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§2.3 Product and Quotient Rules; Higher-Order Derivative
P134 19, 46
19. Differentiate the given function: g (t ) 
t2  t
2t  5
46. Find the second derivative of the given function. In each case, use the appropriate notation for the
second derivative and simplify your answer. (Don’t forget to simplify the first derivative as much as
possible before computing the second derivative.)
1
y  ( x 2  x )( 2 x  )
x
§2.4 The Chain Rule
P147 7, 19, 35, 45, 57, 63
In the following problems, differentiate the given function and simplify your answer.
19. f ( x)  ( x 5  4 x 4  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 with 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.
1 2
x  2 x  39; p( x)   x 2  4 x  80
3
(a) Find the marginal cost and the marginal revenue.
C ( x) 
(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.
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
C
a
,
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 formula? (Your answer
will be in terms of a, b, and c.)
§2.6 Implicit Differentiation and Related Rates
P171 11, 13, 43, 51, 55
In problems 11 and 13, find
dy
by implicit differentiation.
dx
11. y 2  2 xy 2  3x  1  0
13.
x  y 1
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
How fast is the demand x changing with respect to time when the price is $ 7 and is decreasing at the rate
dp
 0.75 .)
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
at the point ( x0 , y0 ) is
x2 y2

1
a2 b2
x0 x y 0 y
 2  1.
a2
b