CALCULUS:GRAPHICAL,...,AP ED.-W/ACCESS
CALCULUS:GRAPHICAL,...,AP ED.-W/ACCESS
6th Edition
ISBN: 9781418300227
Author: Finney
Publisher: SAVVAS L
Question
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Chapter 0.4, Problem 44E

(a.)

To determine

The graphs of the given parametric equations in the parametric interval 0,2π , for k=0 and h=2,1,1,2 in turn; and the role of h .

(a.)

Expert Solution
Check Mark

Answer to Problem 44E

The graphs of the given parametric equations in the parametric interval 0,2π , for k=0 and h=2,1,1,2 in turn have been provided; and it has been determined that h is the x -coordinate of the center of the circle formed by graphing the given parametric equations.

Explanation of Solution

Given:

The parametric equations; x=2cost+h , y=2sint+k .

Concept used:

The parametric equations are graphed for each of the given values of h , k .

Calculation:

The given parametric equations are x=2cost+h and y=2sint+k .

The graphs of these parametric equations in the parametric interval 0,2π are as follows:

When k=0 and h=2:

  CALCULUS:GRAPHICAL,...,AP ED.-W/ACCESS, Chapter 0.4, Problem 44E , additional homework tip  1

When k=0 and h=1:

  CALCULUS:GRAPHICAL,...,AP ED.-W/ACCESS, Chapter 0.4, Problem 44E , additional homework tip  2

When k=0 and h=1:

  CALCULUS:GRAPHICAL,...,AP ED.-W/ACCESS, Chapter 0.4, Problem 44E , additional homework tip  3

When k=0 and h=2:

  CALCULUS:GRAPHICAL,...,AP ED.-W/ACCESS, Chapter 0.4, Problem 44E , additional homework tip  4

It can be seen that h is the x -coordinate of the center of the circle formed by graphing the given parametric equations.

Conclusion:

The graphs of the given parametric equations in the parametric interval 0,2π , for k=0 and h=2,1,1,2 in turn have been provided; and it has been determined that h is the x -coordinate of the center of the circle formed by graphing the given parametric equations.

(b.)

To determine

The graphs of the given parametric equations in the parametric interval 0,2π , for h=0 and k=2,1,1,2 in turn; and the role of k .

(b.)

Expert Solution
Check Mark

Answer to Problem 44E

The graphs of the given parametric equations in the parametric interval 0,2π , for h=0 and k=2,1,1,2 in turn have been provided; and it has been determined that k is the y -coordinate of the center of the circle formed by graphing the given parametric equations.

Explanation of Solution

Given:

The parametric equations; x=2cost+h , y=2sint+k .

Concept used:

The parametric equations are graphed for each of the given values of h , k .

Calculation:

The given parametric equations are x=2cost+h and y=2sint+k .

The graphs of these parametric equations in the parametric interval 0,2π are as follows:

When h=0 and k=2:

  CALCULUS:GRAPHICAL,...,AP ED.-W/ACCESS, Chapter 0.4, Problem 44E , additional homework tip  5

When h=0 and k=1:

  CALCULUS:GRAPHICAL,...,AP ED.-W/ACCESS, Chapter 0.4, Problem 44E , additional homework tip  6

When h=0 and k=1:

  CALCULUS:GRAPHICAL,...,AP ED.-W/ACCESS, Chapter 0.4, Problem 44E , additional homework tip  7

When h=0 and k=2:

  CALCULUS:GRAPHICAL,...,AP ED.-W/ACCESS, Chapter 0.4, Problem 44E , additional homework tip  8

It can be seen that k is the y -coordinate of the center of the circle formed by graphing the given parametric equations.

Conclusion:

The graphs of the given parametric equations in the parametric interval 0,2π , for h=0 and k=2,1,1,2 in turn have been provided; and it has been determined that k is the y -coordinate of the center of the circle formed by graphing the given parametric equations.

(c.)

To determine

A parametrization of the given circle.

(c.)

Expert Solution
Check Mark

Answer to Problem 44E

It has been determined that the parametrization of the given circle is x=5cost+2 , y=5sint3 in the parametric interval 0,2π .

Explanation of Solution

Given:

The circle with radius 5 and center at 2,3 .

Concept used:

The parametrization; x=acost+h , y=asint+k in the parametric interval 0,2π ; gives a circle of radius a whose center is at h,k .

Calculation:

As seen in the previous parts, the parametrization; x=acost+h , y=asint+k in the parametric interval 0,2π ; gives a circle of radius a whose center is at h,k .

The given circle has radius 5 .

Then, here a=5 .

The center of the given circle is at 2,3 .

Then, here h,k=2,3 .

Put these values in x=acost+h , y=asint+k to get,

  x=5cost+2 ,

  y=5sint3

So, the parametrization of the given circle is x=5cost+2 , y=5sint3 in the parametric interval 0,2π .

Conclusion:

It has been determined that the parametrization of the given circle is x=5cost+2 , y=5sint3 in the parametric interval 0,2π .

(d.)

To determine

A parameterization for the given ellipse.

(d.)

Expert Solution
Check Mark

Answer to Problem 44E

It has been determined that the parameterization for the given ellipse is x=5cost3 , y=2sint+4 in the parametric interval 0,2π .

Explanation of Solution

Given:

An ellipse centered at 3,4 with semi-major axis of length 5 parallel to x -axis and semi-minor axis of length 2 parallel to y -axis.

Concept used:

The parametrization of an ellipse centered at h,k with semi-major axis of length a parallel to x -axis and semi-minor axis of length b parallel to y -axis; is given as x=acost+h , y=bsint+k , in the parametric interval 0,2π .

Calculation:

The given ellipse is centered at 3,4 .

Then, here h,k=3,4 .

The length of the semi-major axis of the given ellipse, parallel to the x -axis, is 5 .

Then, here a=5 .

The length of the semi-minor axis of the given ellipse, parallel to the y -axis, is 2 .

Then, here b=2 .

Put these values in x=acost+h , y=bsint+k to get,

  x=5cost3 ,

  y=2sint+4

So, the required parametrization is x=5cost3 , y=2sint+4 in the parametric interval 0,2π .

Conclusion:

It has been determined that the parameterization for the given ellipse is x=5cost3 , y=2sint+4 in the parametric interval 0,2π .

Chapter 0 Solutions

CALCULUS:GRAPHICAL,...,AP ED.-W/ACCESS

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