Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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Please answer these questions!
Thank you!
![12:29
Done
9 of 27
1. Use Riemann sums to compute the integral xdx. Hint. Divide the
interval [0, 1] into n equal parts.
2. Find the integrals using the Fundamental Theorem of Calculus.
(»)
sin xdx.
(b)
cos xdx.
1
-dx.
cos? x
1
dx.
(d)
3. Show that
| Vi+x*dx < vV2.
Hint. Use the fact that if m < f(x) < M, then m(b – a) < S. f(x)dx < M(b – a).
4. Use the Cauchy-Schwarz inequality show
dt
b - a
Vab
5. Prove that the Mean Value Theorem for integrals (Theorem 1.3.6) is a
consequence of the Mean Value Theorem for derivatives applied to the
function F(x) = S" f(t)dt.
6. Let f' be continuous on [a, b). Prove that the Mean Value Theorem for
derivatives, is a consequence of the Mean Value Theorem for integrals
applied to f'.
7. Find the mean value of f (x)
= a.x + b in the interval [x1, x2).
1
8. Find the mean value of f(x) = x³ in the interval [0, 1].
9. Find the mean value of f(x) = V in the interval [0, 1].
%3D
10. Find the derivative F'(x), where
cx³
(a) F(x) =
cos tdt. (b) F(x) = |.
In tdt.
11. Let f(x) = S (1+ sin(sin t))đt. Find (f-1)'(0).
12. Show that
S sin t?dt
lim
1
3
13. Let f : [0, a] → R be a continuous function and
SE SE F(t)dt if x > 0,
g(x) =
f (0)
if x = 0.
Show that g is continuous at x = 0. Moreover, show that if f is differ-
entiable at x = 0, then g is differentiable on [0, a].](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Feb01d057-cd2a-4813-8156-f003bc7ba153%2Fa5cc6504-bc77-4153-937b-a1abba9aca66%2F2ngzztr_processed.png&w=3840&q=75)
Transcribed Image Text:12:29
Done
9 of 27
1. Use Riemann sums to compute the integral xdx. Hint. Divide the
interval [0, 1] into n equal parts.
2. Find the integrals using the Fundamental Theorem of Calculus.
(»)
sin xdx.
(b)
cos xdx.
1
-dx.
cos? x
1
dx.
(d)
3. Show that
| Vi+x*dx < vV2.
Hint. Use the fact that if m < f(x) < M, then m(b – a) < S. f(x)dx < M(b – a).
4. Use the Cauchy-Schwarz inequality show
dt
b - a
Vab
5. Prove that the Mean Value Theorem for integrals (Theorem 1.3.6) is a
consequence of the Mean Value Theorem for derivatives applied to the
function F(x) = S" f(t)dt.
6. Let f' be continuous on [a, b). Prove that the Mean Value Theorem for
derivatives, is a consequence of the Mean Value Theorem for integrals
applied to f'.
7. Find the mean value of f (x)
= a.x + b in the interval [x1, x2).
1
8. Find the mean value of f(x) = x³ in the interval [0, 1].
9. Find the mean value of f(x) = V in the interval [0, 1].
%3D
10. Find the derivative F'(x), where
cx³
(a) F(x) =
cos tdt. (b) F(x) = |.
In tdt.
11. Let f(x) = S (1+ sin(sin t))đt. Find (f-1)'(0).
12. Show that
S sin t?dt
lim
1
3
13. Let f : [0, a] → R be a continuous function and
SE SE F(t)dt if x > 0,
g(x) =
f (0)
if x = 0.
Show that g is continuous at x = 0. Moreover, show that if f is differ-
entiable at x = 0, then g is differentiable on [0, a].
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