Math 1100
Compiled Midterm Exams
13 December, 2013
Test 1
1. Fill in the blanks:
(a) A derivative represents the instantaneous
of a function at a point.
(b) A derivative is the slope of the
tion at a point.
to a func-
(c) The derivative of a revenue function represents
2. Use the graph of f (x) below to compute each of the following. Write DNE if it does not
exist.
(a) f (−1) =
(e) f (2) =
(b)
(f) lim− f (x) =
(c)
lim f (x) =
x→−1−
lim f (x) =
x→−1+
(d) lim f (x) =
x→−1
x→2
(g) lim+ f (x) =
x→2
(h) lim f (x) =
x→2
.
Math 1100
Exam 1, Page 2 of 9
25 September, 2013
3. Determine whether each of the following graphs is continuous at x = 1. If not, explain
why not. Your explanation should reference the definition of continuity and/or the three
conditions required. (Write your answer to the right of each graph.)
(a) .
(b) .
(c) .
Math 1100
Exam 1, Page 3 of 9
25 September, 2013
4. Compute the following limits or state that they do not exist.
(a)
lim 4x3 − 2x2 + 1
x→1
(b)
x2 + 16
x→4 x − 4
lim
(c)
x2 − 9
lim
x→−3 x + 3
5. Use the definition of a derivative (not derivative rules) to compute the derivative of
f (x) = 3x − 1
6. Differentiate each function. Use any method you like.
(a)
y = x4 −
√
1
+ 3x
3
x
(b)
g(x) =
x2 + 1
3x
(c)
f (t) = (t3 − 4t)6
(d)
y = (x2 + 1)5 · (2x + 1)
7. Let y =
√
x + 2x.
(a) Find
dy
.
dx
(b) Find
d2 y
.
dx2
(c) Find
d3 y
.
dx3
8. Evaluate the following limit. Show your work. Your answer will be a function of x and
should be simplified.
(x + h)8 − x8
lim
h→0
h
Math 1100
Exam 1, Page 4 of 9
25 September, 2013
9. Samantha Carter wants to open a restaurant that serves blue Jell-o. She estimates that
her profit in dollars from selling x servings of blue Jell-o will be
P (x) =
−3 2
x + 4x − 40
400
For reference: 202 = 400
(a) What is Carter’s profit on 20 servings of blue Jell-o?
(b) What is Carter’s marginal profit function?
(c) What is Carter’s marginal profit at x = 20?
(d) Use your answer to part (c) to estimate Carter’s profit from 30 servings of blue
Jell-o.
10. Colonel O’Neill is flying an X-301 aircraft. Its height in meters after t seconds is given
by
h(t) = 6t2 + 30t + 15
(a) How high is Colonel O’Neill after 10 seconds?
(b) What is Colonel O’Neill’s velocity after 10 seconds?
(c) What is Colonel O’Neill’s acceleration after 10 seconds?
(d) What do you think would happen if Colonel O’Neill and Samantha Carter teamed
up to fight aliens? You may answer in words or pictures. Use the back of this sheet
if you need to.
Math 1100
Exam 1, Page 5 of 9
25 September, 2013
Test 2
For questions 1-4, assume the graph below represents f (x).
10
5
-15
-10
-5
0
5
10
15
-5
-10
11. Point A is:
13. At point A, f 00 is:
A. A relative maximum
A. positive
B. A relative minimum
B. negative
C. A horizontal point of inflection
C. zero
D. None of the above
14. At point B, the graph is:
12. At point A, f 0 (x) is:
A. increasing and concave up
A. positive
B. increasing and concave down
B. negative
C. decreasing and concave up
C. zero
D. decreasing and concave down
Math 1100
Exam 1, Page 6 of 9
25 September, 2013
For questions 5 and 6, assume the graph below represents f 0 (x).
10
5
-15
-10
-5
0
5
10
15
-5
-10
15. Which point corresponds to a maximum of f (x)? (Remember the graph represents
f 0 (x))
A. Point A
B. Point B
C. Point C
D. Point D
E. None of the above
16. At point D, the graph of f (x) (not the graph of f 0 (x)) is:
A. increasing and concave up
B. increasing and concave down
C. decreasing and concave up
D. decreasing and concave down
Math 1100
17. Find
dy
dx
Exam 1, Page 7 of 9
25 September, 2013
for each relation. Your answers should be simplified enough to be easily readable.
(a)
y = ln(2x2 + 2)
(b)
x
y = e x+1
(c)
x2 y = 2y + 3x + 4
18. A company can produce x children’s Batman costumes at cost C(x) = 5 + 15x. The
1
revenue function from selling x Batman costumes is R(x) = 16x − 100
x2 .
(a) How many Batman costumes should the company produce to maximize profit?
(b) What is the maximum possible profit for the company?
19. The volume of a spherical pumpkin is increasing at the rate of 4 in3 /day. How fast is
the radius of the pumpkin increasing when the radius is 5 inches? (Note: the volume of
a sphere is given by V = 34 πr3 ).
Test 3
20. If F 0 (x) = f (x), then the Fundamental Theorem of Calculus states that:
Z
b
f (x) dx =
a
21. The graph below represents the function g(x) with the lines x = 1 and x = 4. Write an
integral representing the shaded area.
0
Math 1100
Exam 1, Page 8 of 9
25 September, 2013
22. Evaluate the following integral:
Z
"
!
#
r
d
7x
+
3
ln 3
+ 4 dx
dx
x+2
Write a few words explaining your answer.
Note: This is a conceptual problem, not a computational one. You should not need extra
space.
23. Evaluate each indefinite integral. Remember the constant of integration!
(a)
Z
2x(x2 + 4)11 dx
(b)
x3
dx
2x4 − 3
Z
24. Evaluate each definite integral.
(a)
Z
9
1
√ dx
x
1
(b)
1
Z
xe(2x
2 −2)
dx
0
25. Evaluate the integral if it converges.
Z
1
∞
2
dx
x3
26. Find the area of the shaded region.
-5
0
The two equations graphed are y = 1 and y = 3 − 21 x2 .
5
Math 1100
Exam 1, Page 9 of 9
25 September, 2013
27. Find the general solution to the following differential equation.
xy
dy
=1
dx
28. Find the solution to the following differential equation that passes through the point
(3,2).
dy
x2 + 3
=
dx
3y 2
29. Sally wants to breed a particularly fancy type of turtle. Suppose the marginal cost for
x turtles is M C = 4x + 50, the marginal revenue is M R = 500, and the cost of raising
10 turtles is $1000. (The revenue from 0 turtles is 0, of course.)
(a) What is the total revenue function for Sally’s turtles?
(b) What is the total cost function for Sally’s turtles?
(c) What is the total profit function for Sally’s turtles?