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Unformatted text preview: 992 CHAPTER 14 PARTIAL DERIVATIVES 10. 1f (2. l) is a critical point of f and
f..(2. 11./gym. 1) < [11112. ur then f has a saddle point at (2. 1). 11. lff(x. y) = sin x + sin y. then —\/'2_ S Duf(x,y) S ﬁ. 12. Iff(x, y) has two local maxima. then f must have a local
minimum. Exercises 12 Find and sketch the domain of the function.
@f(x. y) = ln(x + y + I)
@ﬂxw) = «4 ~ x: w y2 + v1 ‘ X2 3—4 Sketch the graph of the function. @m. ,1») = 1  4. f(x, y) = x: + (y _ 2): 56 Sketch several level curves of the function. 5. f(.\‘,)’) = V4.1rZ + y1 6. f(.r,y) = e" + y 7. Make a rough sketch of a contour map for the function whose
graph is shown. 8. A contour map of a function f is shown. Use it to make a
rough sketch of the graph of f. 9—10 Evaluate the limit or show that it does not exist. . 2x11 . . va
( 9.) 11m —Z—'—; @ 11m —,'—,
INHHJI x + Zy' whatnot y + 2y 11. A metal plate is situated in the xy—plane and occupies the
rectangle 0 S x $ 10, 0 S y S 8, where x and y are measured
in meters. The temperature at the point (x, y) in the plate is
TU. y), where T is measured in degrees Celsius. Temperatures E Graphing calculator or computer required at equally spaced points were measured and recorded in the table. (:1) Estimate the values of the partial derivatives T.(6, 4)
and T\.(6, 4). What are the units? (b) Estimate the value of Du T(6, 4), where u = (i + j) M5 .
Interpret your result. (c) Estimate the value of T.).(6. 4). 71‘, 896‘90‘86.80 75 I 1
I
I
i 78‘ ~ 10 92 92 91 87 I __.a.r , , ,‘MH “ 12. Find a linear approximation to the temperature function T(x. y)
in Exercise 11 near the point (6, 4). Then use it to estimate the
temperature at the point (5, 3.8). 13—17 Find the ﬁrst partial derivatives. @ftx. y) = (5)” + was 15 ”FIatﬂ) = 42 Mai + B2)
17 S(u, v. w) = u arctan(v\/EJ_) 11+ 21)
[£2 + v2 16. G(.r,y, z) = e”sin(y/:) 14. 9(11, 1)) = 18. The speed of sound traveling through ocean water is a function
of temperature. salinity, and pressure. It has been modeled by
the function C = 1449.2 + 4.6T — 0.055T2 + 0.00029T‘
+ (1.34 — 0.01r)(5 — 35) + 0.016D where C is the speed of sound (in meters per second), T is the
temperature (in degrees Celsius). S is the salinity (the concen
tration of salts in parts per thousand, which means the number
of grams of dissolved solids per 1000 g of water), and D is the
depth below the ocean surface (in meters). Compute aC/BT,
OC/aS, and OC/aD when T = 10°C. S = 35 parts per thousand.
and D = 100 m. Explain the physical signiﬁcance of these
partial derivatives. 19422 Find all second partial derivatives of ,1. 71.1.1) 4; ~111 " .11'1 20. : 2 .1'2' 3‘
@ f(.1'.1'. :1 : .1”1“':"‘ 22. ' I rcost1 we 21) . , (1: (1:
® 11 _ 2 11' + .12” ‘. sho11 that 1— +1— 2 .11 + :.
1)\ ' {)1
24.11: = sin(\' + sin I). show that
2’): 2'13: 2'): {13:
(1.1 {1.1211 ()1 {1.13 25—29 Find equations 01(21) the tangent plane and (11) the normal
line to the given surface at the speciﬁed point. @ : = 3.13 — 13 + 2.1. 11.42.11
26. : Z c" cos 1'. ((1.1). 1)
@13 + 21'2 " 3:1 = 3. (2. "l. l) .\1+\:+:.=1' 3. (1.1.1) @su111)"1+21+3— (2. "1. 0) ED: 30 Use a computer to graph the surface : = x: + 1" and its
tangent plane 21nd no1m211 line at (1.1.2) on the same screen.
Choose the domain and viewpoint so that you get 21 good
view of all three objects. 31. Find the points on the hyperboloid .12 + 41'2  :3 — "4 where the tangent plane is parallel to the plane 21' + 2y + _— = 5'
@ Find (1111111 2111(1 4 “.111 33. Find the linear approxinmtion of the function
1"(1 1. :1" e 1 \_1~ + “3 at the point (2. 3. 4) and use it
to estimate the number( 191‘8 V(3. 01 1 +( (3 .97) 34. The two legs of '21 right triangle are measured as 5 m and
12 m with a possible cum in meastttement of at most (1. )2 cm
in e21ch.1 .lse di11e1entials to estimate the maximum etrot' in
the calculated value of (a) the area of the triangle and (b) the
length of the hypotenuse. 11'11 : 131'3 + :4. where .1' z [I + 3171. 1' : pv". and
: : 1) sin [1, use the Chain Rule to ﬁnd (111/rip. 36. if v = .1'3sin 1' + 1c“. where .1‘ : .1 + 21 and 1' = .11. use the
Chain Rule to find {111/{1.1 and fin/1")! when .1' 1 (1 and r 2 1. 37. Suppose: —/( .t. 1'). wihe1e 1 .2 "21(1'. 1) 1' — "[1(..1 I).
5111.2):3.g.(1.2)= l._q.(1. 2) =4. 11(1. 2126.
11.(1. 2) = *5. 11.11.21"— 10. ,I.( (3. 6) " 7. and .1113. (1) Z 8.
Find 2'):/2').1' and £1:/2')t when .1 = 1 and t 2 2. 38. Use a [tee diagtam to w'tite out the Chain Rule 101 the case
whetc 10 "ﬂ 1‘. u. 11). t" — 1(1). 2]. 1'1. . 1'). u = 11(1) q. 1.1.) and
v I 12(1). 2]. r. .1") are all differentiable functions.
@ 11' : = 1' + f(.1: e 1'3). where f is differentiable. show that
('1: t): 1' — + 1"— :1
('1.\ (11 CHAPTER 14 BEVlEW 993 40 The length .1 ot a side ot a ttiangle 1s inc1e21sing at 21 late ol
3 in/s. the length _1 of another side is decieasing at a late ol
2 111/s. and the contained angle H is ineleasing at a rate oi 0. ()5 tadian/s Ho“ 121st is the then 01 the triangle changing
when .1' = 40111.1 I 511111. and 9 " 77/6.) 41. 1f : : flu. v). 11 here 11 = 11'. L' = 1/1. and f has continuous
second partial derivatives. show that . ('1‘: . 17‘: ('1': (I:
.1" , . 1" , . 41111 , , l 212' ,
(1.1" ' «11' ’ (111 rlt.‘ (IL'
. . «I: (I):
'11 cos(.\'1':) : 1 + .1"\' +_ '" .lind and ,7.
2 ' ’ 111 211' 43. Find the gradient of the function f(.1. 1'. :1 = 1'30” 44. ('21) When is the directional derivative of f a maximum?
(b) When is it a minimum?
(c) When is it 0‘?
(d') When is it half of its maximum value? 45~46 Find the directional derivative of f at the given point in
the indicated direction. @1111):132»". (42 “0) in the direction toward the point (2 "3)
1 _( . 1 .
45 111_)=.1“\'+.1'\/1+:. (1.2.3). in the direction 011 " 21 + j e 2k 47. Find the maximum rate ofchange of _(f .1'. 1') " —.1'3\' + \[1' at the point(2 .1.) In which ditection does it occur? 48. Find the direction in which f(.1'. 1'. :) = :23“ increases most
rapidly at the point (0. 1. 2). What is the maximum rate of
increase? 49. The contour map shows wind speed in knots during Hurri
cane Andrew on August 24. 1992. Use it to estimate the
value of the directional derivative of the wind speed at
Homestead. Florida. in the direction of the eye of the
hurricane. [Homestead
, ' .2 ”Rhvaest 30
1___1._4.__t_4—J
(1 1(1 2(1 .31) 4(1 (1)1s12111ce11111111es) ...
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 Summer '08
 Helton
 Math, Multivariable Calculus

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