For your solutions to problems (15) and (16) below, include:



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.

(b) State the zeros (or roots) of f.

(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.

18. Use the Left-Endpoint Rule with n = 4 subdivisions to approximate [pic].

19. Determine the area of the region in the 4th quadrant bounded by the curves

[pic].

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:

[pic]

(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 [pic].

(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 [pic] 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.

[pic]

A) None B) only I C) only II D) only III

E) only I and II F) only I and III G) only II and III H) I, II, and III

25. Determine whether the series [pic] is convergent, absolutely convergent, 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 [pic]. Provide complete and appropriate justification for your responses.

BONUS #1

Show that the function

[pic]

is a solution to the differential equation

[pic].

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 50 points

Questions 1 through 10

2 points each with no partial credit. No need to show any work on these.

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.

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