5128_Ch03_pp098-184
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140 Chapter 3 Derivatives<br />
47. Finding f from f Let f x 3x 2 .<br />
(a) Compute the derivatives of gx x 3 , hx x 3 2, and<br />
tx x 3 3. g(x) h(x) t(x) 3x 2<br />
(b) Graph the numerical derivatives of g, h, and t.<br />
(c) Describe a family of functions, f x, that have the property<br />
that f x 3x 2 . f(x) must be of the form f (x) x 3 c, where c is<br />
a constant.<br />
(d) Is there a function f such that f x 3x 2 and f 0 0?<br />
If so, what is it? Yes. f (x) x 3<br />
(e) Is there a function f such that f x 3x 2 and f 0 3?<br />
If so, what is it? Yes. f(x) x 3 3<br />
48. Airplane Takeoff Suppose that the distance an aircraft travels<br />
along a runway before takeoff is given by D 109t 2 , where<br />
D is measured in meters from the starting point and t is measured<br />
9. (a) Move forward: 0 t 1 and 5 t 7<br />
move backward: 1 t 5<br />
speed up: 1 t 2 and 5 t 6<br />
slow down: 0 t 1, 3 t 5, and 6 t 7<br />
(b) Positive: 3 t 6<br />
negative: 0 t 2 and 6 t 7<br />
zero: 2 t 3 and 7 t 9<br />
(c) At t 0 and 2 t 3<br />
(d) 7 t 9<br />
10. (a) Left: 2 t 3, 5 t 6<br />
Right: 0 t 1<br />
Standing still: 1 t 2, 3 t 5<br />
29. (d) The maximum occurs when x 106.44. Since x must be an integer,<br />
P(106) 4.924 thousand dollars or $4924.<br />
(e) $13 per package sold, $165 per package sold, $118 per package<br />
sold, $31 per package sold, $6 per package sold, P(300) 0 (on<br />
the order of 10 6 , or $0.001 per package sold)<br />
(f) The limit is 10. Maximum possible profit is $10,000 monthly.<br />
(g) Yes. In order to sell more and more packages, the company might<br />
need to lower the price to a point where they won’t make any additional<br />
profit.<br />
37. (a) It begins at the point (5, 2) moving in the positive direction. After a<br />
little more than one second, it has moved a bit past (6, 2) and it turns<br />
back in the negative direction for approximately 2 seconds. At the end<br />
of that time, it is near (2, 2) and it turns back again in the positive direction.<br />
After that, it continues moving in the positive direction indefinitely,<br />
speeding up as it goes.<br />
37. (e) The velocity starts out positive but decreasing, it becomes negative,<br />
then starts to increase, and becomes positive again and continues to<br />
increase.<br />
39. Since profit revenue cost, using Rule 4 (the “difference rule”), and<br />
taking derivatives, we see that marginal profit<br />
= marginal revenue – marginal cost.<br />
in seconds from the time the brakes are released. If the aircraft<br />
will become airborne when its speed reaches 200 kmh, how<br />
long will it take to become airborne, and what distance will it<br />
have traveled by that time? It will take 25 seconds, and the aircraft<br />
will have traveled approximately 694.444 meters.<br />
Extending the Ideas<br />
49. Even and Odd Functions<br />
(a) Show that if f is a differentiable even function, then f is an<br />
odd function.<br />
(b) Show that if f is a differentiable odd function, then f is an<br />
even function.<br />
50. Extended Product Rule Derive a formula for the derivative<br />
of the product fgh of three differentiable functions.<br />
d df<br />
dg dh<br />
fgh gh f h fg <br />
dx<br />
d x d x dx<br />
49. (a) Assume that f is even. Then,<br />
f (x h) f (x)<br />
f (x) lim <br />
h→0 h<br />
lim f (x h ) f (x)<br />
,<br />
h→0 h<br />
and substituting k h,<br />
lim f(x k) f(x)<br />
<br />
k→0 k<br />
lim f (x k ) f (x)<br />
f(x)<br />
k→0 k<br />
So, f is an odd function.<br />
(b) Assume that f is odd. Then,<br />
f(x) lim f (x h ) f (x)<br />
<br />
h→0 h<br />
lim f(x h) f(x)<br />
,<br />
h→0 h<br />
and substituting k h,<br />
lim f(x k) f(x)<br />
<br />
k→0 k<br />
lim f(x k ) f(x)<br />
f(x)<br />
k→0 k<br />
So, f is an even function.<br />
46. At t 0: 10,000 bacteria/hour<br />
At t 5: 0 bacteria/hour<br />
At t 10: 10,000 bacteria/hour