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Mathematics 117 test three
Wednesday, May 24, 2006
1. Suppose the population P is a function of time t:
(a) If the population is increasing, but the rate of growth is slowing down, then
P 0 (t) is
positive or negative
and P 00 (t) is
positive or negative
(b) If the population graph is increasing and concave down, then
P 0 (t) is
positive or negative
and P 00 (t) is
positive or negative
2. The demand function relating the price p per item to the quantity q sold is given by
q = 2000e−.4p+8
(b) Use the differential to estimate the
change in q if the price increases from
p = 20 to p = 23.
(a) Calculate the differential dq.
(c) Calculate the price p which maximizes the revenue R = pq.
3. Find the absolute maximum value and the absolute minimum value of
f (x) = x3 − 15x + 2 when 0 ≤ x ≤ 3.
4. Sketch the graph of a function which matches the first and second derivative information
below:
− − − − − 0 + + + + + + + + + 0 − − − − − − − − − − − − −−
2
4
f 0 (x)
x
+ + + + + + + + ++ 0 − − − − − − − − − − − − −− 0 + + + + + f 00 (x)
3
6
x
5. Use the first and second derivatives to sketch the graph of
y = x3 − 9x + 2.
Your sketch should show the exact x-values for relative maximum, relative minimum, and
inflection points. It is not necessary to compute y-values for your sketch.
page two
6. Population P (in millions) is given as a function of time t (in centuries after 2000) by
the graph below. Use the graph to answer the questions on this page.
40
35
30
P values
25
20
15
10
5
0
0
1
2
3
4
5
6
7
t values
(a) What is the absolute maximum value of the population P ?
(b) What are the critical values for this function?
(c) At what time(s) does the function attain a relative minimum value?
(d) Over which time intervals is the population P decreasing?
(e) Over which time intervals is the derivative of the population (P 0 ) decreasing?
(f) Over which time intervals is the graph concave up?
(g) Over which time intervals are the first derivative (P 0 ) & the second derivative (P 00 )
both positive?
7. Minimize the surface area of a 32 cubic centimeter rectangular box with a square bottom
and no top. If x represents the length of a side of the square bottom, and y represents the
height of the box, use the formulas
surface area A = x2 + 4xy and volume V = x2 y
Find the values of x and y which minimize the surface area.