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School of Mathematics and Statistics
Carleton University
Math. 69.107
SOLUTIONS TO TEST 2

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This test is out of a Total of 30.

PART I: Multiple Choice Questions
(Choose and CIRCLE only ONE answer)

  1. [2 marks] Let tex2html_wrap_inline151 . Which of the following expressions represents the value of tex2html_wrap_inline153 ?

    2. (a) -1, tex2html_wrap329 0, (c) 1, (d) This derivative does not exist.

  2. [2 marks] Let tex2html_wrap_inline161 . Then tex2html_wrap_inline163 is equal to:

    3. (a) tex2html_wrap_inline165 , (b) tex2html_wrap_inline167 , (c) tex2html_wrap_inline169 , tex2html_wrap331 tex2html_wrap_inline171 .

  3. [2 marks] Let tex2html_wrap_inline173 . Then the function f is increasing on the interval:

    4. tex2html_wrap333 tex2html_wrap_inline177 , (b) tex2html_wrap_inline179 , (c) tex2html_wrap_inline181 , (d) tex2html_wrap_inline183 .

  4. [2 marks] Let tex2html_wrap_inline185 . Then f has points of inflection at:

    5. (a) no point whatsoever, tex2html_wrap329 tex2html_wrap_inline189 , (c) x=0 only, (d) tex2html_wrap_inline193 , only.

  5. [2 marks] Answer TRUE or FALSE:

    6. If tex2html_wrap_inline195 then, for each x, its derivative

    displaymath199

    (a) TRUE, tex2html_wrap329 FALSE

PART II: Show all work here.
No additional pages will be accepted
  1. [5+5 marks] Evaluate the following integrals using any method:

    2. a) tex2html_wrap_inline201 .

    Solution: Let tex2html_wrap_inline203 . Then tex2html_wrap_inline205 . When x=0, u=0 and when tex2html_wrap_inline211 , u=0. and so

    eqnarray54

    Alternately,

    eqnarray62

    Thus,

    eqnarray68

    b) tex2html_wrap_inline215

    Use the Table Method:

    tabular78

    and the final answer can be written as,

    eqnarray89

  2. [5+5 marks] Let tex2html_wrap_inline229 .

    3. a) Determine all the intervals where f is increasing and decreasing.

    Hint: Use the Sign Decomposition Table of f.

    Solution:We know that tex2html_wrap_inline235 .

    The Sign Decomposition Table of tex2html_wrap_inline163 is given by

    tabular100

    It follows that f is increasing if tex2html_wrap_inline289 , that is, when x is in either (-1, 0) or tex2html_wrap_inline295 .

    Similarly, f is decreasing when tex2html_wrap_inline299 , which in this case means that x is in the interval tex2html_wrap_inline303 or in the interval (0, 1). b) In what intervals is f concave up and concave down?

    Solution: In this case, we don' t need the SDT of tex2html_wrap_inline309 since tex2html_wrap_inline311 looks like tex2html_wrap_inline313 Note that tex2html_wrap_inline315 when tex2html_wrap_inline317 or, equivalently, when tex2html_wrap_inline319 . So f is concave up in this case.

    Similarly we can see that f is concave down when tex2html_wrap_inline325 . This makes tex2html_wrap_inline327 a point of inflection!

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Angelo Mingarelli

Fri Nov 20 15:02:14 EST 1998