{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

16.5 Ex 5 - 17

# 16.5 Ex 5 - 17 - 996 C H A P T E R 16 M U LTI P L E I N T E...

This preview shows pages 1–2. Sign up to view the full content.

996 C H A P T E R 16 MULTIPLE INTEGRATION (ET CHAPTER 15) The image of the horizontal line v = c is the set of the points ( x , y ) = ( u , c ) = ( e u , e u + c ) x = e u , y = e c x y = e c x , x > 0 Since u can take any value, x can take any positive value, and hence the image is the ray y = e c x , x > 0. u Φ x c y = e c x y In Exercises 5–12, let ( u , v) = ( 2 u + v, 5 u + 3 v) be a map from the u v -plane to the xy-plane. 5. Show that the image of the horizontal line v = c is the line y = 5 2 x + 1 2 c . What is the image (in slope-intercept form) of the vertical line u = c ? SOLUTION The image of the vertical line u = c is the set of the following points: ( x , y ) = ( c , v) = ( 2 c + v, 5 c + 3 v) x = 2 c + v, y = 5 c + 3 v By the first equation, v = x 2 c . Substituting in the second equation gives y = 5 c + 3 ( x 2 c ) = 5 c + 3 x 6 c = 3 x c Therefore, the image of the line u = c is the line y = 3 x c in the xy -plane. The image of the horizontal line v = c is the set of the following points: ( x , y ) = ( u , c ) = ( 2 u + c , 5 u + 3 c ) x = 2 u + c , y = 5 u + 3 c The first equation implies u = x c 2 . Substituting in the second equation gives y = 5 ( x c ) 2 + 3 c = 5 x 2 + c 2 Therefore, the image of the line v = c is the line y = 5 x 2 + c 2 in the xy -plane. 6. Describe the image of the line through the points ( u , v) = ( 1 , 1 ) and ( u , v) = ( 1 , 1 ) under in slope-intercept form. SOLUTION 1 1 1 1 1 1 Φ x y u u = 1 y = 3 x 1 The line is the vertical line u = 1 in the u v -plane. The image of the line under the linear map ( u , v) = ( 2 u + v, 5 u + 3 v) is the line through the images of the points ( u , v) = ( 1 , 1 ) and ( u

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### What students are saying

• As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

Kiran Temple University Fox School of Business ‘17, Course Hero Intern

• I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

Dana University of Pennsylvania ‘17, Course Hero Intern

• The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

Jill Tulane University ‘16, Course Hero Intern