Calculo

CHAPTER

12

Derivatives for Functions of Two or More Variables
4. a.

12.1 Concepts Review
1. real-valued function of two real variables 2. level curve; contour map 3. concentric circles 4. parallel lines

6

b. 12 c. d.

2
(3cos 6)1/ 2 + 1.44 ≈ 3.1372

5. F (t cos t , sec2 t ) = t 2 cos 2 t sec2 t = t 2 , cos t ≠ 0 6. F ( f (t ), g (t )) = F (ln t 2 , et / 2 )
= exp(ln t 2 ) +(et / 2 ) 2 = t 2 + et , t ≠ 0

Problem Set 12.1
1. a. 5 6

b. 0 c. d. e. f.

7. z = 6 is a plane.

a6 + a 2
2 x2 , x ≠ 0

Undefined

The natural domain is the set of all (x, y) such that y is nonnegative. 2. a. 4 8. x + z = 6 is a plane.

b. 17 c. d. e. f.

17 16
1 + a2 , a ≠ 0
x 3 + x, x ≠ 0

Undefined
9. x + 2y + z = 6 is a plane.

The natural domain is the set of all(x, y) such that x is nonzero.
3. a. b.
sin(2π ) = 0
⎛π⎞ 4sin ⎜ ⎟ = 2 ⎝6⎠

c. d.

⎛π⎞ 16sin ⎜ ⎟ = 16 ⎝2⎠ π2 sin(π2 ) ≈ –4.2469

744

Section 12.1

Instructor’s Resource Manual

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10. z = 6 – x 2 is a parabolic cylinder.

15. z = exp[–( x 2 + y 2 )]

11. x 2 + y 2 + z 2 = 16 , z ≥ 0 is a hemisphere.

16. z =

x2 ,y>0 y

12.

x2 y 2 z 2 + + = 1 , z ≥ 0 is a hemi-ellipsoid. 4 16 16

17. x 2 + y 2 = 2 z; x 2 + y 2 = 2k

13. z = 3 – x 2 – y 2 is a paraboloid.

18. x = zy, y ≠ 0 ; x =ky, y ≠ 0

14. z = 2 – x – y 2

19. x 2 = zy, y ≠ 0; x 2 = ky, y ≠ 0

Instructor’s Resource Manual

Section 12.1

745

© 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from thepublisher.

20. x 2 = –( y – z ); x 2 = –( y – k )

24. ( x – 2)2 + ( y + 3)2 =

16
V2

21. z =

x2 + 1 x2 + y 2

25. a.
, k = 1, 2, 4

San Francisco and St. Louis had a temperature between 70 and 80 degrees Fahrenheit. Drive northwest to get to cooler temperatures, and drive southeast to get warmer temperatures. Since the level curve for 70 runs southwest to northeast, you coulddrive southwest or northeast and stay at about the same temperature. The lowest barometric pressure, 1000 millibars and under, occurred in the region of the Great Lakes, specifically near Wisconsin. The highest barometric pressure, 1025 millibars and over, occurred on the east coast, from Massachusetts to South Carolina. Driving northwest would take you to lower barometric pressure, and drivingsoutheast would take you to higher barometric pressure. Since near St. Louis the level curves run southwest to northeast, you could drive southwest or northeast and stay at about the same barometric pressure.

k = 1: y 2 = 1 or y = ±1 ; two parallel lines k = 2: 2 x 2 + 2 y 2 = x 2 + 1
x2 y 2 + = 1 ; ellipse 1 1
2

b.

c.

k = 4: 4 x 2 + 4 y 2 = x 2 + 1
x2
1 3

+

y2
1 4

= 1;ellipse

26. a.

22. y = sin x + z; y = sin x + k

b.

c. 23. x = 0, if T = 0: ⎛1 ⎞ y 2 = ⎜ – 1⎟ x 2 , if y ≠ 0 . ⎝T ⎠

27. x 2 + y 2 + z 2 ≥ 16 ; the set of all points on and outside the sphere of radius 4 that is centered at the origin 28. The set of all points inside (the part containing the z-axis) and on the hyperboloid of one sheet; x2 y 2 z 2 + – = 1. 9 9 9 29.
x2 y 2 z 2 + + ≤ 1 ;points inside and on the 9 16 1 ellipsoid

746

Section 12.1

Instructor’s Resource Manual

© 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.

30. Points...