(x) + 5)^{3} sin(x)dx ?
(a) −\frac{(cos(x) + 5)^{4}}{5} (b) −3(cos(x) + 5)^{2} (c) −\frac{(cos(x) + 5)^{4}}{4} + C (d) \frac{(cos(x) + 5)^{4}}{4}

(e) \frac{(cos(x) + 5)^{4}}{4} + C

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

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

12. Evaluate the integral ∫_{0}^{2π}(sin x − cos x) dx . The integral is equal to

(a) 0  (b) 2  (c) −1  (d) 1  (e) −2

13. Find the area enclosed by the curves y = x and y = x^{3} , between x = 0 and x = 1 .

(a) \frac{3}{4}  (b) 1  (c) −\frac{1}{4}  (d) \frac{1}{2}  (e) \frac{1}{4}

14. A particle is moving in a straight line with position function s(t) = \frac{t^{3}}{6} . What is the average velocity over the time period [1, 3] .

(a) 4  (b) 13  (c) \frac{13}{6}  (d) \frac{9}{2}  (e) \frac{13}{3}

15. Find the equation of the tangent line to the curve x^{2} + y^{2} = 169 at the point (12, 5) .

(a) y = −2x + 13 (b) y = −2.4x + 33.8 (c) y = \frac{5}{12}x (d) y = 2.4x + 23.8 (e) y = \frac{−5}{12}x

16. Find horizontal and vertical asymptotes to f(x) = \frac{1}{x} + \frac{1}{x + 1} . (a) vertical asymptotes at x = 0, horizontal asymptotes at y = −1

(b) vertical asymptotes at x = 0 and x = 1, horizontal asymptotes at ∞

(c) vertical asymptotes at x = 0, horizontal asymptotes at y = 0

(d) vertical asymptotes at x = 0 and x = −1, horizontal asymptotes at y = 1

(e) vertical asymptotes at x = 0 and x = −1, horizontal asymptotes at y = 0

17. Evaluate the integral ∫_{0}^{1} ^{3} \sqrt{x}(x − 1)dx .

(a) \frac{−3}{7}  (b) \frac{−9}{28}  (c) \frac{3}{7}  (d) −6  (e) \frac{11}{28}

18. Evaluate the integral ∫ (1 − x)\sqrt{2x − x^{2}}dx .

(a) (2x − x^{2})\frac{3}{2} + C  (b) \frac{1}{3}(2x − x^{2})\frac{3}{2} + C  (c) (2 − x)\frac{3}{2} + C  (d) −(2x − x^{2})^{− \frac{1}{2}} + C (e) (2x − x^{2})^{− \frac{1}{2}} + C

19. Which of the following numbers is equal to the value of the limit

lim_{x→4} \frac{\sqrt{x} − 2}{x − 4}
(a) 2  (b) 4  (c) 1/2  (d) 1/4  (e) 1

20. Which of the following limits is equal to the derivative of f(x) = \frac{1}{\sqrt{x + 1}} at x = 3 ?

(a) f′(3) = lim_{h→3} \frac{\frac{1}{\sqrt{1+h}} − \frac{1}{2}}{h} (b) f′(3) = lim_{h→3} \frac{\frac{1}{\sqrt{4+h}} − \frac{1}{2}}{h} (c) f′(3) = lim_{h→0} \frac{\frac{1}{\sqrt{1+x+h}} − \frac{1}{2}}{h}
(d) f′(3) = lim_{h→0} \frac{\frac{1}{\sqrt{1+h}} − \frac{1}{2}}{h}  (e) f′(3) = lim_{h→0} \frac{\frac{1}{\sqrt{4+h}} − \frac{1}{2}}{h}

21. Which one of the following functions F (x) satisfies F′(x) = 1/x for x > 0 ?

(a) F (x) = ∫_{1}^{x} \frac{1}{t + 1}dt − 1  (b) F (x) = \frac{1}{x^{2}}  (c) F (x) = ∫_{1}^{x} \frac{1}{t}dt  (d) F (x) = ∫_{1}^{x} −\frac{1}{t^{2}} dt (e) F (x) = −\frac{1}{x^{2}}

22. The product of two positive numbers is 900 . What is the minimum their sum can be?

(a) 30  (b) 60  (c) −30  (d) −60  (e) 0

23. Evaluate the integral ∫_{−1}^{0}(x + 2)^{3} dx

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

24. A balloon is rising vertically from a point A . It has constant speed of 5m/sec. A girl is watching from a position exactly 80m from A . How fast is the distance between the girl and the balloon increasing when the balloon is 60m high?

(a) 11\frac{m}{s}  (b) 5\frac{m}{s}  (c) 9\frac{m}{s}  (d) 3\frac{m}{s}  (e) 7\frac{m}{s}

25. On what intervals is f(x) = \frac{1}{4x − 7} concave up?

(a) (−∞, \frac{7}{4})  (b) (\frac{7}{4}, ∞)  (c) (−∞, ∞)  (d) (−∞, \frac{7}{4}) ∪ (\frac{7}{4}, ∞)  (e) (0, ∞)

Version 1 color:

Math 119 Calculus, Final December 15, 2000

This Examination contains 25 problems, worth a total of 150 points, on 11 sheets of paper including the front cover. Each Problem is worth 6 points. Please cross × the correct answers for the multiple choice questions. 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. •  b  c  d  e
4. a  b  c  •  e
5. •  b  c  d  e
6. a  b  c  d  •
7. a  b  c  •  e
8. a  b  •  d  e
9. a  •  c  d  e
10. a  b  •  d  e
11. •  b  c  d  e
12. •  b  c  d  e
13. a  b  c  d  •
14. a  b  •  d  e
15. a  •  c  d  e

16. a  b  c  d  •
17. a  •  c  d  e
18. a  •  c  d  e
19. a  b  c  •  e
20. a  b  c  d  •
21. a  b  •  d  e
22. a  •  c  d  e
23. a  •  c  d  e
24. a  b  c  •  e
25. a  •  c  d  e

Total

Math 119 Calculus, Final December 15, 2000

This Examination contains 25 problems, worth a total of 150 points, on 11 sheets of paper including the front cover. Each Problem is worth 6 points. Please cross × the correct answers for the multiple choice questions. 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
13. a  b  c  d  e
14. a  b  c  d  e
15. a  b  c  d  e

16. a  b  c  d  e
17. a  b  c  d  e
18. a  b  c  d  e
19. a  b  c  d  e
20. a  b  c  d  e
21. a  b  c  d  e
22. a  b  c  d  e
23. a  b  c  d  e
24. a  b  c  d  e
25. a  b  c  d  e

Total

Sign your name:

(a) 2xf(x) + \frac{f′(x)}{x^{3}} (b) 2xf(x) − \frac{f′(x)}{x^{3}} (c) −2x^{−3}f′(x) + x^{−2}f(x) (d) \frac{x^{2}f′(x) − 2xf(x)}{x^{4}} (e) \frac{2xf(x) − x^{2}f′(x)}{x^{4}}

2. Assume x ∈ [0, \frac{π}{2} ] . If sin x = \frac{3}{5} then tan x =

(a) \frac{4}{5}  (b) \frac{5}{3}  (c) \frac{4}{3}  (d) \frac{3}{4}  (e) \frac{5}{4}

3. Suppose that f = g ∘ h + h ∘ g , g(0) = 1 , h(0) = 2 , g′(0) = 3 , g′(2) = 5 , h′(0) = 6 , h′(1) = 7 then f′(0) =

(a) 21  (b) 31  (c) 51  (d) 11  (e) 41

4. Which of the following quantities is equal to

lim_{h→0} \frac{(2 + h)^{3} − 8}{h}
(a) 1  (b) 0  (c) f′(h), f(x) = x^{2}  (d) f′(2), f(x) = x^{3}  (e) 3h^{2}

5. The volume V of a cube is increasing at a rate of 300cm^{3} per minute. How fast is the length L of each side increasing where L = 10 cm ?

(a) 1cm/min  (b) 100cm/min  (c) 10cm/min  (d) 3cm/min  (e) 30cm/min

6. Compute the indefinite integral

∫ \frac{2x}{(x^{2} + 1)^{5}} dx =

(a) −\frac{1}{4(x^{2} + 1)^{4}} + C  (b) −\frac{(x^{2} + 1)^{4}}{4} + C  (c) \frac{1}{x^{4}} + C  (d) \frac{1}{4(x^{2} + 1)^{4}} + C (e) \frac{(x^{2} + 1)^{4}}{4} + C

7. The function f(x) = \frac{x^{2}}{x^{2} − 1} is increasing on the interval(s):

(a) (−1, 1) (b) (0, 1) ∪ (1, ∞) (c) (−∞, 0) (d) (−1, 0) ∪ (0, 1) (e) (−∞, −1) ∪ (−1, 0)

8. Which of the following is \frac{f(x + h) − f(x)}{h} if f(x) = x^{2} + 7x ? Note that you are not being asked to compute the limit as h → 0 .

(a) 2xh + h^{2}  (b) 2x + 7 + h  (c) 2x + 7h  (d) 2x + 7  (e) 2x + 7x + h

9. Let

f(x) = (sin(2x))^{3}.

Which of the following is f′(x) ?

(a) 3(sin(2x))^{2} cos(2x) (b) (sin(2x))^{2} (c) −6(sin(2x))^{2} cos(2x) (d) 3 sin(x) cos(x)

(e) 6(sin(2x))^{2} cos(2x)

10. Which of the following equals

∫ (cos(x) + 5)^{3} sin(x)dx ?
(a) −3(cos(x) + 5)^{2} (b) −\frac{(cos(x) + 5)^{4}}{4} + C (c) −\frac{(cos(x) + 5)^{4}}{5}  (d) \frac{(cos(x) + 5)^{4}}{4} + C

(e) \frac{(cos(x) + 5)^{4}}{4}

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

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

12. Evaluate the integral ∫_{0}^{2π}(sin x − cos x) dx . The integral is equal to

(a) −1  (b) −2  (c) 0  (d) 1  (e) 2

13. Find the area enclosed by the curves y = x and y = x^{3} , between x = 0 and x = 1 .

(a) \frac{3}{4}  (b) −\frac{1}{4}  (c) \frac{1}{4}  (d) \frac{1}{2}  (e) 1

14. A particle is moving in a straight line with position function s(t) = \frac{t^{3}}{6} . What is the average velocity over the time period [1, 3] .

(a) 13  (b) \frac{13}{6}  (c) \frac{13}{3}  (d) 4  (e) \frac{9}{2}

15. Find the equation of the tangent line to the curve x^{2} + y^{2} = 169 at the point (12, 5) .

(a) y = \frac{−5}{12}x (b) y = 2.4x + 23.8 (c) y = −2.4x + 33.8 (d) y = \frac{5}{12}x (e) y = −2x + 13

16. Find horizontal and vertical asymptotes to f(x) = \frac{1}{x} + \frac{1}{x + 1} . (a) vertical asymptotes at x = 0, horizontal asymptotes at y = 0

(b) vertical asymptotes at x = 0 and x = −1, horizontal asymptotes at y = 0

(c) vertical asymptotes at x = 0 and x = 1, horizontal asymptotes at ∞

(d) vertical asymptotes at x = 0 and x = −1, horizontal asymptotes at y = 1

(e) vertical asymptotes at x = 0, horizontal asymptotes at y = −1

17. Evaluate the integral ∫_{0}^{1} ^{3} \sqrt{x}(x − 1)dx .

(a) \frac{−9}{28}  (b) −6  (c) \frac{11}{28}  (d) \frac{−3}{7}  (e) \frac{3}{7}

18. Evaluate the integral ∫ (1 − x)\sqrt{2x − x^{2}}dx .

(a) (2 − x)\frac{3}{2} + C  (b) \frac{1}{3}(2x − x^{2})\frac{3}{2} + C  (c) (2x − x^{2})^{− \frac{1}{2}} + C  (d) −(2x − x^{2})^{− \frac{1}{2}} + C (e) (2x − x^{2})\frac{3}{2} + C

19. Which of the following numbers is equal to the value of the limit

lim_{x→4} \frac{\sqrt{x} − 2}{x − 4}
(a) 1/4  (b) 1/2  (c) 2  (d) 1  (e) 4

20. Which of the following limits is equal to the derivative of f(x) = \frac{1}{\sqrt{x + 1}} at x = 3 ?

(a) f′(3) = lim_{h→0} \frac{\frac{1}{\sqrt{1+x+h}} − \frac{1}{2}}{h} (b) f′(3) = lim_{h→3} \frac{\frac{1}{\sqrt{1+h}} − \frac{1}{2}}{h} (c) f′(3) = lim_{h→0} \frac{\frac{1}{\sqrt{1+h}} − \frac{1}{2}}{h}
(d) f′(3) = lim_{h→3} \frac{\frac{1}{\sqrt{4+h}} − \frac{1}{2}}{h}  (e) f′(3) = lim_{h→0} \frac{\frac{1}{\sqrt{4+h}} − \frac{1}{2}}{h}

21. Which one of the following functions F (x) satisfies F′(x) = 1/x for x > 0 ?

(a) F (x) = ∫_{1}^{x} \frac{1}{t + 1}dt − 1  (b) F (x) = −\frac{1}{x^{2}}  (c) F (x) = \frac{1}{x^{2}}  (d) F (x) = ∫_{1}^{x} −\frac{1}{t^{2}} dt (e) F (x) = ∫_{1}^{x} \frac{1}{t}dt

22. The product of two positive numbers is 900 . What is the minimum their sum can be?

(a) 0  (b) −30  (c) 60  (d) 30  (e) −60

23. Evaluate the integral ∫_{−1}^{0}(x + 2)^{3} dx

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

24. A balloon is rising vertically from a point A . It has constant speed of 5m/sec. A girl is watching from a position exactly 80m from A . How fast is the distance between the girl and the balloon increasing when the balloon is 60m high?

(a) 5\frac{m}{s}  (b) 3\frac{m}{s}  (c) 9\frac{m}{s}  (d) 11\frac{m}{s}  (e) 7\frac{m}{s}

25. On what intervals is f(x) = \frac{1}{4x − 7} concave up?

(a) (−∞, \frac{7}{4})  (b) (−∞, \frac{7}{4}) ∪ (\frac{7}{4}, ∞)  (c) (−∞, ∞)  (d) (0, ∞)  (e) (\frac{7}{4}, ∞)

Math 119 Calculus, Final December 15, 2000

This Examination contains 25 problems, worth a total of 150 points, on 11 sheets of paper including the front cover. Each Problem is worth 6 points. Please cross × the correct answers for the multiple choice questions. Books and notes are not allowed. You may use your calculator.

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

1. a  b  c  •  e
2. a  b  c  •  e
3. a  b  •  d  e
4. a  b  c  •  e
5. •  b  c  d  e
6. •  b  c  d  e
7. a  b  c  d  •
8. a  •  c  d  e
9. a  b  c  d  •
10. a  •  c  d  e
11. a  b  •  d  e
12. a  b  •  d  e
13. a  b  •  d  e
14. a  •  c  d  e
15. a  b  •  d  e

16. a  •  c  d  e
17. •  b  c  d  e
18. a  •  c  d  e
19. •  b  c  d  e
20. a  b  c  d  •
21. a  b  c  d  •
22. a  b  •  d  e
23. a  b  c  •  e
24. a  •  c  d  e
25. a  b  c  d  •

Total

(a) −2x^{−3}f′(x) + x^{−2}f(x) (b) \frac{x^{2}f′(x) − 2xf(x)}{x^{4}} (c) \frac{2xf(x) − x^{2}f′(x)}{x^{4}} (d) 2xf(x) + \frac{f′(x)}{x^{3}} (e) 2xf(x) − \frac{f′(x)}{x^{3}}

2. Assume x ∈ [0, \frac{π}{2} ] . If sin x = \frac{3}{5} then tan x =

(a) \frac{3}{4}  (b) \frac{5}{4}  (c) \frac{4}{5}  (d) \frac{4}{3}  (e) \frac{5}{3}

3. Suppose that f = g ∘ h + h ∘ g , g(0) = 1 , h(0) = 2 , g′(0) = 3 , g′(2) = 5 , h′(0) = 6 , h′(1) = 7 then f′(0) =

(a) 51  (b) 31  (c) 21  (d) 41  (e) 11

4. Which of the following quantities is equal to

lim_{h→0} \frac{(2 + h)^{3} − 8}{h}
(a) 1  (b) f′(2), f(x) = x^{3}  (c) 0  (d) 3h^{2}  (e) f′(h), f(x) = x^{2}

5. The volume V of a cube is increasing at a rate of 300cm^{3} per minute. How fast is the length L of each side increasing where L = 10 cm ?

(a) 3cm/min  (b) 30cm/min  (c) 10cm/min  (d) 1cm/min  (e) 100cm/min

6. Compute the indefinite integral

∫ \frac{2x}{(x^{2} + 1)^{5}} dx =

(a) \frac{(x^{2} + 1)^{4}}{4} + C  (b) \frac{1}{x^{4}} + C  (c) −\frac{1}{4(x^{2} + 1)^{4}} + C  (d) \frac{1}{4(x^{2} + 1)^{4}} + C (e) −\frac{(x^{2} + 1)^{4}}{4} + C

7. The function f(x) = \frac{x^{2}}{x^{2} − 1} is increasing on the interval(s):

(a) (0, 1) ∪ (1, ∞) (b) (−∞, −1) ∪ (−1, 0) (c) (−1, 1) (d) (−1, 0) ∪ (0, 1) (e) (−∞, 0)

8. Which of the following is \frac{f(x + h) − f(x)}{h} if f(x) = x^{2} + 7x ? Note that you are not being asked to compute the limit as h → 0 .

(a) 2xh + h^{2}  (b) 2x + 7x + h (c) 2x + 7 + h  (d) 2x + 7  (e) 2x + 7h

9. Let

f(x) = (sin(2x))^{3}.

Which of the following is f′(x) ?

(a) 6(sin(2x))^{2} cos(2x) (b) (sin(2x))^{2} (c) 3(sin(2x))^{2} cos(2x) (d) −6(sin(2x))^{2} cos(2x)

(e) 3 sin(x) cos(x)

10. Which of the following equals

∫ (cos(x) + 5)^{3} sin(x)dx ?
(a) −\frac{(cos(x) + 5)^{4}}{5} (b) −3(cos(x) + 5)^{2} (c) \frac{(cos(x) + 5)^{4}}{4} + C (d) \frac{(cos(x) + 5)^{4}}{4}

(e) −\frac{(cos(x) + 5)^{4}}{4} + C

11. 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) 74  (c) 9  (d) 70  (e) 18

12. Evaluate the integral ∫_{0}^{2π}(sin x − cos x) dx . The integral is equal to

(a) 1  (b) 0  (c) 2  (d) −2  (e) −1

13. Find the area enclosed by the curves y = x and y = x^{3} , between x = 0 and x = 1 .

(a) 1  (b) −\frac{1}{4}  (c) \frac{3}{4}  (d) \frac{1}{2}  (e) \frac{1}{4}

14. A particle is moving in a straight line with position function s(t) = \frac{t^{3}}{6} . What is the average velocity over the time period [1, 3] .

(a) \frac{13}{3}  (b) \frac{13}{6}  (c) \frac{9}{2}  (d) 13  (e) 4

15. Find the equation of the tangent line to the curve x^{2} + y^{2} = 169 at the point (12, 5) .

(a) y = −2x + 13 (b) y = 2.4x + 23.8 (c) y = −2.4x + 33.8 (d) y = \frac{−5}{12}x (e) y = \frac{5}{12}x

16. Find horizontal and vertical asymptotes to f(x) = \frac{1}{x} + \frac{1}{x + 1} . (a) vertical asymptotes at x = 0 and x = 1, horizontal asymptotes at ∞

(b) vertical asymptotes at x = 0, horizontal asymptotes at y = −1

(c) vertical asymptotes at x = 0 and x = −1, horizontal asymptotes at y = 1

(d) vertical asymptotes at x = 0 and x = −1, horizontal asymptotes at y = 0

(e) vertical asymptotes at x = 0, horizontal asymptotes at y = 0

17. Evaluate the integral ∫_{0}^{1} ^{3} \sqrt{x}(x − 1)dx .

(a) \frac{−9}{28}  (b) −6  (c) \frac{11}{28}  (d) \frac{−3}{7}  (e) \frac{3}{7}

18. Evaluate the integral ∫ (1 − x)\sqrt{2x − x^{2}}dx .

(a) (2 − x)\frac{3}{2} + C  (b) −(2x − x^{2})^{− \frac{1}{2}} + C  (c) \frac{1}{3}(2x − x^{2})\frac{3}{2} + C  (d) (2x − x^{2})^{− \frac{1}{2}} + C (e) (2x − x^{2})\frac{3}{2} + C

19. Which of the following numbers is equal to the value of the limit

lim_{x→4} \frac{\sqrt{x} − 2}{x − 4}
(a) 2  (b) 1/2  (c) 4  (d) 1/4  (e) 1

20. Which of the following limits is equal to the derivative of f(x) = \frac{1}{\sqrt{x + 1}} at x = 3 ?

(a) f′(3) = lim_{h→0} \frac{\frac{1}{\sqrt{1+h}} − \frac{1}{2}}{h} (b) f′(3) = lim_{h→0} \frac{\frac{1}{\sqrt{1+x+h}} − \frac{1}{2}}{h} (c) f′(3) = lim_{h→0} \frac{\frac{1}{\sqrt{4+h}} − \frac{1}{2}}{h}
(d) f′(3) = lim_{h→3} \frac{\frac{1}{\sqrt{1+h}} − \frac{1}{2}}{h}  (e) f′(3) = lim_{h→3} \frac{\frac{1}{\sqrt{4+h}} − \frac{1}{2}}{h}

21. Which one of the following functions F (x) satisfies F′(x) = 1/x for x > 0 ?

(a) F (x) = −\frac{1}{x^{2}}  (b) F (x) = ∫_{1}^{x} \frac{1}{t}dt  (c) F (x) = ∫_{1}^{x} −\frac{1}{t^{2}} dt (d) F (x) = \frac{1}{x^{2}} (e) F (x) = ∫_{1}^{x} \frac{1}{t + 1}dt − 1

22. The product of two positive numbers is 900 . What is the minimum their sum can be?

(a) −60  (b) 30  (c) 0  (d) −30  (e) 60

23. Evaluate the integral ∫_{−1}^{0}(x + 2)^{3} dx

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

24. A balloon is rising vertically from a point A . It has constant speed of 5m/sec. A girl is watching from a position exactly 80m from A . How fast is the distance between the girl and the balloon increasing when the balloon is 60m high?

(a) 11\frac{m}{s}  (b) 7\frac{m}{s}  (c) 5\frac{m}{s}  (d) 9\frac{m}{s}  (e) 3\frac{m}{s}

25. On what intervals is f(x) = \frac{1}{4x − 7} concave up?

(a) (−∞, ∞)  (b) (\frac{7}{4}, ∞)  (c) (−∞, \frac{7}{4}) ∪ (\frac{7}{4}, ∞)  (d) (0, ∞)  (e) (−∞, \frac{7}{4})

Math 119 Calculus, Final December 15, 2000

This Examination contains 25 problems, worth a total of 150 points, on 11 sheets of paper including the front cover. Each Problem is worth 6 points. Please cross × the correct answers for the multiple choice questions. Books and notes are not allowed. You may use your calculator.

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

1. a  •  c  d  e
2. •  b  c  d  e
3. •  b  c  d  e
4. a  •  c  d  e
5. a  b  c  •  e
6. a  b  •  d  e
7. a  •  c  d  e
8. a  b  •  d  e
9. •  b  c  d  e
10. a  b  c  d  •
11. a  b  c  •  e
12. a  •  c  d  e
13. a  b  c  d  •
14. a  •  c  d  e
15. a  b  •  d  e

16. a  b  c  •  e
17. •  b  c  d  e
18. a  b  •  d  e
19. a  b  c  •  e
20. a  b  •  d  e
21. a  •  c  d  e
22. a  b  c  d  •
23. •  b  c  d  e
24. a  b  c  d  •
25. a  •  c  d  e

Total