FINAL EXAM (PRACTICE A) MATH 265people.ucalgary.ca/~aswish/Math-265-Final-PracticeA.pdfMATH 265 - FINAL EXAM - PRACTICE A Part I: True/False questions are worth 2 marks each, and multiple

UNIVERSITY OF CALGARYFACULTY OF SCIENCE

DEPARTMENT OF MATHEMATICS & STATISTICS

FINAL EXAM (PRACTICE A)

MATH 265

NAME STUDENT ID

EXAMINATION RULES

1. This is a closed book examination.

2. Calculators are not permitted.

3. The use of personal electronic or communication devices is prohibited.

4. The exam has many questions.

5. Scantron sheets must be filled out during the exam time limit. No additional time will be grantedto fill in scantron form.

6. A University of Calgary Student ID card is required to write the Test. If adequate ID is not present,the Student may be asked to complete an Identification Form.

7. Students late in arriving will not be permitted to write the exam thirty (30) minutes after theexamination has started.

8. No student will be permitted to leave the examination room during the first thirty (30) minutes,nor during the last ten (10) minutes of the examination. Students must stop writing and hand intheir exam immediately when time expires.

9. All inquiries and requests must be addressed to the exams Supervisor.

10. Students are strictly cautioned against:

(a) communicating to other students;

(b) leaving answer papers exposed to view;

(c) attempting to read other students’ examination papers.

11. If a student becomes ill during the course of the examination, he/she must report to the Invigilator,hand in the unfinished paper and request that it be cancelled.

12. Once the examination paper has been handed in for marking, the Student cannot request that theexamination be canceled.

13. Failure to comply with these regulations may result in the rejection of the examination paper.

MATH 265 - FINAL EXAM - PRACTICE A

Part I: True/False questions are worth 2 marks each, and multiple choice questions areworth 4 marks each. For each question, clearly circle your choice on this booklet, andrecord your answer on the scantron sheet provided. Make sure that you answer all thequestions: remember there is no penalty in guessing.

1. True / False.If a function f(x) is continuous at x = a, then f(x) is also di↵erentiable at x = a.

2. True / False.

Suppose f(x) is everywhere continuous and

ˆx

1

f(t) dt = sin x. Then f(⇡) = �1.

3. True / False.For any two functions f(x) and g(x), the derivative of f(x) · g(x) is equal to f

0(x) · g0(x).

4. True / False.Suppose lim

x!2f(x) = 3. Then at least one of the following holds:

• f(x) is continuous at x = 2.

• f(x) is di↵erentiable at x = 2.

• f(2) = 3.

5. True / False.

Assuming x > 1, the equation log

x

(2

x

) =

x

4

has a unique solution of x = 16.

Page 2


6. Find the (natural) domain of the function g(x) =

p5� x+ |3� x|

x

2 � 1

.

(a) (�1,1).

(b) [5,1).

(c) (�1,�1) [ (�1, 1) [ (1,1).

(d) (�1,�1) [ (�1, 1) [ (1, 3].

(e) (�1,�1) [ (�1, 1) [ (1, 5].

7. Which of the following is equal to sin

�1

✓sin

✓3⇡

5

◆◆?

(a) 0.

(b) ⇡/5.

(c) 2⇡/5.

(d) 3⇡/5.

(e) �⇡/5.

8. Evaluate the limit L = lim

x!�1

p4x

2+ 3

x� 3

.

(a) L = 4.

(b) L = �2.

(c) L = �4.

(d) L =

4

3

.

(e) L does not exist.

Page 3


9. Let f be a continuous function on [�2, 7]. If f(�2) = �1 and f(7) = 3, then the Intermediate

Value Theorem guarantees that

(a) f(0) = 0.

(b) f

0(c) =

4

9

for at least one c between �2 and 7.

(c) �1 f(x) 3 for all x between �2 and 7.

(d) f(c) = 1 for at least one c between �2 and 7.

(e) f(c) = 0 for at least one c between �1 and 3.

10. Which of the following does not describe the derivative of a function f(x) at x = a:

(a) lim

x!a

f(x)� f(a)

x� a

.

(b) lim

h!a

f(x+ h)� f(a)

h

.

(c) lim

h!0

f(a+ h)� f(a)

h

.

(d) The slope of the tangent line to the graph y = f(x) at x = a.

(e) The instantaneous rate of change of f(x) at x = a.

11. If f(x) is a function such that lim

x!1

2f(x)� 2f(1)

x� 1

= 0, which of the following must be true?

(a) The limit of f(x) as x approaches 1 does not exist.

(b) f(x) is not defined at x = 2.

(c) The derivative of f(x) at x = 1 is equal to 0.

(d) f(x) is continuous at x = 0.

(e) f(1) = 0.

Page 4


12. Find an equation of the tangent line to f(x) =

✓x+

1

x

◆4

at x = 1.

(a) y = 16.

(b) y = �16.

(c) y = 32x+ 16.

(d) y = �32x+ 16.

(e) None of the above.

13. The Taylor polynomial of degree 2, centred at x = 0, for the function y = e

2xis:

(a) 1 + x+ x

2.

(b) 1 + 2x+ 4x

2.

(c) 1 + 2(x� 2) + 2(x� 2)

2.

(d) 1 + 2x+ 2x

2.


14. Find

dy

dx

if x

2+ 3y

2= 2xy.

(a)

dy

dx

=

1

3

.

(b)

dy

dx

=

x� y

x� 3y

.

(c)

dy

dx

=

�x

3y

.

(d)

dy

dx

=

x� y

�x+ 3y

.

(e)

dy

dx

=

x

x� 2y

.

Page 5


15. Find the derivative of the function f(x) =

psin(⇡x).

(a)

⇡ cos(⇡x)

2

psin(⇡x)

.

(b)

cos(⇡x)

2

psin(⇡x)

.

(c)

⇡ cos(⇡x)psin(⇡x)

.

(d)

12(sin(⇡x))

� 12.

(e)

pcos(⇡x).

16. The derivative of the function (x+ 1)

x+1is:

(a) x

x

.

(b) (x+ 1)

x+1

(c) (x+ 1)

x+1(ln x)

(d) (x+ 1)

x+1(ln(x+ 1) + 1)


17. What is the derivative of f(x) = x

2cos

�1(x)?

(a) � 2xp1� x

2.

(b) 2x cos

�1(x) +

x

2

p1� x

2.

(c) 2x cos

�1(x)� x

2

p1� x

2.

(d) 2x cos

�1(x) + x

2(cos

�2(x)) sin x.

(e) 2x cos

�1(x)� x

2(cos

�2(x)) sin x.

Page 6


18. Which of the following is the di↵erential of y = x cos(2x)?

(a) dy = � sin(2) dx.

(b) dy = �2 sin(2x) dx.

(c) dy = (�x sin(2) + cos(2x)) dx.

(d) dy = (2x sin(2x) + cos(2x)) dx.

(e) dy = (�2x sin(2x) + cos(2x)) dx.

19. Answer the following question given the function

f(x) =

x

x

2+ 1

,

and its derivative

f

0(x) =

�(x

2 � 1)

(x

2+ 1)

2.

(a) f has a local minimum at x = �1 and a local minimum at x = 1.

(b) f has a local maximum at x = �1 and a local maximum at x = 1.

(c) f has a local minimum at x = �1 and a local maximum at x = 1.

(d) f has a local maximum at x = �1 and a local minimum at x = 1.

(e) f has a local minimum at x = 0.

20. Which of the following is the absolute minimum value of f(x) = x+

4

x

on the interval [1, 4]?

(a) 1.

(b) 2.

(c) 3.

(d) 4.

(e) 5.

Page 7


21. If x = 2 is a critical point of f(x) =

ax+ 3

x

2+ 8

, what is the value of the constant a?

(a) 3/2.

(b) 2.

(c) 3.

(d) 6.

(e) 12.

22. Suppose we want to estimate the value of

p3 by applying Newton’s method to the equation

x

2 � 3 = 0 (i.e., using f(x) = x

2 � 3). If we start with the first estimate x1 = 2, what is the

value of the second estimate x2?

(a) 3/2.

(b) 1/4.

(c) 5/4.

(d) 7/4.

(e) 13/8.

23. A company wants to manufacture a cookie tin with a square base and top each of side length

x, and rectangular sides. The material for the sides costs $3/cm

2, and for the top and bottom

the cost is $4/cm

2. The tin is to have a volume of 5/cm

3. The dimensions of the cheapest such

container can be found by:

(a) minimizing the function C(x) = 4x

2+ 60/x

2on (0,1).

(b) minimizing the function C(x) = 4x

2+ 60/x on (0,1).

(c) minimizing the function C(x) = 8x

2+ 60/x on (0,1).

(d) minimizing the function C(x) = 8x

2+ 4/x on (0,1).

(e) minimizing the function C(x) = 8x

2+ 4/x

2on (0,1).

Page 8


24. Determine the concavity of the function f(x) = x

4 � 4x

3.

(a) Concave upward on (�1, 0) and on (2,1), concave downward on (0, 2).

(b) Concave downward on (�1, 0) and on (2,1), concave upward on (0, 2).

(c) Concave upward on (�1, 0) and on (0, 3) and concave downward on (3,1).

(d) Concave downward on (�1, 0) and on (0, 3) and concave upward on (3,1).


25. Suppose that the graph of f passes through the point (1, 3) and that the slope of its tangent

line at (x, f(x)) is 2x. What is the value of f(3)?

(a) 7.

(b) 8.

(c) 9.

(d) 10.

(e) 11.

26. If

ˆ 4

0

f(x) dx = 2 and

ˆ 4

0

g(x) dx = �3, what is the value of

ˆ 4

0

(f(x) + g(x) + 1) dx?

(a) �6.

(b) �3.

(c) 0.

(d) 3.

(e) 6.

Page 9


27. Find the function F (x), given that F

0(x) =

3

x

3� 9

x

7and F (1) = 0.

(a) F (x) = � 3

2x

2+

3

2x

6.

(b) F (x) = � 3

2x

2� 3

2x

6.

(c) F (x) = 3� 3

2x

2+

3

2x

6.

(d) F (x) = 3� 3

2x

2� 3

2x

6.

(e) F (x) = � 9

x

2+

63

x

6.

28. Evaluate

ˆ⇡/2

0

�cos x e

sinx

�dx.

(a) e.

(b) �e.

(c) 1� e.

(d) e� 1.


29. Evaluate

ˆ 1

0

x

2(1� x) dx.

(a) 0.

(b) 1/3.

(c) 1/4.

(d) 1/12.


Page 10


30. Calculate

ˆe

1

ln x

x

dx.

(a)

1

2

.

(b)

e

2 � 1

2

.

(c) 0.

(d)

1

e

.


31. If h(x) =

ˆ 9

x

ln(t

2+ 1) dt, then

(a) h

0(x) = ln(x

2+ 1).

(b) h

0(x) = � ln(x

2+ 1).

(c) h

0(x) = ln(82)� ln(x

2+ 1).

(d) h

0(x) = ln(x

2+ 1)� ln(82).

(e) h

0(x) = 2x ln(x

2+ 1).

32. If F (x) =

ˆx

�1

t

2e

t

dt, then what is the value of (F

�1)

0(0)?

(a) 0.

(b) 1.

(c) e.

(d) 1/e.

(e) It does not exist.

Page 11


33. Determine the area enclosed by the curves x = y and x = y

3.

(a) 0.

(b) 1/2.

(c) 1/3.

(d) 1/4.


34. Which of the following is NOT an improper integral?

(a)

ˆ 1

�1

sin x

x

dx.

(b)

ˆ 1

0

e

�3xcos(x) dx.

(c)

ˆ 3

0

x

(x� 3)

2dx.

(d)

ˆ 2

�2

1

x

2 � 1

dx.

(e)

ˆ 1

�1

1

x

2 � 4

dx.

35. The improper integral

ˆ 1

0

e

�x

dx?

(a) converges to 1.

(b) converges to e� 1.

(c) converges to e.

(d) converges to e+ 1.

(e) diverges.

Page 12

FINAL EXAM (PRACTICE A) MATH 265people.ucalgary.ca/~aswish/Math-265-Final-PracticeA.pdfMATH 265 - FINAL EXAM - PRACTICE A Part I: True/False questions are worth 2 marks each, and multiple

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