5128_Ch03_pp098-184
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<strong>184</strong> Chapter 3 Derivatives<br />
75. Searchlight The figure shows a boat 1 km offshore sweeping<br />
the shore with a searchlight. The light turns at a constant rate,<br />
dudt 0.6 radsec.<br />
(a) How fast is the light moving along the shore when it reaches<br />
point A? 0.6 km/sec<br />
(b) How many revolutions per minute is 0.6 radsec?<br />
<br />
1 km<br />
x<br />
A<br />
18/ <br />
5.73 revolutions/min<br />
(a) Estimate the initial number of students infected with<br />
measles. P(0) 1.339, so initially, one student was infected<br />
(b) About how many students in all will get the measles? 200<br />
(c) When will the rate of spread of measles be greatest? What is<br />
this rate? After 5 days, when the rate is 50 students/day<br />
79. Graph the function f x tan 1 tan 2x in the window<br />
p, p by 4, 4. Then answer the following questions.<br />
(a) What is the domain of f ?<br />
(b) What is the range of f ?<br />
x k , where k is an odd integer<br />
4<br />
(2,2)<br />
(c) At which points is f not differentiable?<br />
(d) Describe the graph of f .<br />
80. If x 2 y 2 1, find d 2 ydx 2 at the point 2, 3. 1/(33)<br />
76. Horizontal Tangents The graph of y sin x sin x<br />
appears to have horizontal tangents at the x-axis. Does it? Yes<br />
77. Fundamental Frequency of a Vibrating Piano String<br />
We measure the frequencies at which wires vibrate in cycles<br />
(trips back and forth) per sec. The unit of measure is a hertz:<br />
1 cycle per sec. Middle A on a piano has a frequency 440 hertz.<br />
For any given wire, the fundamental frequency y is a function<br />
of four variables:<br />
r: the radius of the wire;<br />
l: the length;<br />
d: the density of the wire;<br />
T: the tension (force) holding the wire taut.<br />
With r and l in centimeters, d in grams per cubic centimeter, and<br />
T in dynes (it takes about 100,000 dynes to lift an apple), the<br />
fundamental frequency of the wire is<br />
1<br />
y 2 rl <br />
T<br />
p<br />
d<br />
.<br />
If we keep all the variables fixed except one, then y can be<br />
alternatively thought of as four different functions of one<br />
variable, yr, yl, yd, and yT . How would changing<br />
each variable affect the string’s fundamental frequency? To find<br />
out, calculate yr, yl, yd, and yT .<br />
78. Spread of Measles The spread of measles in a certain school<br />
is given by<br />
200<br />
Pt ,<br />
1 e5t<br />
where t is the number of days since the measles first appeared,<br />
and Pt is the total number of students who have caught the<br />
measles to date.<br />
<br />
1 T<br />
77. y(r) 2r 2 l , so increasing r decreases the frequency.<br />
d<br />
1 T<br />
y(l) 2r l 2 , so increasing l decreases the frequency.<br />
d<br />
1 T<br />
y(d) 4 rl <br />
d 3, so increasing d decreases the frequency.<br />
1<br />
y(T) 4rl Td <br />
, so increasing T increases the frequency.<br />
79. (c) Where it’s not defined, at x k , k an odd integer<br />
4<br />
(d) It has period 2 and continues to repeat the pattern seen in this window.<br />
AP* Examination Preparation<br />
You may use a graphing calculator to solve the following<br />
problems.<br />
81. A particle moves along the x-axis so that at any time t 0 its<br />
position is given by x(t) t 3 12t 5.<br />
(a) Find the velocity of the particle at any time t.<br />
(b) Find the acceleration of the particle at any time t.<br />
(c) Find all values of t for which the particle is at rest.<br />
(d) Find the speed of the particle when its acceleration is zero.<br />
(e) Is the particle moving toward the origin or away from the origin<br />
when t 3? Justify your answer.<br />
e<br />
82. Let y x e<br />
x<br />
. 2<br />
(a) Find d y<br />
.<br />
dx<br />
d<br />
(b) Find 2 y<br />
. dx<br />
2<br />
(c) Find an equation of the line tangent to the curve at x 1.<br />
(d) Find an equation of the line normal to the curve at x 1.<br />
(e) Find any points where the tangent line is horizontal.<br />
83. Let f (x) ln (1 x 2 ).<br />
(a) State the domain of f .<br />
(b) Find f (x).<br />
(c) State the domain of f .<br />
(d) Prove that f (x) 0 for all x in the domain of f .<br />
Additional Answers:<br />
)x ln x<br />
21. 2(ln x <br />
x<br />
(2 2 x )[x 3 ln 2 x ln 2 1]<br />
22. or<br />
(x 2 1) 3/2<br />
(2x)2<br />
x<br />
1 x ln 2 x x <br />
x 2 1<br />
2 1<br />
u<br />
24. u 2 u<br />
u<br />
4 ⏐u ⏐ 1 u 2<br />
t<br />
25. ⏐t⏐ t 2 1<br />
sec1 t <br />
2<br />
1t<br />
26. 2 2t2<br />
2t cot 1 2t<br />
1 4t2