40. The gravitational force exerted by the earth on a unit mass at a
distance r from the center of the planet is
where M is the mass of the earth, R is its radius, and G is the
gravitational constant. Is F a continuous function of r ?
For what value of the constant is the function continuous
on ?
42. Find the values of and that make continuous everywhere.
43. Which of the following functions has a removable disconti-
nuity at ? If the discontinuity is removable, find a function
that agrees with for and is continuous at .
(a) ,
(b) ,
(c) ,
44. Suppose that a function is continuous on [0, 1] except at
0.25 and that and . Let N ! 2. Sketch two
possible graphs of , one showing that might not satisfy
the conclusion of the Intermediate Value Theorem and one
showing that might still satisfy the conclusion of the
Intermediate Value Theorem (even though it doesn’t satisfy the
hypothesis).
45. If , show that there is a number such
that .
46. Suppose is continuous on and the only solutions of the
equation are and . If , explain
why .
47–50 Use the Intermediate Value Theorem to show that there is a
root of the given equation in the specified interval.
, 48.
,
49. , 50. , !0, 1.4"tan x ! 2x!0, 1"cos x ! x
!0, 1"
s
3
x
! 1 " x!1, 2"x
4
! x " 3 ! 0
47.
f !3" ' 6
f !2" ! 8x ! 4x ! 1f !x" ! 6
$1, 5%f
f !c" ! 1000
cf !x" ! x
2
! 10 sin x
f
ff
f !1" ! 3f !0" ! 1
f
a !
&
f !x" ! * sin x +
a ! 2f !x" !
x
3
" x
2
" 2x
x " 2
a ! 1f !x" !
x
4
" 1
x " 1
ax " af
ta
f
f !x" !
x
2
" 4
x " 2
ax
2
" bx ! 3
2x " a ! b
if x
(
2
if 2
(
x
(
3
if x ) 3
fba
f !x" !
)
cx
2
! 2x
x
3
" cx
if x
(
2
if x ) 2
!"$, $"
fc
41.
if r ) R
GM
r
2
F!r" !
GMr
R
3
if r
(
R
SECTION 2.5 CONTINUITY
|| ||
107
51–52 (a) Prove that the equation has at least one real root.
(b) Use your calculator to find an interval of length 0.01 that con-
tains a root.
51. 52.
;
53–54 (a) Prove that the equation has at least one real root.
(b) Use your graphing device to find the root correct to three deci-
mal places.
53. 54.
55. Prove that is continuous at if and only if
56. To prove that sine is continuous, we need to show that
for every real number . By Exercise 55
an equivalent statement is that
Use (6) to show that this is true.
57. Prove that cosine is a continuous function.
58. (a) Prove Theorem 4, part 3.
(b) Prove Theorem 4, part 5.
59. For what values of is continuous?
60. For what values of is continuous?
Is there a number that is exactly 1 more than its cube?
62. If and are positive numbers, prove that the equation
has at least one solution in the interval .
63. Show that the function
is continuous on .
64. (a) Show that the absolute value function is contin-
uous everywhere.
(b) Prove that if is a continuous function on an interval, then
so is .
&
f
&
f
F!x" !
&
x
&
!"$, $"
f !x" !
)
x
4
sin!1#x"
0
if x " 0
if x ! 0
!"1, 1"
a
x
3
! 2x
2
" 1
!
b
x
3
! x " 2
! 0
ba
61.
t!x" !
)
0
x
if x is rational
if x is irrational
tx
f !x" !
)
0
1
if x is rational
if x is irrational
fx
lim
h l 0
sin!a ! h" ! sin a
alim
x l a
sin x ! sin a
lim
h l 0
f !a ! h" ! f !a"
af
s
x " 5
!
1
x ! 3
x
5
" x
2
" 4 ! 0
x
5
" x
2
! 2x ! 3 ! 0cos x ! x
3