5128_Ch03_pp098-184

140 Chapter 3 Derivatives
47. Finding f from f Let f x 3x 2 .
(a) Compute the derivatives of gx x 3 , hx x 3 2, and
tx x 3 3. g(x) h(x) t(x) 3x 2
(b) Graph the numerical derivatives of g, h, and t.
(c) Describe a family of functions, f x, that have the property
that f x 3x 2 . f(x) must be of the form f (x) x 3 c, where c is
a constant.
(d) Is there a function f such that f x 3x 2 and f 0 0?
If so, what is it? Yes. f (x) x 3
(e) Is there a function f such that f x 3x 2 and f 0 3?
If so, what is it? Yes. f(x) x 3 3
48. Airplane Takeoff Suppose that the distance an aircraft travels
along a runway before takeoff is given by D 109t 2 , where
D is measured in meters from the starting point and t is measured
9. (a) Move forward: 0 t 1 and 5 t 7
move backward: 1 t 5
speed up: 1 t 2 and 5 t 6
slow down: 0 t 1, 3 t 5, and 6 t 7
(b) Positive: 3 t 6
negative: 0 t 2 and 6 t 7
zero: 2 t 3 and 7 t 9
(c) At t 0 and 2 t 3
(d) 7 t 9
10. (a) Left: 2 t 3, 5 t 6
Right: 0 t 1
Standing still: 1 t 2, 3 t 5
29. (d) The maximum occurs when x 106.44. Since x must be an integer,
P(106) 4.924 thousand dollars or $4924.
(e) $13 per package sold, $165 per package sold, $118 per package
sold, $31 per package sold, $6 per package sold, P(300) 0 (on
the order of 10 6 , or $0.001 per package sold)
(f) The limit is 10. Maximum possible profit is $10,000 monthly.
(g) Yes. In order to sell more and more packages, the company might
need to lower the price to a point where they won’t make any additional
profit.
37. (a) It begins at the point (5, 2) moving in the positive direction. After a
little more than one second, it has moved a bit past (6, 2) and it turns
back in the negative direction for approximately 2 seconds. At the end
of that time, it is near (2, 2) and it turns back again in the positive direction.
After that, it continues moving in the positive direction indefinitely,
speeding up as it goes.
37. (e) The velocity starts out positive but decreasing, it becomes negative,
then starts to increase, and becomes positive again and continues to
increase.
39. Since profit revenue cost, using Rule 4 (the “difference rule”), and
taking derivatives, we see that marginal profit
= marginal revenue – marginal cost.
in seconds from the time the brakes are released. If the aircraft
will become airborne when its speed reaches 200 kmh, how
long will it take to become airborne, and what distance will it
have traveled by that time? It will take 25 seconds, and the aircraft
will have traveled approximately 694.444 meters.
Extending the Ideas
49. Even and Odd Functions
(a) Show that if f is a differentiable even function, then f is an
odd function.
(b) Show that if f is a differentiable odd function, then f is an
even function.
50. Extended Product Rule Derive a formula for the derivative
of the product fgh of three differentiable functions.
d df
dg dh
fgh gh f h fg
dx
d x d x dx
49. (a) Assume that f is even. Then,
f (x h) f (x)
f (x) lim
h→0 h
lim f (x h ) f (x)
,
h→0 h
and substituting k h,
lim f(x k) f(x)
k→0 k
lim f (x k ) f (x)
f(x)
k→0 k
So, f is an odd function.
(b) Assume that f is odd. Then,
f(x) lim f (x h ) f (x)
h→0 h
lim f(x h) f(x)
,
h→0 h
and substituting k h,
lim f(x k) f(x)
k→0 k
lim f(x k ) f(x)
f(x)
k→0 k
So, f is an even function.
46. At t 0: 10,000 bacteria/hour
At t 5: 0 bacteria/hour
At t 10: 10,000 bacteria/hour