MATH 317: SPRING 2016
EXAM 1
RULES
• No books or notes or calculators allowed.
• No bathroom breaks until after you have completed and turned in your test.
• Out of consideration for your classmates, do not make disturbing noises during the exam.
• Phones and other electronic devices must be off or in silent mode.
NAME:
Solutions
SID #:
Cheating will not be tolerated. If there is any indication that a student may have given or received unauthorized aid on this test, the case will be handed over to the Dean’s Office. When
you finish the exam, please sign the statement on the last page of the test acknowledging
that you understand this policy.
- scratch -
Part 1. Complete the table by blacking out letters corresponding to correct answers. If the
expression you are asked to compute is not defined or does not appear in the list of possible
answers, choose “none of these.”
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)
1
, then the value of g(f (−1)) is
(3pts) 1. If f (x) = (x − 1)3 and g(x) = √9−2x
√
√
(a) −1/ 7
(b) 1/ 17
(c) 1/5
(d)1/3
(3pts) 2. Simplify the expression
(a) 2r2 s6 t4
√
3
(e) none of these
√
8r6 s4 t6 . (Assume that r, s, and t are positive.)
(b) 8r2 s2 t3
(c) 2r2 s6 t3
(d)2r2 s2 t3
(e) none of these
(3pts) 3. Find the domain of the function
√
x−6
g(x) =
.
x(x − 8)
(a) [6, 8) ∪ (8, ∞) (b) [6, ∞) (c) (8, ∞) (d) (−∞, 0) ∪ (0, 8) ∪ (8, ∞) (e) (−∞, ∞)
(3pts) 4. Which function(s) has its domain identical with its range? (select all that apply)
√
(e) none of these
(a) f (x) = 1/x
(b) g(x) = x
(c) h(x) = x2
(d) i(x) = x
(3pts) 5. Find the set of all x values where the function
f (x) =
(a) (−∞, ∞)
x2 − 4
x−2
is continuous.
(b) (−∞, −2)
(c) (2, ∞)
(d) (−∞, 2) ∪ (2, ∞)
Score for this page:
(e) none of these
out of 15
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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)
(3pts) 6. The number of acres in a region cleared for farming follows the formula A = f (t) = 2t2
where t is the number of years since the region started to be farmed and t ranges from
t = 0 to t = 10. Find the average rate of change in the number of acres cleared for
farming between t = 1 and t = 4, with appropriate units.
(a) 10 acres/year
(b) 30 acres
(c) 10 years/acre
(d) 30 years
(e) 0.10 acres/year
(3pts) 7. The derivative of f (x) at x = a is denoted f 0 (a) and represents which of the following?
Circle the letter(s) next to the correct choice(s); circle all that apply!)
1. The average rate of change of f (x) over a small interval to the right of a.
2. The instantaneous rate of change of f (x) at the point x = a.
3. The limit of f (x) as x approaches a.
4. The slope of the line tangent to the graph of f (x) at the point (a, f (a)).
5. The slope of the line connecting the x-axis to the point (a, f (a)).
(3pts) 8. If g(x) = 2x3 + 1, then g 0 (−2) =
(a) −12
(b) −11
(c) 0
(d) 12
(e) 24
(d) 0.2 ln x
(e) 0.2 x−0.8
(3pts) 9. If y = x0.2 , then y 0 =
(a) ln(0.2) x0.2
(b) x−0.8
(c) 0.2 x−1.2
(3pts) 10. If f (t) = 10t , then f 0 (t) =
(a) 10t ln t
(b) 10t ln(10)
(c) t10t−1
(d) t10 ln 10
Score for this page:
(e) none of these
out of 15
- scratch -
problem
answer choice
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)
x3
, then f 0 (x) =
(3pts) 11. If f (x) = x+1
(a)
3x2 (x+1)−x3
(x+1)2
(b)
x3 −3x2 (x+1)
(x+1)2
(c) 3x2 (x + 1) + x3
(d) 3x2 ln(x + 1)
1
0
(3pts) 12. If f (x) = √
3 x , then f (x) =
(a) − 13 x−2/3
(b) 32 x−2/3
(c) 13 x2/3
(d) −3x−4
(e) − 13 x−4/3
(3pts) 13. If g(t) = ln(et − 1), then g 0 (t) =
(a)
et
et −1
(b)
t
et −1
t
(c) − ete−1
1
et −1
(d)
(e)
1
ln(et −1)
(3pts) 14. If f (x) = sin(π e2x ), then f 0 (0) =
(a) −2π
(b) −1
(c) 0
(d) 1
(e) 2π
(3pts) 15. If y = tan7 (x) then y 0 =
(a) 7 tan6 (x) (b) 7 tan6 (x) sec2 (x) (c) −7 tan6 (x) csc2 (x) (d) tan6 x(7 tan2 x + 1)(e) π/2
Score for this page:
out of 15
Part 2.
(5pts) 16. Which of the following graphs (a)–(d) could represent the slope at every point of the
function graphed in Figure 2.8?
Score for this page:
out of 5
(5pts) 17. Suppose
f 0 (x) < 0, for 0 < x < 2, for 4 < x < 5, and for 6 < x.
f 0 (x) > 0, for x < 0, for 2 < x < 4, and for 5 < x < 6.
Which of the graphs (a)–(d) could be the graph of f (x)? (Select all that apply! )
Score for this page:
out of 5
18. All parts of this question concern the function f (x) =
(2pts)
1
x
(a) Compute the limit of f (x) = 1/x as x approaches infinity, as x approaches zero
from the right, and as x approaches negative infinity. (You don’t need to show your
work for this part.)
Answers: lim f (x) =
x→∞
lim f (x) =
x→0+
lim f (x) =
x→−∞
(2pts)
(b) Find the instantaneous rate of change of the function f (x) = 1/x at an arbitrary
point x = x0 by computing the following limit. (For full credit, show your work.)
f (x0 + h) − f (x0 )
h→0
h
lim
(2pts)
(c) Find the instantaneous rate of change of the function at the point x0 = 1/2 by
plugging 1/2 into the result you obtained in part (ii).
(2pts)
(d) Using this and your answer to part (iii), write down the equation
of the line tangent
1
1
to the graph of x at the point where (x, f (x)) = 2 , 2 . Put your answer in the
form y = mx + b.
Score for this page:
out of 8
19. Evaluate each limit. If the limit does not exist, write DNE. (Show your work!)
(4pts)
(a)
2x2 + 1
x→2 x2 + 5x − 4
lim
Answer:
(4pts)
(b)
lim
x→2 x2
1
−4
Answer:
(4pts)
(c)
x2 + 4x + 4
x→−2
x4 − 16
lim
Answer:
(4pts)
(d)
1 − 3y 2
y→∞ 2y 2 + 5y
lim
Answer:
Score for this page:
out of 16
(6pts) 20. Consider the graph of a function shown below. Identify the domain of the function and
the set of values at which the function is continuous. (Circle letters next to correct
answers.)
The domain of the function:
x values where the function is continuous:
1. [−4, 8)
1. [−4, 8)
2. [−4, 4) ∪ (4, 8)
2. [−4, 4) ∪ (4, 8)
3. (−∞, −4) ∪ [8, ∞)
3. (−∞, −4) ∪ [8, ∞)
4. [−4, −2) ∪ (−2, 2) ∪ [2, 6) ∪ (6, 8)
4. [−4, −2) ∪ (−2, 2) ∪ [2, 6) ∪ (6, 8)
5. [−4, −2)∪(−2, 2)∪(2, 4)∪(4, 6)∪(6, 8)
5. [−4, −2)∪(−2, 2)∪(2, 4)∪(4, 6)∪(6, 8)
Score for this page:
out of 6
MATH 215
EXAM 1
SPRING 2017
Cheating will not be tolerated. If there is any indication that a student may have given or
received unauthorized aid on this test, the case will be handed over to the Dean’s Office.
When you finish the exam, please sign the following statement acknowledging that you understand this policy:
“On my honor as a student I,
nor received unauthorized aid on this exam.”
, have neither given
(print name clearly)
Date:
Signature:
2017-02-16
Page:
3
5
7
8
9
10
11
12
Total
Points:
15
15
15
5
5
8
16
6
85
Score: