11DIFFERENTIATION - Department of Mathematics

758 11 DIFFERENTIATION
vertical takeoff mode, its altitude (in feet) was
h(t) 1
16 t 4 t 3 4t 2 (0 t 8)
a. Find an expression for the craft’s velocity at time t.
b. Find the craft’s velocity when t 0 (the initial velocity),
t 4, and t 8.
c. Find an expression for the craft’s acceleration at
time t.
d. Find the craft’s acceleration when t 0, 4, and 8.
e. Find the craft’s height when t 0, 4, and 8.
34. U.S. CENSUS According to the U.S. Census Bureau, the
number of Americans aged 45 to 54 will be approximately
N(t) 0.00233t 4 0.00633t 3 0.05417t 2
1.3467t 25
million people in year t, where t 0 corresponds to
the beginning of 1990. Compute N(10) and N(10) and
interpret your results.
Source: U.S. Census Bureau
35. A IR P URIFICATION During testing of a certain brand of
air purifier, it was determined that the amount of smoke
remaining t min after the start of the test was
A(t) 0.00006t 5 0.00468t 4 0.1316t 3
1.915t 2 17.63t 100
percent of the original amount. Compute A(10) and
A(10) and interpret your results.
Source: Consumer Reports
S OLUTIONS TO S ELF-CHECK E XERCISES 11.5
In Exercises 36–39, determine whether the
statement is true or false. If it is true, explain
why it is true. If it is false, give an example to
show why it is false.
36. If the second derivative of f exists at x a, then
f (a) [f(a)] 2 .
37. If h fg where f and g have second-order derivatives,
then
h(x) f (x)g(x) 2f(x)g(x) f(x)g(x)
38. If f(x) is a polynomial function of degree n, then
f (n1) (x) 0.
39. Suppose P(t) represents the population of bacteria at
time t and suppose P(t) 0 and P(t) 0; then the
population is increasing at time t but at a decreasing rate.
40. Let f be the function defined by the rule f(x) x 7/3 .
Show that f has first- and second-order derivatives at all
points x, in particular at x 0. Show also that the third
derivative of f does not exist at x 0.
41. Construct a function f that has derivatives of order up
through and including n at a point a but fails to have
the (n 1)st derivative there.
Hint: See Exercise 40.
42. Show that a polynomial function has derivatives of all
orders.
Hint: Let P(x) a 0x n a 1x n1 a 2x n2 a n be a
polynomial of degree n, where n is a positive integer and a0,
a 1,...,a n are constants with a 0 0. Compute P(x), P(x),....
1. f(x) 10x4 9x2 2x 6
f (x) 40x3 18x 2
f (x) 120x2 18
2. We write f(x) (1 x) 1 and use the general power rule, obtaining
2 d
f(x) (1)(1 x)
dx (1 x) (1 x)2 (1)
(1 x) 2
1
(1 x) 2