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Winter Term, 2011

Name:

Section:

WILFRID LAURIER UNIVERSITY

Waterloo, Ontario

Mathematics 110* Introduction to Differential andIntegral Calculus

Midterm 2 February 2, 2011

Instructor:

Section A (MWF 9:30 am): S. HuSection B (MWF 10:30 am): S. Hu

Section C (MWF 11:30 am): A. Allison

Time Allowed: 80 minutes

Total Value: 80 marks

Number of Pages: 10 plus cover page

Instructions:

Non-programmable, non-graphing calculators are permitted. No other aids are

allowed.

Check that your test paper has no missing, blank, or illegible pages.

Answer in the spaces provided. Please note that questions are printed on

both sides of the test pages.

Show all your work. Insufficient justification will result in a loss of marks.

Write your student number in the space provided on the next page.

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MA110* Midterm Test 2 Page 2 of 10

2. Sketch the graph ofy = f(x) where f(x) = (x + 1)ex, following the steps outlined below:

a) Find the domain off(x).[1 mark]

b) Find the intercepts ofy = f(x).[2 marks]

c) Find the asymptotes ofy = f(x).[2 marks]

d) Find the critical value(s) off(x).[2 marks]

e) Find the intervals of increase and decrease of y = f(x).[2 marks]

f) Find any local maxima and minima of y = f(x), if they exist.[1 mark]

#2 continued on the next page

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MA110* Midterm Test 2 Page 3 of 10

#2 continued

g) Find the second order derivative off(x).[1 mark]

h) Determine the concavity and the points of inflection for y = f(x).[3 marks]

i) Use the information obtained above (from a) to h)) to sketch the graph of y = f(x) on[3 marks]

the following axes. Label clearly all the special points and asymptotes. (You may usethe following values for the powers of e: e 2.71828, e1 0.36788, e2 0.13533.)

x

y

2 1 1 2 3 4

3

2

1

1

2

O

Over

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MA110* Midterm Test 2 Page 4 of 10

3. Find the tangent line ofx3y + tan1 x = sin y at (0, ).[6 marks]

4. Let f(x) = x(x2+1), then f(1) = 1.

a) Find f(x).[4 marks]

b) Find the approximate value off(0.9) = 0.91.81 using the linear approximation off(x) at[3 marks]a = 1.

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MA110* Midterm Test 2 Page 5 of 10

5. Suppose that the height, depth, and width of a staircase is, respectively, 4 meters, 3 meters,[10 marks]and w meters. A rectangular box of width w meters is to fit under the staircase. Find thedimensions (height and depth) of the box so that it is as large as possible.

?

?

3 m

4 m

Over

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MA110* Midterm Test 2 Page 6 of 10

6. Follow the steps below and show that y = x3 and y = ex intersect exactly once.

a) Show that y = x3 and y = ex intersect at least once.[6 marks](Hint: Intermediate value theorem.)

b) Show that y = x3 and y = ex cannot intersect more than once.[4 marks](Hint: Mean value theorem or Rolles theorem.)

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MA110* Midterm Test 2 Page 7 of 10

7. Evaluate

20

4 x2dx by interpreting it as an area.[4 marks]

8. Solve the following equation for y = f(x), where f(1) = 1:[4 marks]

2sinx +4

x2 + 1 1

x+ 3x2 = y.

Over

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MA110* Midterm Test 2 Page 9 of 10

This page is intentionally left blank.

You may use it to continue any question for which you need more space, in

which case, be sure to indicate clearly, in the original location and here, that the

work continues here.

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MA110* Midterm Test 2 Page 10 of 10

You may use the space below for rough work, in which case, you could tear this

page off. Or leave it attached if you want to use it to continue any question for which

you need more space. In that case, be sure to indicate clearly, in the original location

and here, that your work continues here.