Download calcu2013Assign3

Chapter 2 Differentiation: Basic Concepts
§2.4 The Chain Rules
P147
7, 19, 35, 45, 63
In Problems 7, use the chain rule to compute the derivative
7. y 
dy
and simplify your answer.
dx
1
; u  x2  9
u
In the following problems, differentiate the given function and simplify your answer.
19. f ( x)  ( x 5  4 x 3  7)8
35. f ( y) 
3y  1
1 4y
In the Problem 45, find all values of x  c so that the tangent line to the graph of f (x ) at (c, f (c )) will
be horizontal.
45. f ( x) 
x
(3x  2) 2
63. MAMMALIAN GROWTH Observations show that the length L in millimeters (mm) from
nose to tip of tail of a Siberian tiger can be estimated using the function L  0.25w 2.6 , where w
is the weight of the tiger in kilograms(kg). Furthermore, when a tiger is less than 6 months old, its
weight (kg) can be estimated in terms of its age A in days by the function w  3  0.21A .
a. At what rate is the length of a Siberian tiger increasing with respect to its weight when it
weighs 60 kg?
b. How long is a Siberian tiger when it is 100 days old? At what rate is its length increasing with
respect to time at this age?
§2.5 The Marginal Analysis and Approximations Using Increments
P159
3, 12, 19, 25, 27
In this problem, C (x) is the total cost of producing x units of a particular commodity and
p (x ) is the price at which all x units will be sold. Assume p (x ) and C (x) are in dollars.
(a) Find the marginal cost and the marginal revenue.
(b) Use marginal cost to estimate the cost of producing the forth unit.
(c) Find the actual cost of producing the fourth unit.
(d) Use marginal revenue to estimate the revenue derived from the sale of the fourth unit.
(e) Find the actual revenue derived from the sale of the fourth unit.
1
1 2
x  2 x  39; p( x)   x 2  4 x  80
3
12 ． MARGINAL ANALYSIS
A manufacturer’s
3. C ( x) 
total monthly revenue is
R(q)  240q  0.05q dollars when q units are produced and sold during the month. Currently,
the manufacturer is producing 80 units a month and is planning to increase the monthly output by
1 unit.
a. Use marginal analysis to estimate the additional revenue that will be generated by the
production and sale of the 81st unit.
b. Use the revenue function to compute the actual additional revenue that will be generated by the
production and sale of the 81st unit.
2
19. PRODUCTION At a certain factory, the daily output is Q( K )  600 K 1/ 2 units, where K
denotes the capital investment measured in units of $1,000. The current capital investment is
$900,000. Estimate the effect that an additional capital investment of $800 will have on the daily
output.
1
25. MANUFACTURING A manufacturer’s total cost is C (q )  q 3  642q  400 dollars
6
when q units are produce. The current level of production is 4 units. Estimate the amount by
which the manufacturer should decrease production to reduce the cost by $130.
27. CARDIAC OUTPUT Cardiac output is the volume (cubic centimeters) of blood pumped
by a person’s heart each minute. One way of measuring cardiac output C is by Fick’s formula
a
C
x b
where x is the concentration of carbon dioxide in the blood entering the lungs from the right heart
and a and b are positive constants. If x is measured as x=c with a maximum error of 3%, what is
the maximum percentage error that can be incurred by measuring cardiac output with Fick’s
fomula? (Your answer will be in terms of a, b, and c.)
§2.6 Implicit Differentiation and Related Rates
P171
11, 13, 26, 43, 51, 55
In problems 11 and 13, find
dy
by implicit differentiation.
dx
11. y 2  2 xy2  3x  1  0
13.
x  y 1
In problem 26, find the equation of the tangent line to the given curve at the specified point.
26. ( x 2  2 y) 3  2 xy2  64; (0,2)
43. DEMAND RATE When the price of a certain commodity is p dollars per unit, consumers
demand x hundred units of the commodity, where
75 x 2  17 p 2  5300
2
How fast is the demand x changing with respect to time when the price is $ 7 and is decreasing
dp
 0.75 .)
at the rate of 75 cents per month? (That is,
dt
51. MANUFACTURING At a certain factory, output Q is related to inputs x and y by the
equation
Q  2 x 3  3x 2 y 2  (2  y ) 3
If the current levels of input are x  30 and y  20 , use calculus to estimate the change in input
y that should be made to offset a decrease of 0.8 units in input x so that output will be
maintained at it s current level.
55. Show that the tangent line to curve
x2 y2

1
a2 b2
x0 x y 0 y
 2  1.
a2
b
at the point ( x0 , y0 ) is
f
gf '  fg '
( )' 
g
g2
Extra credit:
1. Prove the Product Rule:
from first principle (i.e. starting from the definition of df/dx).
2. Use the Product Rule and the General Power Rule to derive the Quotient Rule:
3. Use the Product Rule and the method of Implicit Differentiation to derive the Quotient Rule.
3