11DIFFERENTIATION - Department of Mathematics

716 11 DIFFERENTIATION
In Exercises 35–38, find the derivative of each
of the given functions and evaluate f(x) at the
given value of x.
35. f(x) (2x 1)(x 2 3); x 1
36. f(x)
37. f(x)
2x 1
; x 2
2x 1
x
x4 2x 2 ; x 1
1
38. f(x) (x 2x)(x 3/2 x); x 4
In Exercises 39–42, find the slope and an equation
of the tangent line to the graph of the function
f at the specified point.
39. f(x) (x 3 1)(x 2 2); (2, 18)
40. f(x)
41. f(x)
3
2 x 4
;2,
x 1
x 1
x 2 ; (1, 1)
1
9
1 2x1/2 5
42. f(x) ;4, 3/2 1 x
43. Find an equation of the tangent line to the graph of the
function f(x) (x 3 1)(3x 2 4x 2) at the point (1, 2).
44. Find an equation of the tangent line to the graph of the
function f(x) 3x
x 2 at the point (2, 3).
2
45. Let f(x) (x 2 1)(2 x). Find the point(s) on the
graph of f where the tangent line is horizontal.
46. Let f(x) x
x 2 . Find the point(s) on the graph of f
1
where the tangent line is horizontal.
47. Find the point(s) on the graph of the function f(x)
(x 2 6)(x 5) where the slope of the tangent line is
equal to 2.
48. Find the point(s) on the graph of the function f(x)
x 1
where the slope of the tangent line is equal to 1/2.
x 1
49. Astraight line perpendicular to and passing through the
point of tangency of the tangent line is called the normal
to the curve. Find the equation of the tangent line and
the normal to the curve y 1
at the point (1, ).
2 1 x
50. C ONCENTRATION OF A D RUG IN THE B LOODSTREAM The concentration
of a certain drug in a patient’s bloodstream t hr
after injection is given by
C(t) 0.2t
t2 1
a. Find the rate at which the concentration of the drug
is changing with respect to time.
b. How fast is the concentration changing hr, 1 hr, and
2 hr after the injection?
51. C OST OF R EMOVING T OXIC W ASTE Acity’s main well was
recently found to be contaminated with trichloroethylene,
a cancer-causing chemical, as a result of an abandoned
chemical dump leaching chemicals into the water.
Aproposal submitted to the city’s council members indicates
that the cost, measured in millions of dollars, of
removing x percent of the toxic pollutant is given by
C(x) 0.5x
100 x
Find C(80), C(90), C(95), and C(99) and interpret
your results.
52. D RUG D OSAGES Thomas Young has suggested the following
rule for calculating the dosage of medicine for children
1 to 12 yr old. If a denotes the adult dosage (in
milligrams) and if t is the child’s age (in years), then the
child’s dosage is given by
D(t) at
t 12
Suppose that the adult dosage of a substance is 500 mg.
Find an expression that gives the rate of change of a
child’s dosage with respect to the child’s age. What is
the rate of change of a child’s dosage with respect to his
or her age for a 6-yr-old child? For a 10-yr-old child?
53. E FFECT OF B ACTERICIDE The number of bacteria N(t) ina
certain culture t min after an experimental bactericide
is introduced obeys the rule
N(t) 10,000
2000
2 1 t
Find the rate of change of the number of bacteria in the
culture 1 minute after and 2 minutes after the bactericide
is introduced. What is the population of the bacteria
in the culture 1 min and 2 min after the bactericide
is introduced?
54. D EMAND F UNCTIONS The demand function for the Sicard
wristwatch is given by
50
d(x)
0.01x 2 (0 x 20)
1
where x (measured in units of a thousand) is the quantity
demanded per week and d(x) is the unit price in dollars.