4. a) For 0≤x≤ 1, show that < x² √1+x < x² b) Use the estimate in a) and integrate it to prove that 32/1/2 = S/² x² 3√2 √1+x 3 -dr < c) By estimating 2x/sinx on the interval [/6, 7/2], prove that 477² 9 277² 9 ≤ π/6 2x sin a
4. a) For 0≤x≤ 1, show that < x² √1+x < x² b) Use the estimate in a) and integrate it to prove that 32/1/2 = S/² x² 3√2 √1+x 3 -dr < c) By estimating 2x/sinx on the interval [/6, 7/2], prove that 477² 9 277² 9 ≤ π/6 2x sin a
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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![### Calculus Problem Set
#### Problem 4:
**a)** For \( 0 \leq x \leq 1 \), show that
\[
\frac{x^2}{\sqrt{2}} \leq \frac{x^2}{\sqrt{1 + x}} \leq x^2.
\]
**b)** Use the estimate in part (a) and integrate it to prove that
\[
\frac{1}{3\sqrt{2}} \leq \int_0^1 \frac{x^2}{\sqrt{1+x}} \, dx \leq \frac{1}{3}.
\]
**c)** By estimating \(\frac{2x}{\sin x}\) on the interval \([\pi/6, \pi/2]\), prove that
\[
\frac{2\pi^2}{9} \leq \int_{\pi/6}^{\pi/2} \frac{2x}{\sin x} \, dx \leq \frac{4\pi^2}{9}.
\]
#### Explanation of Diagrams or Graphs:
There are no diagrams or graphs included in this problem set. The exercise involves manipulating and evaluating integrals based on given conditions.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F1f647d9f-6880-40dd-a686-e7d1674263c8%2F36cee35b-9870-4f61-a8bc-19e55a317f83%2F6jnoqk9_processed.png&w=3840&q=75)
Transcribed Image Text:### Calculus Problem Set
#### Problem 4:
**a)** For \( 0 \leq x \leq 1 \), show that
\[
\frac{x^2}{\sqrt{2}} \leq \frac{x^2}{\sqrt{1 + x}} \leq x^2.
\]
**b)** Use the estimate in part (a) and integrate it to prove that
\[
\frac{1}{3\sqrt{2}} \leq \int_0^1 \frac{x^2}{\sqrt{1+x}} \, dx \leq \frac{1}{3}.
\]
**c)** By estimating \(\frac{2x}{\sin x}\) on the interval \([\pi/6, \pi/2]\), prove that
\[
\frac{2\pi^2}{9} \leq \int_{\pi/6}^{\pi/2} \frac{2x}{\sin x} \, dx \leq \frac{4\pi^2}{9}.
\]
#### Explanation of Diagrams or Graphs:
There are no diagrams or graphs included in this problem set. The exercise involves manipulating and evaluating integrals based on given conditions.
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