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CalculusCalculus (MindTap Course List)8th Edition

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Chapter 3.7, Problem 24E
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Calculus (MindTap Course List)

8th Edition
James Stewart
ISBN: 9781285740621

Solutions

Chapter
Section
Problem 1E
Problem 2E
Problem 3E
Problem 4E
Problem 5E
Problem 6E
Problem 7E
Problem 8E
Problem 9E
Problem 10E
Problem 11E
Problem 12E
Problem 13E
Problem 14E
Problem 15E
Problem 16E
Problem 17E
Problem 18E
Problem 19E
Problem 20E
Problem 21E
Problem 22E
Problem 23E
Problem 24E
Problem 25E
Problem 26E
Problem 27E
Problem 28E
Problem 29E
Problem 30E
Problem 31E
Problem 32E
Problem 33E
Problem 34E
Problem 35E
Problem 36E
Problem 37E
Problem 38E
Problem 39E
Problem 40E
Problem 41E
Problem 42E
Problem 43E
Problem 44E
Problem 45E
Problem 46E
Problem 47E
Problem 48E
Problem 49E
Problem 50E
Problem 51E
Problem 52E
Problem 53E
Problem 54E
Problem 55E
Problem 56E
Problem 57E
Problem 58E
Problem 59E
Problem 60E
Problem 61E
Problem 62E
Problem 63E
Problem 64E
Problem 65E
Problem 66E
Problem 67E
Problem 68E
Problem 69E
Problem 70E
Problem 71E
Problem 72E
Problem 73E
Problem 74E
Problem 75E
Problem 76E
Problem 77E
Problem 78E
Problem 79E
Problem 80E
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Calculus (MindTap Course List)

8th Edition
James Stewart
ISBN: 9781285740621
Textbook Problem

Find, correct to two decimal places, the coordinates of the point on the curve y = sin x that is closest to the point ( 4 , 2 ) .

To determine

To find:

The coordinates of the point on the curve y=sin⁡x closest to the point (4,2) correct to two decimal places.

Explanation

1) Concept:

For finding the closest point, use the distance formula and then apply the first derivative test.

2) Test:

First derivative test:

Suppose c is a critical number of a continuous function f defined on an interval.

(a) If f'(x)>0 for all x<c and f'(x)<0 for all x>c, then f(c) is the absolute maximum value of f.

(b) If f'(x)<0 for all x<c and f'(x)>0 for all x>c, then f(c) is the absolute minimum value of f.

3) Formula:

Distance formula: The distance between two points x1,y1 and (x2,y2) is

d=x2-x12+y2-y12

4) Given:

i) The curve y=sin⁡x

ii) The curve y=sin⁡x is closest to the point (4,2)

5) Calculation:

The distance between point (4,2) and point (x,y) is

d=x-42+y-22

If (x,y) lies on the curve y=sin⁡x then x=sin-1⁡y

The expression for d becomes

d=x-42+sin⁡x-22

For easy simplification instead of minimizing d, minimize its square

d2=fx=x-42+sin⁡x-22

There are no restrictions on x,

The domain is all real numbers.

Differentiating fx with respect to x,

f'x=2x-4+2sin⁡x-2cos⁡x

=2x-4+2sin⁡x-2cos⁡x

Therefore,

f'x=2x-4+2sin⁡x-2cos⁡x

f'x=0

2x-4+2sin⁡x-2cos⁡x=0

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Chapter 3.7,
Problem 23E
Chapter 3.7,
Problem 25E
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Sect-3.1 P-1ESect-3.1 P-2ESect-3.1 P-3ESect-3.1 P-4ESect-3.1 P-5ESect-3.1 P-6ESect-3.1 P-7ESect-3.1 P-8ESect-3.1 P-9ESect-3.1 P-10E
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Elementary Geometry For College Students, 7e

Test the series for convergence or divergence. 15. k=12k13k+1kk

Single Variable Calculus: Early Transcendentals

Minimum Length The wall of a building is to be braced by a beam that must pass over a parallel fence 5 feet hig...

Calculus of a Single Variable

For Questions 1 and 2, fill in the blank with an appropriate word. When simplifying expressions that involve in...

Trigonometry (MindTap Course List)

True or False Label each of the following statements as either true or false. The regular pentagon possesses on...

Elements Of Modern Algebra

An archery set at The Sports Authority in Mason, Ohio, has a retail price of $129.95. The sales tax in Ohio is ...

Contemporary Mathematics for Business & Consumers

Let and g(x) = x + 3. Then (g ∘ h)(x) = ______.

Study Guide for Stewart's Single Variable Calculus: Early Transcendentals, 8th

True or False: Every series must do one of these: converge absolutely, converge conditionally, or diverge.

Study Guide for Stewart's Multivariable Calculus, 8th

Average Value In Exercises 51-56, find the average value of f(x,y) over the plane region R. f(x,y)=2xy R:. rect...

Calculus: Early Transcendental Functions (MindTap Course List)

For Review Exercises 30 and 31, write the converse, inverse, and contrapositive of each statement. If two angle...

Elementary Geometry for College Students

Give a point-slope form, b the slope-intercept form, and c the general equation of a line.

Finite Mathematics for the Managerial, Life, and Social Sciences

Finding a Limit of a Trigonometric Function In Exercises 63-74, find the limit of the trigonometric function. l...

Calculus (MindTap Course List)

Volume In Exercises 19-24, use a triple integral to find the volume of the solid bounded by the graphs of the e...

Multivariable Calculus

Evaluating a Triple Iterated Integral Using Technology In Exercises 11 and 12, use a computer algebra system to...

Calculus

Perform the operations and simplify. z+4z-6

College Algebra (MindTap Course List)

Determine the following numbers with fractional exponents. 31. 1961/2

Mathematics For Machine Technology

The National Association of Home Builders provided data on the cost of the most popular home remodeling project...

STATISTICS F/BUSINESS+ECONOMICS-TEXT

Explain how the Pearson correlation describes the relationship between two variables and predicts the general a...

Research Methods for the Behavioral Sciences (MindTap Course List)

A customer satisfaction survey is planned. The company carrying out the survey plans to contact 50 passengers s...

Introduction To Statistics And Data Analysis

Consider the following data for two independent random samples taken from two normal populations. a. Compute th...

Statistics for Business & Economics, Revised (MindTap Course List)

Describe the goal of single-case research and explain how single-case designs are related to other experimental...

Research Methods for the Behavioral Sciences (MindTap Course List)

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