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CalculusCalculus10th Edition

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Chapter 10.4, Problem 54E
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Calculus

10th Edition
Ron Larson + 1 other
ISBN: 9781285057095

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Chapter
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Problem 1E
Problem 2E
Problem 3E
Problem 4E
Problem 5E
Problem 6E
Problem 7E
Problem 8E
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Problem 11E
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Problem 13E
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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
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Problem 31E
Problem 32E
Problem 33E
Problem 34E
Problem 35E
Problem 36E
Problem 37E
Problem 38E
Problem 39E
Problem 40E
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Problem 42E
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Problem 44E
Problem 45E
Problem 46E
Problem 47E
Problem 48E
Problem 49E
Problem 50E
Problem 51E
Problem 52E
Problem 53E
Problem 96E
Problem 97E
Problem 54E
Problem 55E
Problem 56E
Problem 57E
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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
Problem 81E
Problem 82E
Problem 83E
Problem 84E
Problem 85E
Problem 86E
Problem 87E
Problem 88E
Problem 89E
Problem 90E
Problem 91E
Problem 92E
Problem 93E
Problem 94E
Problem 95E
Problem 98E
Problem 99E
Problem 100E
Problem 101E
Problem 102E
Problem 103E
Problem 104E
Problem 105E
Problem 106E
Problem 107E
Problem 108E
Problem 109E
Problem 110E
Problem 111E
Problem 112E
Problem 113E
Problem 114E
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Calculus

10th Edition
Ron Larson + 1 other
ISBN: 9781285057095
Textbook Problem

Distance Formula

(a) Verify that the Distance Formula for the distance between

the two points ( r 1 , θ 1 ) and ( r 2 , θ 2 ) in polar coordinates is d = r 1 2 + r 2 2 − 2 r 1 r 2 cos ( θ 1 − θ 2 ) .

(b) Describe the positions of the points relative to each other for θ 1 = θ 2 , Simplify the Distance Formula for this case. Is the simplification what you expected? Explain.

(c) Simplify the Distance Formula for θ 1 − θ 2 = 90 ∘ . Is the simplification what you expected? Explain.

(d) Choose two points on the polar coordinate system and find the distance between them. Then choose differentpolar representations of the same two points and apply the Distance Formula again. Discuss the result.

(a)

To determine

To prove: That the distance formula for the distance between the two points given as, (r1,θ1) and (r2,θ2) in terms of polar coordinates is d=r12+r22−2r1r2cos(θ1−θ2).

Explanation

Given:

The given polar coordinates are (r1,θ1) and (r2,θ2).

Formula used:

The formula to calculate the distance between two points d=(x2−x1)2+(y2−y1)2.

Proof:

Polar coordinates are given as, (r1,θ1) and (r2,θ2). Converts it into Cartesian co-ordinates, (x1,y1)=(r1cosθ1,r1sinθ1) and (x2,y2)=(r2cosθ2,r2sinθ2).

Now, apply the distance formula between the points as shown below,

d=(x2−x1)2+(y2−y1)2

Substitute the values into the above equation,

d=(r2cosθ2−r1cosθ1)2+(r2sinθ2−r1sinθ1)2=r12(cosθ1)2+r22(cosθ2

(b)

To determine

To calculate: The position of points relative to each other for θ1=θ2. Also, simplify the distance formula for it..

(c)

To determine

To calculate: The distance formula for (θ1−θ2)=90°. Also, explain the result.

(d)

To determine

To calculate: The distance between any two arbitrary points. Also, discuss their result.

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Counterexample Find a counterexample to show that the following conjecture is false. Conjecture: For all number...

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It can be shown that the length of a diagonal of a right rectangular prism with dimensions l, w, and h is given...

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Verifying a Formula In Exercises 33-36, use the method of partial fractions to verify the integration formula. ...

Calculus: Early Transcendental Functions (MindTap Course List)

Points of Intersection In Exercises 17-20, apply Newtons Method to approximate the x-value(s) of the indicated ...

Calculus of a Single Variable

Use the properties of integrals to verify the inequality. /4/2sinxxdx22

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CHOOSING ASTRONAUTS The National Aeronautics and Space Administration NASA announced that it will choose two as...

Finite Mathematics for the Managerial, Life, and Social Sciences

In Exercise 23 to 30, determine the value of x so that line l will be parallel to line m. m6=x29 m2=x(x1)

Elementary Geometry for College Students

Let F(x, y, z) = (2x + 3y + z)i + (z − 2y)j + (x + 3y2 + xz + yz)k and the surface S be that portion of the gra...

Study Guide for Stewart's Multivariable Calculus, 8th

For f(x) = tanh1 2x, f(x) = a) 2(sech1 2x)2 b) 2(sech2 2x)1 c) 214x2 d) 114x2

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

Using Initial Conditions In Exercises 17-22, solve the differential equation by the method of undetermined coef...

Multivariable Calculus

Self Check Evaluate k=25(3k+2).

College Algebra (MindTap Course List)

Consider a sample with data values of 10, 20, 12, 17, and 16. Compute the z-score for each of the five observat...

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

Briefly explain how random assignment attempts to keep participant characteristics such as age or gender from b...

Research Methods for the Behavioral Sciences (MindTap Course List)

The average percent defective product of a manufacturing plant is 1.20%. On a particular day 50 pieces were rej...

Mathematics For Machine Technology

The article Bicyclists and Other Cyclists (Annals of Emergency Medicine [2010]: 426) reported that in 2008, the...

Introduction To Statistics And Data Analysis

Fifteen percent of Australians smoke. By introducing tough laws banning brand labels on cigarette packages, Aus...

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The following list contains several variables that differentiate college students. a. Select two variables from...

Research Methods for the Behavioral Sciences (MindTap Course List)

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