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MA110* Midterm Test 1 Page 1 of 9

Student Number:

1. Solve the following inequality and equation:

a)2x + 1

x + 1

1[5 marks]

b)|2x + 1||x + 1| = 1[5 marks]

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MA110* Midterm Test 1 Page 2 of 9

2. Find the following limits (simplify as much as possible):

a) limx

2 + sec2 x

ln(e2x sinx) + x[4 marks]

b) limx

(x2 + 1 x)[4 marks]

c) limx0

sin(5x)

sin(7x)[4 marks]

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MA110* Midterm Test 1 Page 3 of 9

3. Sketch the graph ofy = |(x 1)2 3| using graphical operations, starting from the graph of[8 marks]y = x2. Make sure to

label the intermediate graphs by the corresponding functions,

label the x- and y-intercepts of each graph,

sketch the final graph separately.

4. The graph ofy = f(x) is given below. Sketch on the same axis the graph off(x). Make sure[6 marks]to label clearly where (a) f(x) does not exist and (b) f(x) = 0.

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MA110* Midterm Test 1 Page 4 of 9

5. Let f(x) =x2 3x + 2x2 + x 2 . Find the following:

a) The domain off(x).[2 marks]

b) The asymptotes (both horizontal and vertical ones if they exist) of the graph ofy = f(x).DO NOT sketch the graph.[8 marks]

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MA110* Midterm Test 1 Page 5 of 9

6. Let f(x) =2x + 1

x + 2. Find the points where the tangent line of the graph of y = f(x) is per-[10 marks]

pendicular to 3y + x 7 = 0.

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MA110* Midterm Test 1 Page 6 of 9

7. Suppose that |f(x) ln(x + 1)| tan2 x for x near 0. Show that limx0

f(x) = 0.[10 marks]

(Hint: Squeeze theorem.)

8. Show that the graphs ofy = x3 and y = cos x intersect at least once between x = 0 and x = .[6 marks](Hint: Intermediate value theorem.)

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MA110* Midterm Test 1 Page 7 of 9

9. The displacement (or position) of a particle moving along a straight line at time t 0 ( in[8 marks]seconds) is given by s(t) = et cos t (in meters). Find the times when the acceleration a(t) is 0.

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MA110* Midterm Test 1 Page 8 of 9

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 1 Page 9 of 9

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.