James Stewart-Calculus_ Early Transcendentals-Cengage Learning (2015)

350 Chapter 4 Applications of Differentiation
36. Use Newton’s method to find the absolute maximum value
of the function f sxd − x cos x, 0 < x < , correct to six
decimal places.
37. Use Newton’s method to find the coordinates of the
inflection point of the curve y − x 2 sin x, 0 < x < ,
correct to six decimal places.
38. Of the infinitely many lines that are tangent to the curve
y − 2sin x and pass through the origin, there is one that
has the largest slope. Use Newton’s method to find the
slope of that line correct to six decimal places.
39. Use Newton’s method to find the coordinates, correct to six
decimal places, of the point on the parabola y − sx 2 1d 2
that is closest to the origin.
40. In the figure, the length of the chord AB is 4 cm and the
length of the arc AB is 5 cm. Find the central angle ,
in radians, correct to four decimal places. Then give the
answer to the nearest degree.
A
5 cm
4 cm
¨
B
Replacing i by x, show that
48xs1 1 xd 60 2 s1 1 xd 60 1 1 − 0
Use Newton’s method to solve this equation.
42. The figure shows the sun located at the origin and the earth
at the point s1, 0d. (The unit here is the distance between
the centers of the earth and the sun, called an astronomical
unit: 1 AU < 1.496 3 10 8 km.) There are five locations
L 1, L 2, L 3, L 4, and L 5 in this plane of rotation of the earth
about the sun where a satellite remains motionless with
respect to the earth because the forces acting on the satellite
(including the gravitational attractions of the earth and the
sun) balance each other. These locations are called libration
points. (A solar research satellite has been placed at one of
these libration points.) If m 1 is the mass of the sun, m 2 is the
mass of the earth, and r − m 2ysm 1 1 m 2d, it turns out that
the x-coordinate of L 1 is the unique root of the fifth-degree
equation
psxd − x 5 2 s2 1 rdx 4 1 s1 1 2rdx 3 2 s1 2 rdx 2
1 2s1 2 rdx 1 r 2 1 − 0
and the x-coordinate of L 2 is the root of the equation
psxd 2 2rx 2 − 0
Using the value r < 3.04042 3 10 26 , find the locations of the
libration points (a) L 1 and (b) L 2.
41. A car dealer sells a new car for $18,000. He also offers to
sell the same car for payments of $375 per month for five
years. What monthly interest rate is this dealer charging?
To solve this problem you will need to use the formula
for the present value A of an annuity consisting of n equal
payments of size R with interest rate i per time period:
L∞
sun
y
L¢
earth
L¡ L
x
A − R i
f1 2 s1 1 id 2n g
L£
A physicist who knows the velocity of a particle might wish to know its position at a
given time. An engineer who can measure the variable rate at which water is leaking
from a tank wants to know the amount leaked over a certain time period. A biologist who
knows the rate at which a bacteria population is increasing might want to deduce what
the size of the population will be at some future time. In each case, the problem is to
find a function F whose derivative is a known function f. If such a function F exists, it
is called an antiderivative of f.
Definition A function F is called an antiderivative of f on an interval I if
F9sxd − f sxd for all x in I.
Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s).
Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.