MthSc 103 Test #3 Spring 2011 Version A 3.6 Student’s Printed …mthsc.clemson.edu/ug/MthSc103/Test3_VA_AK.pdf · 2011. 11. 5. · of academic integrity. No part of this test may

MthSc 103 Test #3 Spring 2011

Version A JIT 6.1, 8.2; Calculus 2.6, 3.1 – 3.6

Page 1 of 13

Student’s Printed Name: _______________________ CUID:___________________

Instructor: ______________________ Section # :_________

You are not permitted to use a calculator on any portion of this test. You are not allowed to use

any textbook, notes, cell phone, laptop, PDA, or any technology on either portion of this test. All

devices must be turned off while you are in the testing room.

During this test, any communication with any person (other than the instructor or his designated

proctor) in any form, including written, signed, verbal, or digital, is understood to be a violation

of academic integrity.

No part of this test may be removed from the testing room.

Read each question very carefully. In order to receive full credit for the free response portion of

the test, you must:

1. Show legible and logical (relevant) justification which supports your final answer.

2. Use complete and correct mathematical notation.

3. Include proper units, if necessary.

4. Give exact numerical values whenever possible.

You have 90 minutes to complete the entire test.

On my honor, I have neither given nor received inappropriate or unauthorized information

at any time before or during this test.

Student’s Signature: ________________________________________________ Do not write below this line.

Free Response

Problem

Possible

Points

Points

Earned

Free

Response

Problem

Possible

Points

Points

Earned

1abc 5 3d 5

1d 3 4ab 6

2a 8 4cd 4

2bc 5 5 1

3a 5 Free

Response 52

3b 5 Multiple

Choice 48

3c 5 Test Total 100

ANSWER KEY AND

GRADING GUIDELINES



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Multiple Choice. There are 18 multiple choice questions. Each question is worth 2 – 3

points and has one correct answer. The multiple choice problems will count 48% of the

total grade. Use a number 2 pencil and bubble in the letter of your response on the

scantron sheet for problems 1 – 18. For your own record, also circle your choice on your

test since the scantron will not be returned to you. Only the responses recorded on your

scantron sheet will be graded. You are NOT permitted to use a calculator on any portion of

this test.

#1. Given that , determine . (3 pts.)

a)

b)

c)

d)

Questions #2and #3 refer to the graph of and the five points labeled A – E shown

here:

#2. Determine all labeled points at which the derivative function is negative.

(3 pts.)

a) B

b) B, D and E

c) B and C

d) D and E

#3. Determine all labeled points at which the derivative function is zero.

(3 pts.)

a) D

b) A and D

c) A and C

d) A, C and D

A

B

C

D

E



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Questions #4 - #7 pertain to the graph of shown here:

#4. Identify ALL points of discontinuity for . (3 pts.)

a)

b)

c)

d)

#5. Identify ALL points at which is left-continuous but not continuous. (2 pts.)

a)

b)

c)

d)

#6. Choose the phrase below that completes a true statement:

At , the function is _______________. (3 pts.)

a) both continuous and differentiable.

b) continuous but not differentiable.

c) neither continuous nor differentiable.

d) differentiable but not continuous.

#7. Which one of the following is an interval of continuity for ? (3 pts.)

a)

b)

c)

d)



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#8. Determine the equation of the line tangent to at . (3 pts.)

a)

b)

c)

d)

#9. For

, determine .

(3 pts.)

a)

b)

c)

d)

#10. The function

can be decomposed as where

(3 pts.)

a)

b)

c)

d)

#11. Determine all values of at which

has a horizontal tangent line.

(3 pts.)

a)

b)

c)

d)



Page 5 of 13

Questions #12 – #14 pertain to the following situation:

A police car leaves a patrol station at 8 a.m., heading south on a straight north-south highway.

The position of the police car hours after 8 a.m. is given by miles, with positive position corresponding to points north of the station.

#12. Determine the location of the police car at 10 a.m. (3 pts.)

a) At 10 a.m., the police car is 116 miles north of the station.

b) At 10 a.m., the police car is 300 miles south of the station.

c) At 10 a.m., the police car is 116 miles south of the station.

d) At 10 a.m., the police car is 300 miles north of the station.

#13. Determine the velocity and acceleration of the police car at 10 a.m. (2 pts.)

a) At 10 a.m., the police car is heading northward at 52 miles per hour and is

speeding up at a rate of 10 miles per hour squared.

b) At 10 a.m., the police car is heading southward at 52 miles per hour and is

slowing down at a rate of 10 miles per hour squared.

c) At 10 a.m., the police car is heading northward at 52 miles per hour and is

slowing down at a rate of 10 miles per hour squared.

d) At 10 a.m., the police car is heading southward at 52 miles per hour and is

speeding up at a rate of 10 miles per hour squared.

#14. At what time(s) does the police car have zero acceleration? (2 pts.)

a) 3:33 p.m. only

b) 8:00 a.m, 8:20 a.m, 3:33 p.m. and 5:00 p.m.

c) 8:20 a.m. only

d) 8:00 a.m. only



Page 6 of 13

#15. The graph of is shown. Identify the graph of .

(3 pts.)

a)

b)

c)

d)

#16. Determine .

(2 pts.)

a)

b)

c)

d)



Page 7 of 13

#17. Determine

(2 pts.)

a)

b)

c)

d)

#18. Determine

(2 pts.)

a)

b)

c)

d)



Page 8 of 13

Free Response. The Free Response questions will count 52% of the total grade. Read each

question carefully. In order to receive full credit you must show legible and logical

(relevant) justification which supports your final answer. Give answers as exact answers.

You are NOT permitted to use a calculator on any portion of this test.

1. Let

a. [3 points] Determine .

b. [1 points] For what value(s) of is continuous at ?

Award full point if answer here matches the limit obtained in (a) with an arithmetic error but

otherwise correct work.

c. [1 points] If , what type of discontinuity exists at ? Circle your

answer.

Jump Removable Oscillating Infinite

All or nothing; do not award points based on incorrect work in (a).

d. [3 points] What type of discontinuity exists at Support your answer with

an appropriate limit argument.

Limit argument:

OR

Type of discontinuity (circle your answer):

Jump Removable Oscillating Infinite

Work on Problem Points Awarded

Shows limit using correct piece of piecewise defined function, either explicitly or implicitly 1

Simplifies by cancelling factor of 1

Evaluates limit by direct substitution after cancellation 1

Notes:

Deduct ½ point for misuse of notation

Deduct ½ point for arithmetic error after substitution

Direct substitution into correct piece without cancelling (“limit DNE” or “infinite limit)

award 1 point total

Limit is “a” 0 points



Page 9 of 13

2. Let .

a. [8 points] Use the limit definition of the derivative to determine You will

receive no credit for using derivative rules to determine .

b. [2 points] Determine

c. [3 points] State the point-slope equation of the line tangent to at .


Shows limit from left or limit from right ½

Correct notation for infinite limit calculation ½

Correct infinite limit based on whether from left or right 1

Correct conclusion regarding type of limit 1

Notes:

Do not award point for correct conclusion based on incorrect work.


Shows limit as h 0 1

Correct setup for difference quotient, either explicitly or implicitly 1

Correct evaluation of 1

Correct expansion of 1

Distribution of across ½

Distribution of across ½

Simplification of numerator 1

Factoring and cancelling 1

Correct limit calculation after simplification 1

Notes:


Substitution of 4 into correct equation for 1

Correct arithmetic after substitution 1

Notes: Award full credit for derivative found using derivative rules even if no points were awarded in

part (a).



Page 10 of 13

3. [5 points each] Determine each of the following derivatives. You do not need to

simplify. Use notation appropriate to the problem statement to express your answer as an

equation, not an expression.

a.

b.

c.


Recognition (implicit or explicit) that 1

Correct evaluation of to determine ½

Point-slope form used ½

Correct value of slope used in equation for tangent line ½

Correct value of used in equation for tangent line ½

Notes:


Rewriting

as OR correct setup for chain rule 1

Correct derivative of OR correct derivatives in chain rule 1

Correct use of difference and constant multiple rules, either implicitly or explicitly 1

Correct derivative of 1

Notation 1

Notes:


Correct setup for product rule 2



Correct notation 1

Notes:


Correct setup for quotient rule 2



Correct notation 1

Notes:



Page 11 of 13

d.

4. A rock is propelled upwards from a cliff. Its position (in meters above the ground) seconds after it is thrown is given by .

a. [4 points] How long after the rock is thrown does it reach its maximum height?

Show all work and include units with your answer.

The rock reaches maximum height when . Solving for :

The rock reaches its maximum height 2 seconds after it is thrown.


Correct setup for chain rule 2



Correct notation 1

Notes:


Recognizing (implicitly or explicitly) that we need to solve 1

Recognizing (implicitly or explicitly) that ½

Correctly determining 1

Correctly solving for 1

Correct units ½

Notes: If is determined in another part of the problem, those points may be awarded here, even if

the work is not shown in this part.



Page 12 of 13

b. [2 points] What is the maximum height attained by the rock? Show all work and

include units with your answer.

The rock reaches its maximum height after 2 seconds. The height at that time is:

The rock reaches a maximum height of 9 meters above the ground.

c. [2 points] When does the rock strike the ground? Show all work and include

units with your answer.

The rock strikes the ground when its height above ground is 0 meters; that is, the

rock hits the ground when . We must solve for :

Since we cannot have negative time in this context, we have only one solution:

The rock strikes the ground 5 seconds after it is thrown upwards.

d. [2 points] What is the velocity of the rock on impact? Show all work and include

units with your answer.

The rock strikes the ground with a velocity of meters per second. (Or, the

rock is moving downwards at 6 m/s on impact.)


Evaluating at the time found in part (a) 1

Correct arithmetic in the function evaluation ½

Correct units ½

Notes:


Recognizing (implicitly or explicitly) that we need to solve ½

Correctly solving ½

Eliminating the negative solution and choosing ½

Correct units ½

Notes:

If solutions to are determined in another part of the problem, those points may be

awarded here, even if the work is not shown in this part.

No credit awarded for evaluating instead of solving



Page 13 of 13

5. (1 pt) Check to make sure your Scantron form meets the following criteria. If any of the items

are NOT satisfied when your Scantron is handed in and/or when your Scantron is processed one

point will be subtracted from your test total.

My scantron:

□ is bubbled with firm marks so that the form can be machine read;

□ is not damaged and has no stray marks (the form can be machine read)

□ has 18 bubbled in answers;

□ has MthSc 103 and my Section number written at the top;

□ has my Instructor’s name written at the top;

□ has Test No. 3 written at the top;

□ has Test Version A both written at the top and bubbled in below my CUID;

□ and shows my correct CUID both written and bubbled in.


Evaluating at the time found in part c 1

Correct arithmetic ½

Correct units ½

Notes:

Documents

MthSc 103 Test #3 Spring 2011 Version A 3.6 Student’s Printed …mthsc.clemson.edu/ug/MthSc103/Test3_VA_AK.pdf · 2011. 11. 5. · of academic integrity. No part of this test may