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Mathematics Examination: Calculus II at Dawson College, Exams of Calculus

The final examination questions for the calculus ii course offered at dawson college's mathematics department. The exam covers various topics including differentiation, integration, limits, and series. Students are required to find derivatives, evaluate integrals, determine limits, and evaluate taylor polynomials. The document also includes problems related to functions, sales, and supply. The exam consists of 14 questions with varying marks.

Typology: Exams

2012/2013

Uploaded on 02/12/2013

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DAWSON COLLEGE
MATHEMATICS DEPARTMENT
Final Examination
Mathematics 201-203-DW
Calculus II Social Science /Commerce
Instructors: C. Farnesi, I. Rajput, S. Shahabi
Date: Tuesday, December 21, 2010
1. [5marks]Findfxif
f’x4x53x42x
x2
and f4200.
2. [5marks]Findtheaverage value of the function
gx10 3
x
over the interval 4, 15.
3. [5marks] The sales of a certain movie DVD is changing at the rate of
S’x18 000 x2
x34
where Sis the sales in dollars, and xis the number of days elapsed since the DVD was
released. On its release date, the sales were 50 000$. What will the sales be 7 days after
its release?
4. [5marks] Each week, the quantity demanded x(in units of a hundred) of a certain
commodity is related to the unit price p(in dollars) by the supply function
p3x29x50
If the market price is set at 260$, find the producers’ surplus.
5. [5marks] Use the limit definition (Riemann Sums) of the definite integral to evaluate
0
3
x32x21dx
1
pf3
pf4
pf5

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DAWSON COLLEGE

MATHEMATICS DEPARTMENT

Final Examination

Mathematics 201-203-DW Calculus II Social Science / Commerce Instructors: C. Farnesi, I. Rajput, S. Shahabi Date: Tuesday, December 21, 2010

  1. [ 5 marks ] Find fx  if

f ’ x  

4 x^5  3 x^4  2 x x^2 and f  4   200.

  1. [ 5 marks ] Find the average value of the function

gx   10  3 x over the interval 4, 15.

  1. [ 5 marks ] The sales of a certain movie DVD is changing at the rate of

S ’ x   18 000 x^2 x^3  4 where S is the sales in dollars, and x is the number of days elapsed since the DVD was released. On its release date, the sales were 50 000$. What will the sales be 7 days after its release?

  1. [ 5 marks ] Each week, the quantity demanded x (in units of a hundred) of a certain commodity is related to the unit price p (in dollars) by the supply function p  3 x^2  9 x  50 If the market price is set at 260$, find the producers’ surplus.
  2. [ 5 marks ] Use the limit definition (Riemann Sums) of the definite integral to evaluate

0

3 x^3  2 x^2  1 dx

  1. [ 5 marks ] Find the area of the region completely enclosed by the graphs of y 1  x^2  3 x  2 and y 2  7 x  3
  2. [ 20 marks ] Solve the following integrals:

a.

 x^3  x^2  3  10 dx

b.

 3 sin^3 x ^ ^ cos^ x 4 ^ sec^2 ^2 x ^ dx

c.

 18 x^2 ln x dx

d.

x^2  12 x  20 x^2  (^4)  x  (^2) 

dx

  1. [ 5 marks ] Use Simpson’s Rule with n  6 to approximate the value of the definite integral

1

(^10 3) x 3 x  1 dx

(Round your answer to 3 decimal places.)

  1. [ 5 marks ] Find the fourth Taylor Polynomial of the function fx   x^4  2 ln x at x  1.
  2. [ 5 marks ] Evaluate the limit

lim x  1

x  2  3 e

x  1  4 sin x  1  4 ln x  2 x^2  2

Answers

  1. fx   x^4  x^3  4 x
  1. Average Value  9. 6395
  2. Sales  $ 249 535. 23
  3. PS  $ 90 650
    1. 25
  5. Area  36 units 2
  6. a) 241  x^2  3 ^12  223  x^2  3 ^11  C

b)  cos 3 x   4 sin x 4

2 tan 2 x   C

c) x^3 6 ln x  1  C

d) 3 ln| x^2  4|  5 ln| x  2|  C

    1. 311
  2. P 4  x   1  6  x  1   5  x  1 ^2  14 3  x^ ^1 

2  x^ ^1 

4

3 8

  1. (^) ln 3^1
  2. True ( y is a solution of the DE)
  3. y^2  3 y  ^18 2 x  1 2
  1. Sum  4
  2. a) divergent by the test for divergence (since lim n  a (^) n   9  0)

b) convergent by comparison test (compare to the bigger convergent p -series with p  5 2

c) convergent by integral test (since  2

x e ^ x^ (^2)  1 dx  12 e ^3 )