Name: Instructor:

Math 119, Calculus Fall Semester 2000 Exam 1 Thursday, Sept 21

This Examination contains 16 problems, worth a total of 100 points, on 9 sheets of paper including the front cover. The first 12 problems (Section A) are multiple choice with no partial credit, and each is worth 5 points. Record your answers to these problems by placing an × through one letter for each problem below:

1. \underline{\overline{a}} \underline{\overline{b}} \underline{\overline{c}} \underline{\overline{d}} \underline{\overline{e}}
2. \underline{\overline{a}} \underline{\overline{b}} \underline{\overline{c}} \underline{\overline{d}} \underline{\overline{e}}
3. \underline{\overline{a}} \underline{\overline{b}} \underline{\overline{c}} \underline{\overline{d}} \underline{\overline{e}}
4. \underline{\overline{a}} \underline{\overline{b}} \underline{\overline{c}} \underline{\overline{d}} \underline{\overline{e}}
5. \underline{\overline{a}} \underline{\overline{b}} \underline{\overline{c}} \underline{\overline{d}} \underline{\overline{e}}
6. \underline{\overline{a}} \underline{\overline{b}} \underline{\overline{c}} \underline{\overline{d}} \underline{\overline{e}}

7. \underline{\overline{a}} \underline{\overline{b}} \underline{\overline{c}} \underline{\overline{d}} \underline{\overline{e}}
8. \underline{\overline{a}} \underline{\overline{b}} \underline{\overline{c}} \underline{\overline{d}} \underline{\overline{e}}
9. \underline{\overline{a}} \underline{\overline{b}} \underline{\overline{c}} \underline{\overline{d}} \underline{\overline{e}}
10. \underline{\overline{a}} \underline{\overline{b}} \underline{\overline{c}} \underline{\overline{d}} \underline{\overline{e}}
11. \underline{\overline{a}} \underline{\overline{b}} \underline{\overline{c}} \underline{\overline{d}} \underline{\overline{e}}
12. \underline{\overline{a}} \underline{\overline{b}} \underline{\overline{c}} \underline{\overline{d}} \underline{\overline{e}}

The last 4 problems (Section B) 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.

Sign the pledge: “On my honor, I have neither given nor received unauthorized aid on this Exam”:

GOOD LUCK

Part A: Multiple Choice Problems

1. (5 pts.) 1. Which of the following lines is parallel to the line y = \frac{3}{2}x + 5 ?

a) y = 3 b) 2y + 3x − 5 = 0 c) 3y − 2x − 5 = 0 d) 3y + 2x − 11 = 0 e) 2y − 3x − 7 = 0

2. (5 pts.) What is the domain of the function f(x) = \sqrt[4]{8 − 4x^{2}} ?

a) −2 < x < 2 b) x ≤ −\sqrt{2} or x ≥ \sqrt{2} c) −\sqrt{2} ≤ x ≤ \sqrt{2} d) All real numbers e) x ≤ −2 or x ≥ 2.

3. (5 pts.) Find the radius of the circle

2x^{2} + 12x + 2y^{2} + 4y = −16

a) 0  b) 2  c) \sqrt{−16}  d) 4  e) \sqrt{2}.

4. (5 pts.) Which of the following is the value of the limit

lim_{x→4} \frac{x^{2} − 16}{\sqrt{x} − 2} ?

a) 32  (b) 16  (c) 8  (d) 2  (e) 1.

5. (5 pts.) The height f(t) in feet of a ball t seconds after it is thrown vertically into the air is given by the formula f(t) = 64t − 16t^{2}. Find the average velocity (in feet per second) of the ball over the time period from t = 1 to t = 2.

a) 64  b) −32  c) 48  d) 16  e) −48.

6. (5 pts.) Let

f(x) = \{ \begin{matrix}4x^{2} − c, & if x < 1 \\ cx, & if x ≥ 1 \end{matrix}

where c is a constant. What value of c makes f(x) continuous?

a) -2  b) 0  c) 1  d) 2

e) No value of c makes f(x) continuous.

7. (5 pts.) A particle moves so that its position at time t is given by

s(t) = 2t^{2} + \frac{4}{\sqrt[3]{t^{2}}} .

Which of the following is an expression for the instantaneous velocity of the particle at time t?

a) 4t + 12t^{\frac{1}{3}}  b) 4t − \frac{8}{3}t^{− \frac{5}{3}}  c) 4t + \frac{8}{3}t^{− \frac{5}{3}}
d) 4t − 12t^{\frac{5}{3}} e) 4t − 12t^{− \frac{1}{3}} .

8. (5 pts.) If f(x) = \frac{1}{\sqrt{x − 1}}, which of the following equals \frac{f(3 + h) − f(3)}{h}?

a) \frac{−1}{(\sqrt{3} + \sqrt{3 + h})\sqrt{3}\sqrt{3 + h}} b) \frac{−1}{(\sqrt{2} + \sqrt{2 + h})\sqrt{2}\sqrt{2 + h}}

c) \frac{−1}{(\sqrt{2} − \sqrt{2 + h})\sqrt{2}\sqrt{2 + h}} d) \frac{−\sqrt{2 + h}}{(\sqrt{2} + \sqrt{2 + h})\sqrt{2}}

e) \frac{1}{(\sqrt{2} + \sqrt{2 + h})\sqrt{2}\sqrt{2 + h}}

9. (5 pts.) If h(x) = \frac{x − 1}{x^{2} + 1}, find h′(x).

a) \frac{x^{2} + 2x + 1}{x^{2} + 1} b) \frac{x^{2} + 2x + 1}{(x^{2} + 1)^{2}} c) \frac{−x^{2} + 2x + 1}{(x^{2} + 1)^{2}}

d) \frac{−x^{2} + 2x + 1}{x^{2} + 1} e) \frac{3x^{2} + 2x + 1}{(x^{2} + 1)^{2}}

10. (5 pts.) Let f(x) denote any differentiable function of x. Which of the following is an expression for the derivative g′(x) of the function g(x) = x^{3}f(x)?

a) x^{3}f′(x) + 3x^{2}f(x) b) 3x^{2}f′(x) + x^{3}f(x) c) x^{3}f′(x) + x^{2}f(x) d) x^{2}f′(x) + x^{3}f(x) e) x^{3}f′(x) + x^{3}f(x).

11. (5 pts.) If sin(θ) = \frac{3}{4}, and π/2 < θ < π, what is cos(θ)?

a) −\frac{1}{4}  b) \frac{\sqrt{7}}{4}  c) −\frac{\sqrt{7}}{4}  d) \frac{1}{4}

e) Cannot be determined

12. (5 pts.) Let the number x of heartbeats of a mouse after t seconds be given by the following values:

\begin{matrix}t & 17 & 18 & 19 & 20 & 21 & 22 & 23 \\ x & 25 & 29 & 34 & 40 & 48 & 57 & 68 \end{matrix}

Which of the following is the best estimate for the rate of heartbeats of the mouse when t = 20?

a) 4 beats/second  b) 7 beats/second  c) 11 beats/second

d) 40 beats/second e) 0 beats/second

Part B: Partial Credit Problems

13. (10 pts.) Consider the graph of the function f(x) pictured below.

For each of the following, either give the value of the limit, or indicate why it does not exist.

a) lim_{x→5^{+}} f(x)

b) lim_{x→5^{−}} f(x)

c) lim_{x→5} f(x)

d) lim_{x→2}f(x)

14. (10 pts.) Let f(x) = \frac{1}{x + 2}.

a) Find the slope m(h) of the secant line joining the points (1, f(1)) and (1 + h, f(1 + h))

on the graph of y = f(x). Answer:

b) Using your answer to (a), compute the value of the limit

lim_{h→0} m(h)

using only algebra and the limit laws. Answer:

15. (10 pts.) Let f(x) = | x | − | 1 − x |.

a) Complete the following table of values of the function f(x).

x  -2 -1 0 1 2
f(x)

b) Graph y = f(x) on the interval −2 ≤ x ≤ 2. Make sure to include a scale on the x and y axes.

16. (10 pts.) Find the equation of the tangent line to y = \sqrt{x − 1} at the point (5, 2).

Answer: