Math22-Sum11-HW#2 - 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

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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 1-2 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): 5-6 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. @ : = 3.1-3 — 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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