Name ID # Section # Seat #

Mahidol University International CollegeTrimester 1 Academic Year 2016-17

ICNS 103 Fundamental Mathematics Final Exam

Saturday, 10 December 2016 12.00 - 13.50 80 points

Directions Solve the following problems using the bottom of each page or any blank space for scratch-work.Answer the questions according to the instructions in each part. Write your name, ID number, section,and seat number in the space provided on each page. A calculator is NOT allowed for this exam.

• This exam paper contains 80 points + 5 bonus points.

• Unless specified otherwise, you must show detailed work to get full credits

(Questions begin on next page.)

Page 1 scratch-work

Name ID # Section # Seat #

Problem 1 8 points

SCORE

Answer the following questions. No partial credits are given. Points are awarded based on final answers only.

1.1 limx!5�

�x

5� x

= (1 pt.)

1.2 Let f(x) = 2x� 3. Then limx!2

f(1) = (1 pt.)

1.3 Let f(x) = 22 + 2x. Thendy

dx

���x=1

= (1 pt.)

1.4 A cost function is c = 4q2 + 9q + 3. The marginal cost when q = 5 is (1 pt.)

1.5 Let f(x) = (2x2 � 4)(x5 � 5x). Then f

0(0) = (2 pts.)

1.6 An equation of the tangent line to the curve y = 4x2 � 6x � 5 at the point (�1, 5) is (circle the correctanswer). (2 pts.)

A. y = �14x� 9

B. y = 14x+ 71

C. y = �14x+ 19

D. y = (8x� 6)(x+ 1) + 5

E. y = 8x� 6

Page 2 scratch-work

Name ID # Section # Seat #

Problem 2 8 points

SCORE

2.1 Evaluate the rate of change of the following function when x = 0:

f(x) = e

(2x+1)2.

Choose the correct answer from the given multiple choices below. You must show your work that is alignedwith the chosen choice. No partial credit is to be given for this question. (2 pts.)

A. 0

B. 1

C. e

D. 2e

E. 4e

2.2 A company manufactures two products, X and Y , and the joint cost function for these products is givenby c = 3xy + e

x + (x+ y)3, where c is the total cost, in dollar, of producing x units of X and y units of Y .

(6 pts.)

(a) Determine the marginal cost with respect to y when x = 1 and y = 2.

(b) From the above condition, should the manufacturer produce the 3rd unit of Y when X is held constantat 1 unit if it can sell Y product for $25 each? Give your reason.

Page 3 scratch-work

Name ID # Section # Seat #

There is no question on this page. You may use it as a scratch area.This page will be ignored by the grader. If you write some work that needs to be graded here, please indicate

it on the other side.

Page 4 scratch-work

Name ID # Section # Seat #

Problem 3 9 points

SCORE

3.1 Di↵erentiate y = 2ex + e

2 + lnx+ x

e. (2 pts.)

3.2 A manufacturer has determined that, for a certain product, the average cost in dollar per unit is given by

c̄ = 2q2 � 42q + 240 +100

q

,

where q is the number of units produced and 2 q 10. (7 pts.)

(a) At what level within the interval [2, 10] should the production be fixed at in order to minimize thetotal cost? What is the minimum total cost?

(b) If the production level were allowed to be within the interval [0, 10], what value of q would minimizethe total cost?

Page 5 scratch-work

Name ID # Section # Seat #

There is no question on this page. You may use it as a scratch area.This page will be ignored by the grader. If you write some work that needs to be graded here, please indicate

it on the other side.

Page 6 scratch-work

Name ID # Section # Seat #

Problem 4 20 points

SCORE

4.1 Suppose that the demand equation for a monopolist’s product is p = 80� 7q and the average-cost function

is c = 10 +11

q

, where q is number of units, and both p and c are expressed in dollars per unit. Determine

the price at which maximum profit occurs. Give a reason why the obtained value yields the maximum.

(5 pts.)

4.2 Let f(x) = x

3 + 6x2 + (3 � 2x)3. Determine where the function is concave up or concave down. Find thex-values where points of inflection occur. (3 pts.)

4.3 If

Za

0(1 + x) dx = 4, then what are the possible values of a ? (3 pts.)

Page 7 scratch-work

Name ID # Section # Seat #

4.4 Determine the definite integral Z 8

1(5 3px)2 � 10x dx.

(3 pts.)

4.5 Let f(x) = x

5 � 10x4 + 25x3.

(a) Find all the intercepts of the graph of the function f. (2 pts.)

(b) Find intervals on which the function f is increasing or decreasing; Find the coordinates of all relativeextrema of f. (3 pts.)

(c) Sketch the graph of the function f based on the information obtained from the previous questions.

(1 pt.)

Page 8 scratch-work

Name ID # Section # Seat #

Problem 5 20 points

SCORE

5.1 Compute the following indefinite integrals. (9 pts.)

(a)

Zln(

px)

lnxdx

(b)

Zx+ 2

x

dx

(c)

Z3x 5

px� e

2dx

5.2 If a function f satisfies

Ze

x lnx dx = f(x) + C, find the value of f 0(e). (1 pt.)

Page 9 scratch-work

Name ID # Section # Seat #

5.3 The marginal cost function for a certain product is given by

dc

dq

=pq

q4 + q

pq.

Find the total-cost function if the fixed cost is $4. (4 pts.)

5.4 Find the indefinite integral:

Z3x5

(x3 + 1)2dx. (6 pts.)

Page 10 scratch-work

Name ID # Section # Seat #

Problem 6 20 points

SCORE

6.1 Region A is bounded above and below by the curves y =px, y =

1

x

. The left boundary is the intersection

point between the two curves and the right boundary is the vertical line x = 4. Sketch the two curves.Indicate this region and find its area. (6 pts.)

6.2 Evaluate

Z 3

0

e

px+1

2px+ 1

dx (7 pts.)

Page 11 scratch-work

Name ID # Section # Seat #

6.3 Given the demand function p = 400� 2q, and the supply function p = q + 100.

(a) Find the price at which the market equilibrium occurs. (2 pts.)

(b) Find the consumers’ surplus under market equilibrium. You may use any method to calculate the area.

(5 pts.)

Page 12 scratch-work