146
8.
y
6. y
Chapter 2 The Derivative
=
(tanx ± 1)
i.
25. Assuming that the equator is a circle whose radius is ap
proximately 4000 miles, how much longer than the equator would
a concentric, coplanar circle be if each point on it were 2 feet
above the equator? Use differentials.
24. The interior of an open cylindrical tank is 12 feet in diam
eter and 8 feet deep. ‘The bottom is copper and the sides are steel.
Use differentials to find approximately how many gallons of
waterproofing paint are needed to apply a 0.05-inch coat to the
steel part of the inside of the tank (1 gallon
231 cubic inches).
23. ‘The outside diameter of a thin spherical shell is 12 feet. If
the shell is 0.3 inch thick, use differentials to approximate the vol
ume of the region interior to the shell.
22. All six sides of a cubical metal box are 0.25 inch thick, and
the volume of the interior of the box is 40 cubic inches. Use dif
ferentials to find the approximate volume of metal used to make
the box.
121. Approximate the volume of material in a spherical shell
of inner radius 5 centimeters and outer radius 5.125 centimeters
(see Example 3).
20.
18.
hi Problems 18—20, use differentials to approximate the given
number (see .Exanzple 2). Compare ivith calculator values.
Figure 5a shows both the graph of the function f and the linear approximation L
over the interval [0, ‘r]. We can see that the approximation is good near rr/2, but
not so good as you move away from ir/2. Figures 5b and c also show plots of the
functions L and f over
=
(xbo
+
=
. Find the value of dy in each case.
3
x
(b) x = —1,dx = 0.75
In Problems 1—8, find dy.
10. Let y = f(x)
(a) x = 0.5,dx = 1
11. For the function defined in Problem 10, make a careful
drawing of the graph off for —1.5
x
1.5 and the tangents to
the curve at x = 0.5 and x = —1; on this drawing label dy and dx
for each of the given sets of data in parts (a) and (b).
13. For the function defined in Problem 12, make a careful
drawing (as in Problem 11) for —3
x < 0 and 0 < x
3.
y,
E1 14. For the data of Problem 10, find the actual changes in y,
that is, iy.
E1 15. For the data of Problem 12, find the actual changes in
that is, y.
—
16. If y = x
2
3, find the values of y and dv in each case.
(a) x = 2 and dx = x = 0.5
E(b) x = 3anddx = ix = —0.12
x + 2x, find the values of y and dy in each case.
17. If y = 4
(a) x = 2 and dx = Lx = 1
26. The period of a simple pendulum of length L feet is given
2wVL7 seconds. We assume that g, the acceleration due
by T
to gravity on (or very near) the surface of the earth, is 32 feet per
second per second. If the pendulum is that of a clock that keeps
good time when L
4 feet, how much time will the clock gain in
24 hours if the length of the pendulum is decreased to 3.97 feet?
cen
27. The diameter of a sphere is measured as 20 + 0.1
error
timeters. Calculate the volume and estimate the absolute
and the relative error (see Example 4).
28. A cylindrical roller is exactly 12 inches long and its diam
volume with
eter is measured as 6 + 0.005 inches. Calculate its
error.
an estimate for the absolute error and the relative
29. The angle 0 between the two equal sides of an isosceles
triangle measures 0.53 ± 0.005 radian. The two equal sides are
third
exactly 151 centimeters long. Calculate the length of the
relative error.
side with an estimate for the absolute error and the
30. Calculate the area of the triangle of Problem 29 with an
estimate for the absolute error and the relative error. Hint:
A = absin0.
M on a closed inter
dy
y/ds
id
I
31. It can be shown that if 2
val with c and c -I- zx as end points, then
—
—
2.v + 11 when
2
Find, using differentials, the change in y = 3x
x increases from 2 to 2.001 and then give a bound for the error
that you have made by using differentials.
32. Suppose that f is a function satisfying f(I) = 10, and
f’(1.02) = 12. Use this information to approximate f(1.02).
.
=
v’iZ
/c
v
2
‘no
=
ino(1
—
2
2 N/
v
S and
33. Suppose f is a function satisfying f(3)
f’(3.05) = Use this information to approximate [(3.05).
34. A conical cup, 10 centimeters high and g centimeters wide
at the top, is filled with water to a depth of 9 centimeters. An ice
cube 3 centimeters on a side is about to be dropped in. Use dif
ferentials to decide whether the cup will overflow.
35. A tank has the shape of a cylinder with hemispherical
ends. If the cylindrical part is 100 centimeters long and has an out
side diameter of 20 centimeters, about how much paint is required
to coat the outside of the tank to a thickness of 1 millimeter?
36. Einstein’s Special Theory of Relativity says that an ob
ject’s mass 02 is related to its velocity v by the formula
02
2.10 Chapter Review
Concepts Test
Respond with true or false to each of the following assertions. Be
prepared to justify your answer
1. The tangent line to a curve at a point cannot cross the
curve at that point.
2. The tangent line to a curve can touch the curve at only
One
point.
y
x is differ
3. The slope of the tangent line to the curve
ent at every point of the curve.
4. The slope of the tangent line to the curve y = cos s is dif
ferent at every point on the curve.
5. It is possible for the velocity of an object to be increasing
While its speed is decreasing.
Section 2.10 Chapter Review
147
Here 010 is the rest mass and c is the speed of light. Use differen
tials to determine the percent increase in mass of an obj ect when
its velocity increases from 0.9c to 0.92c.
=
2 cos s at a
x
rr/2, [0. ir
ata=2,(0,3]
2
x
in Problems 37—44, find the linear approximation to the given
functions at the specified points. Plot the function and its linear
approximation over the indicated interval.
37. f(x)
38. g(x)
1
6
3x + 4ata’3,[0,
3. h(x) =sinsatao,HT,ir]
=
,
[0. 1)
0,(—nj2,ir/2)
=
0,[—1, 1]
40. F(x)
=
) at a
2
x
=
\/Tata
—
xsecxata
x/(l
=
=
41. f(x)
42. g(x)
43. h(x)
, tO,wj
2
44. (3(x) = s -f sin 2x,ata = w/
45. Find the linear approximation to f(s) = ,ns -f b at an
arbitrary a. What is the relationship between f(s) and L(x)?
4.6. Show that for every a > 0 the linear approximation L(.x)
L(x) for all
to the function f(.t) = Vx at a satisfies f(x)
47. Show that for every a the linear approximation L(.x) to
= x
at a satisfies L(x) < f(s) for alix.
the function f(.x)
(1 -i- x) at
Find a linear approximation to f(s)
x = 0, whereft is anynumber. ForvariousvaluesOf,PI0t f(s)
and its linear approximation L(x). For what values of a does the
linear approximation always overestimate f(s)? For what values
of a does the linear approximation always underestimate f(x)?
—
h—’O
0 and (b) lim
h
c(h)
—
=
0.
2. y; dy
—
149. Suppose f is differentiable. If we use the approximation
f(s) + f’(x)h the error is e(h) = f(x -I-h)
f(x + h)
f’(x)h.Show that
f(s)
(a) liine(h)
Answers to Concepts Review: 1. f’(x) dx
3. ix is small 4. larger; smaller
=
.
4
5w
=
=
g(x) for alix.
f(s) is hori2ontal
Df(g(x)).
7. If the tangent line to the graph of y
x, then
then D,.v
=
6. It is possible for the speed of an object to be increasing
while its velocity is decreasing.
at x
=
=
,
5
ir
f’(g(x))
0.
c, then f’ (c)
f’(x) g’(x) for alls, then f(s)
8. If
9. If g(s)
10. If y
11. If f’ (c) exists, then f is continuous ate.
0 and yet
has a tangent line at s
12. The graph of y =
Dy does not exist there.
13. The derivative of a product is always the product of the
derivatives.