5128_Ch03_pp098-184

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