Thoms calculus 11ed_ism_ch14

Thomas' Calculus, 11th Edition

Info icon This preview shows pages 1–6. Sign up to view the full content.

View Full Document Right Arrow Icon
CHAPTER 14 PARTIAL DERIVATIVES 14.1 FUNCTIONS OF SEVERAL VARIABLES 1. (a) Domain: all points in the xy-plane (b) Range: all real numbers (c) level curves are straight lines y x c parallel to the line y x œ œ (d) no boundary points (e) both open and closed (f) unbounded 2. (a) Domain: set of all (x y) so that y x 0 y x ß   Ê   (b) Range: z 0   (c) level curves are straight lines of the form y x c where c 0 œ   (d) boundary is y x 0 y x, a straight line È œ Ê œ (e) closed (f) unbounded 3. (a) Domain: all points in the xy-plane (b) Range: z 0   (c) level curves: for f(x y) 0, the origin; for f(x y) c 0, ellipses with center ( 0) and major and minor ß œ ß œ axes along the x- and y-axes, respectively (d) no boundary points (e) both open and closed (f) unbounded 4. (a) Domain: all points in the xy-plane (b) Range: all real numbers (c) level curves: for f(x y) 0, the union of the lines y x; for f(x y) c 0, hyperbolas centered at ß œ œ „ ß œ Á (0 0) with foci on the x-axis if c 0 and on the y-axis if c 0 ß (d) no boundary points (e) both open and closed (f) unbounded 5. (a) Domain: all points in the xy-plane (b) Range: all real numbers (c) level curves are hyperbolas with the x- and y-axes as asymptotes when f(x y) 0, and the x- and y-axes ß Á when f(x y) 0 ß œ (d) no boundary points (e) both open and closed (f) unbounded 6. (a) Domain: all (x y) (0 y) ß Á ß (b) Range: all real numbers (c) level curves: for f(x y) 0, the x-axis minus the origin; for f(x y) c 0, the parabolas y cx minus the ß œ ß œ Á œ # origin (d) boundary is the line x 0 œ
Image of page 1

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

View Full Document Right Arrow Icon
864 Chapter 14 Partial Derivatives (e) open (f) unbounded 7. (a) Domain: all (x y) satisfying x y 16 ß # # (b) Range: z   " 4 (c) level curves are circles centered at the origin with radii r 4 (d) boundary is the circle x y 16 # # œ (e) open (f) bounded 8. (a) Domain: all (x y) satisfying x y 9 ß Ÿ # # (b) Range: 0 z 3 Ÿ Ÿ (c) level curves are circles centered at the origin with radii r 3 Ÿ (d) boundary is the circle x y 9 # # œ (e) closed (f) bounded 9. (a) Domain: (x y) (0 0) ß Á ß (b) Range: all real numbers (c) level curves are circles with center ( 0) and radii r 0 (d) boundary is the single point (0 0) ß (e) open (f) unbounded 10. (a) Domain: all points in the xy-plane (b) Range: 0 z 1 Ÿ (c) level curves are the origin itself and the circles with center (0 0) and radii r 0 ß (d) no boundary points (e) both open and closed (f) unbounded 11. (a) Domain: all (x y) satisfying 1 y x 1 ß Ÿ Ÿ (b) Range: z Ÿ Ÿ 1 1 # # (c) level curves are straight lines of the form y x c where 1 c 1 œ Ÿ Ÿ (d) boundary is the two straight lines y 1 x and y 1 x œ œ (e) closed (f) unbounded 12. (a) Domain: all (x y), 0 ß B Á (b) Range: z 1 1 # # (c) level curves are the straight lines of the form y cx, c any real number and x 0 œ Á (d) boundary is the line x 0 œ (e) open (f) unbounded 13. f 14. e 15. a 16. c 17. d 18. b
Image of page 2
Section 14.1 Functions of Several Variables 865 19. (a) (b) 20. (a) (b) 21. (a) (b) 22. (a) (b)
Image of page 3

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

View Full Document Right Arrow Icon
866 Chapter 14 Partial Derivatives 23. (a) (b) 24. (a) (b) 25. (a) (b)
Image of page 4
Section 14.1 Functions of Several Variables 867 26. (a) (b) 27. (a) (b) 28. (a) (b) 29. f(x y) 16 x y and 2 2 2 z 16 2 2 2 6 6 16 x y x y 10 ß œ ß Ê œ œ Ê œ Ê œ # # # # # # # # Š Š Š È È È È 30. f(x y) x 1 and (1 0) 1 1 0 x 1 0 x 1 or x 1 ß œ ß Ê D œ œ Ê œ
Image of page 5

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

View Full Document Right Arrow Icon
Image of page 6
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    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.

    Student Picture

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

  • Left Quote Icon

    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.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    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.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern