Download Math 180 #012

Math 180 #003
UNM
Summer 2012
Elements of Calculus I: Worksheet 4
1. Let C ( x ) be the cost function in dollars from producing x items. Match each question 1-4
with its proper solution A-D.
(1) What is the cost from producing 100 items?
(2) At what level of production will the marginal cost be $100/item?
(3) What is the marginal cost from producing 100 items?
(4) For what level of production will the cost be $100?
(A) Compute C '(100) .
(B) Set C '( x)  100 and solve for x.
(C) Set C ( x)  100 and solve for x.
(D) Compute C (100) .
2. Suppose the cost for a company to produce x refrigerators is given by
C ( x)  8000  200 x  0.2 x 2 , 0  x  400 , where C is in dollars.
(a) Find the fixed cost.
(b) Find the marginal cost when producing 250 refrigerators. Interpret your result.
(c) Find the change in cost from producing the 250th to the 251st refrigerator.
Compare your answer to part (b).
3. The relationship between the unit price p in dollars and the quantity demanded x of
certain model of speakers is p  0.02 x  400 for 0  x  20, 000 . The cost function for
producing x speakers is C ( x)  100 x  200, 000 , where C is in dollars.
(a) Find the revenue function R ( x )
(b) Find the marginal revenue when 2000 speakers are sold. Interpret your result.
(c) Find the profit function P ( x ) .
(d) Find the marginal profit when 2000 speakers are sold. Interpret your result.
4. When a store charges $400 per laptop, 50 laptops were sold per month. For every $5
decrease in price, 1 more laptop is sold per month.
(a) Find the demand function p ( x ) .
(b) Find the revenue function R ( x ) .
(c) Find the marginal revenue when 50 laptops are sold and interpret your result.
(d) Find the number of laptops sold so that the marginal revenue is $500/laptop.
5. Let a and b be constants with s  7a bx 2 
ds
(a)
.
dx
d 2s
(b) 2 .
dx
2a 2 x
a
. Find:

b
bx
d 3s
(c) 3 .
dx
6. Given f ( x ) , let g ( x)  x 2 f ( x) . Find g ''( x) .
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7. Let P (t ) represent the population of bacteria in a certain culture, where t represents time.
Match each mathematical situation 1-4 with its proper description A-D and its graphical
representation I-IV.
(1) P '(t )  0 and P ''(t )  0 .
(2) P '(t )  0 and P ''(t )  0 .
(3) P '(t )  0 and P ''(t )  0 .
(4) P '(t )  0 and P ''(t )  0 .
(A) The population is increasing, but at a decreasing rate.
(B) The population is decreasing and the rate is decreasing.
(C) The population is increasing and the rate is increasing.
(D) The population is decreasing, but at an increasing rate.
[I]
[II]
P(t)
[III]
P(t)
t
[IV]
P(t)
P(t)
t
t
t
8. The distance covered by a car after t sec is given by s(t )  t 3  8t 2  20t for 0  t  6 ,
where s is in feet.
(a) Find the expression a(t ) for the car's acceleration at any time t.
(b) Is the car accelerating or decelerating after 2.5 seconds?
9. The number of major crimes committed in a certain city between 1988 and 1995 is
approximated by the function N (t )  0.1t 3  1.5t 2  100 for 0  t  7 , where N (t )
denotes the number of crimes committed in year t, with t  0 corresponding to 1988.
(a) Find N '(6) and interpret your result. Would the population like this result?
(b) Find N ''(6) and interpret your result. Would the population like this result?
10. Let f ( x)  x 4/3 . Show that the first derivative exists at x  0 , but the second derivative
does not exist at x  0 .
11. Find a function whose first and second derivatives exist at x  0 , but the third derivative
does not exist at x  0 .
12. Classify the following statements as true or false:
d2
d2
2
f
(
cx
)

c
f
''(
cx
)
(a)
.
(c)


dx 2
dx 2
(b)
d2
 f ( x) g ( x)  f ''( x) g ''( x) .
dx 2
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(d)
 f ( x)  f ''( x)
 g ( x)   g ''( x) .


d n1 n
x   0 .
dx n1  