Recall, from Problem 14, that f 2 0 if f(x) > 0 for all x in [a, b]. (a) Give an example where f(x) > 0 for all x, and f(x) > 0 for some x in [a, b], and f = 0. (b) Suppose f(x) > 0 for all x in [a, b] and f is continuous at xo in [a, b] and f(xo) > 0. Prove that f> 0. Hint: It suffices to find one lower sum L(f, P) which is positive. (c) Suppose f is integrable on [a, b] and f(x) > 0 for all x in [a, b]. Prove that f > 0. Hint: You will need Problem 31; indeed that was one reason for including Problem 31.
Recall, from Problem 14, that f 2 0 if f(x) > 0 for all x in [a, b]. (a) Give an example where f(x) > 0 for all x, and f(x) > 0 for some x in [a, b], and f = 0. (b) Suppose f(x) > 0 for all x in [a, b] and f is continuous at xo in [a, b] and f(xo) > 0. Prove that f> 0. Hint: It suffices to find one lower sum L(f, P) which is positive. (c) Suppose f is integrable on [a, b] and f(x) > 0 for all x in [a, b]. Prove that f > 0. Hint: You will need Problem 31; indeed that was one reason for including Problem 31.
Chapter6: Exponential And Logarithmic Functions
Section6.1: Exponential Functions
Problem 60SE: The formula for the amount A in an investmentaccount with a nominal interest rate r at any timet is...
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Solve question no 32
![8:01 AM Ø
VOLTE I 4 17%
Lo ti snonun
muous Tunçtion g With la
function with this property, and then a continuous one. A picture will
help immensely.
29. (a) Show that if s1 and s, are step functions on [a, b], then s1 + s2 is also.
(b) Prove, without using Theorem 5, that (s1 + s2) =
(c) Use part (b) (and Problem 27) to give an alternative proof of
f s1 + s2.
Theorem 5.
30. Suppose that f is integrable on [a, b]. Prove that there is a number x in
[a, b] such that j = [" j. Show by example that it is not always possi-
ble to choose x to be in (a, b).
*31.
The purpose of this problem is to show that if f is integrable on [a, b],
then f must be continuous at many points in [a, b].
(a) Let P = {to, . .. , tn} be a partition of [0, 1] with U(f, P)
L(f, P) < b - a. Prove that for some i we have M; - m; < 1.
(b) Prove that there are numbers a, and bị with a < a, < bị < b and
sup {f(x): a1 < x < bi} - inf {{(x): a1 < x < b1} < 1. (You can
choose [a1, b1] = [t;-1, t4] from part (a) unless i = 1 or n; and in these
two cases a very simple device solves the problem.)
(c) Prove that there are numbers ag and b2 with a1 < a2 < bz < bị and
sup { f(x): az < x < b2} – inf {f(x): a2 < x < b2} < }.
(d) Continue in this way to find a sequence of intervals In = [an, bn]
such that sup {f(x): x in In} – inf {f(x): x in I,} < 1/n. Apply the
Nested Intervals Theorem (Problem 8-14) to find a point x at which
f is continuous.
(e) Prove that f is continuous at infinitely many points in [a, b].
*32.
Recall, from Problem 14, that f 2 O if f(x) > 0 for all x in [a, b].
(a) Give an example where f(x) > 0 for all x, and f(x) > 0 for some x in
[a, b], and f = 0.
(b) Suppose f(x) > 0 for all x in [a, b] and f is continuous at x, in [a, b]
and f(xo) > 0. Prove that f > 0. Hint: It suffices to find one
lower sum L(f, P) which is positive.
(c) Suppose f is integrable on [a, b] and f(x) > 0 for all x in [a, b].
Prove that "ƒ > 0. Hint: You will need Problem 31 ; indeed that
was one reason for including Problem 31.
*33. (a) Suppose that f is continuous on [a, b} and ig = 0 for all con-
tinuous functions g on [a, b]. Prove that f = 0. (This is easy; there
is an obvious g to choose.)
(b) Suppose f is continuous on [a, b] and that ° fg = 0 for those con-
tinuous functions g on [a, b] which satisfy the extra conditions
g(a) = g(b) = 0. Prove that f = 0. (This innocent looking fact is
the coleulun of variations: co the Surzented
an impentant lem](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F49e60f1d-a03d-4abd-b02e-d7681ad9c98e%2F40011896-639c-4553-8a60-62fb439effeb%2Fxyhu735_processed.png&w=3840&q=75)
Transcribed Image Text:8:01 AM Ø
VOLTE I 4 17%
Lo ti snonun
muous Tunçtion g With la
function with this property, and then a continuous one. A picture will
help immensely.
29. (a) Show that if s1 and s, are step functions on [a, b], then s1 + s2 is also.
(b) Prove, without using Theorem 5, that (s1 + s2) =
(c) Use part (b) (and Problem 27) to give an alternative proof of
f s1 + s2.
Theorem 5.
30. Suppose that f is integrable on [a, b]. Prove that there is a number x in
[a, b] such that j = [" j. Show by example that it is not always possi-
ble to choose x to be in (a, b).
*31.
The purpose of this problem is to show that if f is integrable on [a, b],
then f must be continuous at many points in [a, b].
(a) Let P = {to, . .. , tn} be a partition of [0, 1] with U(f, P)
L(f, P) < b - a. Prove that for some i we have M; - m; < 1.
(b) Prove that there are numbers a, and bị with a < a, < bị < b and
sup {f(x): a1 < x < bi} - inf {{(x): a1 < x < b1} < 1. (You can
choose [a1, b1] = [t;-1, t4] from part (a) unless i = 1 or n; and in these
two cases a very simple device solves the problem.)
(c) Prove that there are numbers ag and b2 with a1 < a2 < bz < bị and
sup { f(x): az < x < b2} – inf {f(x): a2 < x < b2} < }.
(d) Continue in this way to find a sequence of intervals In = [an, bn]
such that sup {f(x): x in In} – inf {f(x): x in I,} < 1/n. Apply the
Nested Intervals Theorem (Problem 8-14) to find a point x at which
f is continuous.
(e) Prove that f is continuous at infinitely many points in [a, b].
*32.
Recall, from Problem 14, that f 2 O if f(x) > 0 for all x in [a, b].
(a) Give an example where f(x) > 0 for all x, and f(x) > 0 for some x in
[a, b], and f = 0.
(b) Suppose f(x) > 0 for all x in [a, b] and f is continuous at x, in [a, b]
and f(xo) > 0. Prove that f > 0. Hint: It suffices to find one
lower sum L(f, P) which is positive.
(c) Suppose f is integrable on [a, b] and f(x) > 0 for all x in [a, b].
Prove that "ƒ > 0. Hint: You will need Problem 31 ; indeed that
was one reason for including Problem 31.
*33. (a) Suppose that f is continuous on [a, b} and ig = 0 for all con-
tinuous functions g on [a, b]. Prove that f = 0. (This is easy; there
is an obvious g to choose.)
(b) Suppose f is continuous on [a, b] and that ° fg = 0 for those con-
tinuous functions g on [a, b] which satisfy the extra conditions
g(a) = g(b) = 0. Prove that f = 0. (This innocent looking fact is
the coleulun of variations: co the Surzented
an impentant lem
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