MAT 146
Semester Exam Part II
Name___________________
100 points (Part II: 50 points)
Calculator Used___________________
Impact on Course Grade: approximately 30%
Score___________________
Questions (17) through (26) are each worth 5 points. See the grading rubric for further details.
17. The graph of a 4th–
degree polynomial
function f is shown here.
Use it to determine
responses to the
following questions.
(a) State the global
minimum of f. ______
k
m
a
b
(b) State the zeros (or
roots) of f.
______
c
d
e
f
p
r
(c) State all intervals over which f is increasing.
______________
(d) State all intervals over which f is concave down.
______________
(e) State all intervals over which f is negative.
______________
5
18. Use the Left-Endpoint Rule with n = 4 subdivisions to approximate
 12x
3
2
dx .
19. Determine the area of the region in the 4th quadrant bounded by the curves
y  x 4  x 2 and y  5x 2  5x .
Include a sketch to illustrate the situation, show your steps leading to solution, and
express your solution as an exact rational number.
20. Assume that the daily consumption of electric power (in millions of kilowatt-hours)
of a certain city has the following probability density function:
x

1 xe  3 if x  0
p(x)  9

if x  0
0
(a) What is the probability that the city’s daily power consumption will range between
6 million and 9 million kilowatt-hours?
(b) If the city’s power supply has a daily capacity of 12 million kilowatt-hours, what is
the probability that the available power supply will be inadequate on any given day?
(c) State the two characteristics of p(x) that assure it is a probability density function.
21. Consider the differential equation


dy
 3xy 2 1 x 2 .
dx
(a) Determine the general solution to the differential equation.
(b) Determine the particular solution that satisfies the initial condition y(1) = 4.
22. According to Newton’s Law of Cooling, the temperature T of a warm object
decreases at a rate proportional to the difference between T and the temperature T0 of its
surroundings. If the room temperature is 70° F, and we know in this room it takes
2 minutes for a cup of hot coffee whose initial temperature is 200° F to cool down to
180° F, determine how long it will take for the coffee to cool from 200° F to 100° F.
Express your solution to the nearest hundredth of a minute.
23. State and solve an inequality, involving w, to describe the conditions under which

the series

k1
1
k w 1
will diverge.
24. Which of the three infinite series shown here will converge? Choose the most
appropriate response from (A) through (H) and then briefly explain your choice.

(I)

n1
A) None
E) only I and II
1
n4

(II)

n1
B) only I
F) only I and III
n2
(III)
2n 2 +1

  2
n
n1
C) only II
G) only II and III
D) only III
H) I, II, and III

2n 2 1
25. Determine whether the series 
is convergent, absolutely convergent,
3
5n

n

4
n1
or divergent. Provide complete and appropriate justification for your response by using
one or more tests for convergence.
26. Determine the radius of convergence and the interval of convergence for the power
series given by f (x ) 
your responses.

1n
n1
n

x n . Provide complete and appropriate justification for
BONUS #1
Show that the function
1n x 2n
f x   
n0 2n!

is a solution to the differential equation
f  x  f  x  0 .
BONUS #2
Two bicyclists are 40 miles apart, riding toward each other on a straight line. Each
bicyclist travels at 20 miles per hour. At the very instant the cyclists are 40 miles apart, a
fly starts at one bicyclist and flies toward the other bicyclist at 60 miles per hour. When it
reaches that bike, it turns around and flies back to the other bike. It continues flying back
and forth in this manner until the bicyclists meet.
Determine (a) the distance flown on each leg of the fly’s journey and (b) then create an
infinite series and calculate its value to determine the total distance flown.
Calculus II
MAT 146
Semester Exam
Total Points:
100
Impact of Exam on Semester Grade:
Approximately 30%
Evaluation Criteria
State any numerical solutions as exact values in rational expressions reduced to lowest
terms. If approximations are required, express as a decimal value rounded accurately to
the nearest thousandth of a unit.
Part I: No Calculators
Questions 1 through 10
2 points each with no partial credit. No need to show any work on these.
50 points
Questions 11 through 16
5 points each. Partial credit is possible. Show all steps leading to your solutions. Be clear,
complete, and accurate.
11) 3 pts: complete steps to solution; 2 pts: correct solution
12) 3 pts: complete steps to solution; 1 pt: correct integral evaluation; 1 pt: correct numerical solution
13) 2 pts: set up correct integral; 2 pts: complete steps to solution; 1 pt: correct numerical result
14) 4 pts: complete and accurate integral set-up; 1 pt: appropriate sketch
15) 3 pts: identify desired solutions; 2 pts: explain your choice
16) 3 pts: identify divergent series; 2 pts: explain your choice
Part II: Calculators Allowed
50 points
Questions 17 through 26 are worth 5 points each. Partial credit is possible. Show all steps
leading to your solutions. Be clear, complete, and accurate.
17) 1 pt each
18) 3 pts: correct set-up and application of the Left-Endpoint Rule; 2 pts: correct numerical approximation
19) 2 pts: complete steps to solution; 2 pts: correct numerical result; 1 pt: appropriate sketch
20) (a) and (b) 2 pts each: correct numerical solutions; (c) 1 pt: correct two criteria
21) (a) 3 pts (b) 2 pts
22) 3 pts: correct set-up and application of Newton’s Law; 2 pts: correct numerical result
23) 5 pts for correct inequality involving w
24) 3 pts: correctly identifying status of each series; 2 pts: explanations
25) 1 pt: correctly state convergence, absolute convergence, or divergence; 4 pts: complete and appropriate justification
26) 3 pts: radius of convergence; 2 pts: interval of convergence
Bonus #1: 10 points: Show complete, accurate, and justified response.
Bonus #2: 5 points: Show complete, accurate, and justified response.