Explanation Determine the derivatives of the given function and use that to determine the value of the second derivate in the interval given Given: Let f ( x ) = sin ( x 2 ) and I = ∫ 0 1 f ( x ) d x Calculation: Given f ( x ) = sin ( x 2 ) Hence f ' ( x ) = 2 x cos ( x 2 ) ; f ' ' ( x ) = 2 x ( − sin ( x 2 ) .2 x ) + 2 cos ( x 2 ) = 2 cos ( x 2 ) − 4 x 2 sin ( x 2 ) To determine (b) To show that Error ( M N ) is at most 1 4 N 2 To determine (c) An N such that | 1 − M N | ≤ 10 − 3

3rd Edition

ISBN: 9781464125263

Author: Jon Rogawski, Colin Adams

Publisher: W.H. Freeman & Co

See similar textbooks

Concept explainers

Limits

When it comes to calculus, limits are considered to be a very important topic of discussion. The main importance of limit lies in expressing the value of a function as it gets closer to a certain value. Also, limits can help you to understand other topic…

Videos

Question

Chapter 8.9, Problem 53E

To determine

(a)

To check that $f'' (x) = 2 \cos (x^{2}) - 4 x^{2} \sin (x^{2})$ and then show that $| f'' (x) \leq 6 |$ for $x \in [0, 1]$ using the hint that $| 2 \cos (x^{2}) | \leq 2$ and $| 4 x^{2} \sin (x^{2}) | \leq 4$ for $x \in [0, 1]$

To determine

(b)

To show that Error ( $M_{N}$ ) is at most $\frac{1}{4 N^{2}}$

To determine

(c)

An $N$ such that $| 1 - M_{N} | \leq 10^{- 3}$

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 8.9, Problem 52E

Chapter 8.9, Problem 54E

Students have asked these similar questions

Let f(x) = sinx. Show that f(x+h)-f(x) h (sinh) COS x = sin x (1-cos h).

Let f(x) = (3 sin z + 4 cos a)tan = (2),f

If f(x) = cos(2x), then f"

Chapter 8 Solutions

Calculus - Standalone book

Ch. 8.1 - Prob. 1PQ Ch. 8.1 - Prob. 2PQ Ch. 8.1 - Prob. 3PQ Ch. 8.1 - Prob. 1E Ch. 8.1 - Prob. 2E Ch. 8.1 - Prob. 3E Ch. 8.1 - Prob. 4E Ch. 8.1 - Prob. 5E Ch. 8.1 - Prob. 6E Ch. 8.1 - Prob. 7E

Knowledge Booster

$Calculus$

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, calculus and related others by exploring similar questions and additional content below.

(a) To check that f ' ' ( x ) = 2 cos ( x 2 ) − 4 x 2 sin ( x 2 ) and then show that | f ' ' ( x ) ≤ 6 | for x ∈ [ 0 , 1 ] using the hint that | 2 cos ( x 2 ) | ≤ 2 and | 4 x 2 sin ( x 2 ) | ≤ 4 for x ∈ [ 0 , 1 ]

Solution Summary: The author explains how to determine the derivatives of the given function and the second derivate in the interval given.

Concept explainers

Videos

(a)

To check that $f'' (x) = 2 \cos (x^{2}) - 4 x^{2} \sin (x^{2})$ and then show that $| f'' (x) \leq 6 |$ for $x \in [0, 1]$ using the hint that $| 2 \cos (x^{2}) | \leq 2$ and $| 4 x^{2} \sin (x^{2}) | \leq 4$ for $x \in [0, 1]$

(b)

To show that Error ( $M_{N}$ ) is at most $\frac{1}{4 N^{2}}$

(c)

An $N$ such that $| 1 - M_{N} | \leq 10^{- 3}$

Want to see the full answer?

Chapter 8 Solutions

(a) To check that f ' ' ( x ) = 2 cos ( x 2 ) − 4 x 2 sin ( x 2 ) and then show that | f ' ' ( x ) ≤ 6 | for x ∈ [ 0 , 1 ] using the hint that | 2 cos ( x 2 ) | ≤ 2 and | 4 x 2 sin ( x 2 ) | ≤ 4 for x ∈ [ 0 , 1 ]

Solution Summary: The author explains how to determine the derivatives of the given function and the second derivate in the interval given.

Concept explainers

Videos

(a) To check that f''(x)=2cos(x2)−4x2sin(x2) and then show that |f''(x)≤6| for x∈[0,1] using the hint that |2cos(x2)|≤2 and |4x2sin(x2)|≤4 for x∈[0,1]

(b) To show that Error (MN) is at most 14N2

(c) An Nsuch that |1−MN|≤10−3

Want to see the full answer?

Chapter 8 Solutions

(a)

To check that $f'' (x) = 2 \cos (x^{2}) - 4 x^{2} \sin (x^{2})$ and then show that $| f'' (x) \leq 6 |$ for $x \in [0, 1]$ using the hint that $| 2 \cos (x^{2}) | \leq 2$ and $| 4 x^{2} \sin (x^{2}) | \leq 4$ for $x \in [0, 1]$

(b)

To show that Error ( $M_{N}$ ) is at most $\frac{1}{4 N^{2}}$

(c)

An $N$ such that $| 1 - M_{N} | \leq 10^{- 3}$