MATHEMATICS 184 (Section 921) - TERM EXAM #4
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INSTRUCTIONS: Notes, books and calculators are not permitted. Answers may receive no credit if accompanying work is not provided. When indicated, simplify answers as much as possible, and be sure to clearly indicate the final answer for each question. Please use the back of the pages if you need extra space.

9 1.
A new company, ACME Re-fried Beans Inc. is trying to determine the most economical cylindrical can in which to package their product. If the can must be cylindrical, and must have a volume of 54
π cm
3
, what are the dimensions (height and radius) of the can that would require the least material?
Hint: The cylinder is cut from sheet metal as 2 circles (top and bottom) and a rectangular piece that is rolled up and glued end to end
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7 2.
Find the global maximum and the global minimum (if they exist) of the function y = 3x
5 − 5x
3 − 1 on the interval [ − 1
2
, 2 ] . Provide both x and y coordinates.
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2
3. For the function y =
1 x 2 − 1
− 1
(a) Determine the domain and the range
3 (b) Find the coordinates of all x and y-intercepts
3 (c) Find the coordinates of all local extrema (indicate which are maxima and which are minima)
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4 (d) Determine the intervals of concavity (ie. On which intervals is the function concave up?
on which is it concave down?)
2 (e) Identify any horizontal asymptotes
3 (f) Identify any vertical asymptotes
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2 (g) Using the information from parts (a)-(f), sketch the function y = below.
x 2
1
− 1
− 1 on the grid
4 y
2
–4 –2
0
–2
–4
2 x
4
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5 4. (a) Use the local linearization of the function f ( x ) = ln ( 2x + 1 ) to estimate ln ( 1 .
01 ) near x = 0
2
(b) Compute the following limits: i.
lim x → ∞ e
− x ln ( x )
3 ii.
lim x → 0 sin ( x ) − x x 3
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