CHAPTER PARTIAL DERIVATIVES. FUNCTIONS OF SEVERAL VARIABLES. (a) Domain: all points in the -plane (b) Range: all real numbers (c) level curves are straight lines c parallel to the line (d) no boundar points (e) both open and closed (f) unbounded 2. (a) Domain: set of all (ß) so that 0 Ê (b) Range: z 0 (c) level curves are straight lines of the form c here c 0 (d) boundar is È 0 Ê, a straight line (e) closed (f) unbounded. (a) Domain: all points in the -plane (b) Range: z 0 (c) level curves: for f(ß) 0, the origin; for f(ß) c 0, ellipses ith center (!ß0) and major and minor aes along the - and -aes, respectivel (d) no boundar points (e) both open and closed (f) unbounded. (a) Domain: all points in the -plane (b) Range: all real numbers (c) level curves: for f(ß) 0, the union of the lines ; for f(ß) c Á 0, hperbolas centered at (0ß0) ith foci on the -ais if c 0 and on the -ais if c 0 (d) no boundar points (e) both open and closed (f) unbounded 5. (a) Domain: all points in the -plane (b) Range: all real numbers (c) level curves are hperbolas ith the - and -aes as asmptotes hen f(ß) hen f(ß) 0 (d) no boundar points (e) both open and closed (f) unbounded Á 0, and the - and -aes 6. (a) Domain: all (ß) Á (0ß) (b) Range: all real numbers (c) level curves: for f(ß) 0, the -ais minus the origin; for f(ß) c Á 0, the parabolas c minus the origin (d) boundar is the line 0
86 Chapter Partial Derivatives (e) open (f) unbounded 7. (a) Domain: all (ß) satisfing 6 (b) Range: z (c) level curves are circles centered at the origin ith radii r (d) boundar is the circle 6 (e) open (f) bounded 8. (a) Domain: all (ß) satisfing Ÿ 9 (b) Range: 0 Ÿ z Ÿ (c) level curves are circles centered at the origin ith radii r Ÿ (d) boundar is the circle 9 (e) closed (f) bounded 9. (a) Domain: (ß) Á (0ß0) (b) Range: all real numbers (c) level curves are circles ith center (!ß 0) and radii r 0 (d) boundar is the single point (0ß 0) (e) open (f) unbounded 0. (a) Domain: all points in the -plane (b) Range: 0 z Ÿ (c) level curves are the origin itself and the circles ith center (0ß0) and radii r 0 (d) no boundar points (e) both open and closed (f) unbounded. (a) Domain: all (ß) satisfing Ÿ Ÿ (b) Range: Ÿ z Ÿ (c) level curves are straight lines of the form c here Ÿ c Ÿ (d) boundar is the to straight lines and (e) closed (f) unbounded 2. (a) Domain: all (ß), B Á 0 (b) Range: z (c) level curves are the straight lines of the form c, c an real number and Á 0 (d) boundar is the line 0 (e) open (f) unbounded. f. e 5. a 6. c 7. d 8. b
Section. Functions of Several Variables 865 9. (a) (b) 20. (a) (b) 2. (a) (b) 22. (a) (b)
866 Chapter Partial Derivatives 2. (a) (b) 2. (a) (b) 25. (a) (b)
Section. Functions of Several Variables 867 26. (a) (b) 27. (a) (b) 28. (a) (b) 29. f(ß) 6 and Š 2È2ßÈ2 Ê z 6 Š 2È2 Š È2 6 Ê 6 6 Ê 0 0. f(ß) È and (ß0) Ê D È 0 Ê 0 Ê or '. f(ß) dt at Š È2ßÈ2 Ê z tan tan ; at Š È2ßÈ2 Ê z tan È2 tan Š È2 t 2 tan È 2 Ê tan tan 2 tan È2 _ 2. f(ß)! 2 Š at (ß2) Ê z ; at (ß2) Ê z 2 Ê 2 Ê 2 2 n0 Ê 2 n Š
868 Chapter Partial Derivatives.. 5. 6. 7. 8. 9. 0.. f(ßßz) È ln z at (ßß) Ê È ln z; at (ßß) Ê È ( ) ln 2 Ê Èln z 2
Section. Functions of Several Variables 869 2. f(ß ß z) ln a z b at ( ß ß ) Ê ln a z b; at ( ß ß ) Ê ln ( 2 ) ln Ê ln ln a z b Ê z _ n ( ) _ n ( ) n0 n! zn n! zn n0 Ðln 8ÑÎ ln 2 ÐÑÎz z. g(ßßz)! at (ln 2ßln ß) Ê! e ; at (ln 2ßln ß) Ê e e e 2 Ê 2 e Ê ln 2 ' ' d) dt z. g(ßßz) È È at 0ß ß2 Ê sin sec t È ˆ c ) d c d È ) 2t t z ÐÑÎz Ðln2lnÑÎ 2 sin sin sec z sec Š È 2 Ê sin sin sec z ; at ˆ 0ß ß2 Ê sin sin 0 sec 2 Ê sin sin sec z 5. f(ßßz) z and 20 t, t, z 20 Ê (20 t)(t)(20) along the line Ê 00t 20t d d d Ê dt 00 0t; dt 0 Ê 00 0t 0 Ê t 0 and dt 0 for all t Ê es, maimum at t 0 Ê 20 0 0, 0, z 20 Ê maimum of f along the line is f(0ß0ß20) (0)(0)(20) 2000 6. f(ßßz) z and t, t 2, z t 7 Ê (t )(t 2) (t 7) t t 5 along the line d d d Ê dt 2t ; dt 0 Ê 2t 0 Ê t 2 and dt 2 for all t Ê es, minimum at t 2 Ê 2, 2 2 0, and z 2 7 9 Ê minimum of f along the line is f(ß0ß9) ()(0) 9 9 7. ˆ Th (290 K)(6.8 m) 2.86 m Ê must be (2.86) 6 m south of Nantucet d Î 5 K/m Î 8. The graph of f( ß ß$ ß %) is a set in a five-dimensional space. It is the set of points ( ß ß$ ß% ßf( ß ß$ ß %)) for ( ß ß$ ß %) in the domain of f. The graph of f( ß ß$ ßá ß n) is a set in an (n )-dimensional space. It is the set of points ( ß ß$ ßá ßnßf( ß ß$ ßá ß n)) for ( ß ß$ ßá ß n) in the domain of f. 9-52. Eample CAS commands: Maple: ith( plots ); f := (,) -> *sin(/2) + *sin(2*); domain := =0..5*Pi; domain := =0..5*Pi; 0,0 := *Pi,*Pi; plotd( f(,), domain, domain, aes=boed, stle=patch, shading=zhue, title=9(a) (Section.) ); plotd( f(,), domain, domain, grid=[50,50], aes=boed, shading=zhue, stle=patchcontour, orientation=[-90,0], title=9(b) (Section.) ); (b) L := evalf( f(0,0) ); (c) plotd( f(,), domain, domain, grid=[50,50], aes=boed, shading=zhue, stle=patchcontour, contours=[l], orientation=[-90,0], title=9(c) (Section.) ); 5-56. Eample CAS commands: Maple: eq := *ln(^2+^2+z^2)=; implicitplotd( eq, =-2..2, =-2..2, z=-2..2, grid=[0,0,0], aes=boed, title=5 (Section.) );
870 Chapter Partial Derivatives 57-60. Eample CAS commands: Maple: := (u,v) -> u*cos(v); := (u,v) ->u*sin(v); z := (u,v) -> u; plotd( [(u,v),(u,v),z(u,v)], u=0..2, v=0..2*pi, aes=boed, stle=patchcontour, contours=[($0..)/2], shading=zhue, title=57 (Section.) ); 9-60. Eample CAS commands: Mathematica: (assigned functions and bounds ill var) For 9-52, the command ContourPlot dras 2-dimensional contours that are z-level curves of surfaces z = f(,). Clear[,, f] f[_, _]:= Sin[/2] Sin[2] min= 0; ma= 5; min= 0; ma= 5; {0, 0}={, }; cp= ContourPlot[f[,], {, min, ma}, {, min, ma}, ContourShading Ä False]; cp0= ContourPlot[[f[,], {, min, ma}, {, min, ma}, Contours Ä {f[0,0]}, ContourShading Ä False, PlotStle Ä {RGBColor[,0,0]}]; Sho[cp, cp0] For 5-56, the command ContourPlotD ill be used and requires loading a pacage. Write the function f[,, z] so that hen it is equated to zero, it represents the level surface given. For 5, the problem associated ith Log[0] can be avoided b reriting the function as 2 + 2 +z2 - e/ <<Graphics`ContourPlotD` Clear[,, z, f] 2 2 2 f[_, _, z_]:= z Ep[/] ContourPlotD[f[,, z], {, 5, 5}, {, 5, 5}, {z, 5, 5}, PlotPoints Ä {7, 7}]; For 57-60, the command ParametricPlotD ill be used and requires loading a pacage. To get the z-level curves here, e solve and in terms of z and either u or v (v here), create a table of level curves, then plot that table. <<Graphics`ParametricPlotD` Clear[,, z, u, v] ParametricPlotD[{u Cos[v], u Sin[v], u}, {u, 0, 2}, {v, 0, 2p}]; zlevel= Table[{z Cos[v], z sin[v]}, {z, 0, 2,.}]; ParametricPlot[Evaluate[zlevel],{v, 0, 2}];.2 LIMITS AND CONTINUITY. lim Ð Ä Ð00 5 (0) 0 5 5 2 0 0 2 0 2. lim È 0 Ð Ä Ð0 È. lim È È È2 2È6 Ð Ä Ð. lim Ð Ä Ð2ßÑ 6 6 Š ˆ ˆ 5. lim sec tan (sec 0) ˆ tan ()() Ð Ä ˆ 0ß
Section.2 Limits and Continuit 87 0 0 $ $ 6. lim cos Š cos Š cos 0 Ð Ä Ð00 00 0 ln2 ln 7. lim e e e 2 Ð Ä Ð0ln2 8. lim ln ln () () ln 2 Ð Ä Ð e sin sin sin 9. lim lim e e lim Ð Ä Ð00 a bˆ Ð Ä Ð00! ˆ Ä 0 0. lim cos ˆ $ cos ˆ $ È È()() cos 0 Ð Ä Ð sin sin 0 0. lim 0 Ð Ä Ð0 2 2. lim 2 Ð Ä ˆ ß0 sin 0 sinˆ 2 cos (cos 0) 2 ( ). lim lim lim ( ) ( ) 0 Ð Ä Ð Ð Ä Ð Ð Ä Ð Á ( )( ). lim lim lim ( ) ( ) 2 Ð Ä Ð Ð Ä Ð Ð Ä Ð Á 22 ()(2) 5. lim lim lim ( 2) ( 2) Ð Ä Ð Ð Ä Ð Ð Ä Ð Á Á 6. lim lim lim Ð Ä Ð2ßÑ Ð Ä Ð2ßÑ ( )( ) Ð Ä Ð2ßÑ ( ) (2 ) Á, Á Á, Á Á 2È2È ˆ ÈÈ ˆ ÈÈ2 7. lim È lim lim 2 Ð Ä Ð00 È ˆ È È È Ð Ä Ð00 È Ð Ä Ð00 Á Á Š È0È02 2 Note: (ß) must approach (0ß0) through the first quadrant onl ith Á. ˆ È2 ˆ È2 8. lim È lim lim 2 Ð Ä Ð22 2 È ˆ È Ð Ä Ð22 2 Ð Ä Ð22 Á Á Á Š È222 22 È22 È22 2 È22 È22 È2 9. lim lim ˆ ˆ lim Ð Ä Ð20 Ð Ä Ð20 Ð Ä Ð20 2 Á 2 Á È(2)(2) 0 2 2 ÈÈ ÈÈ ÈÈ ÈÈ ÈÈ 20. lim lim ˆ ˆ lim Ð Ä Ð Ð Ä Ð Ð Ä Ð Á Á
872 Chapter Partial Derivatives ÈÈ 2 2 2. lim T Ä Ð ß 2 9 z 2 2 Š 22. lim T Ä ÐßßÑ 2 z 2()( ) ( )( ) 2 z ( ) 2. lim a sin cos sec z b a sin cos b sec 0 2 T Ä Ð0 ß 2. lim tan (z) tan ˆ 2 tan TĈ ß ß2 ˆ 2 c2 c2ð0ñ 25. lim ze cos 2 e cos 2 ()()() T Ä Ðß0 26. lim ln È z ln È0 ( 2) 0 ln È ln 2 TÄÐßß 0 2 0Ñ 27. (a) All (ß ) (b) All (ß) ecept (0ß0) 28. (a) All (ß) so that Á (b) All (ß ) 29. (a) All (ß) ecept here 0 or 0 (b) All (ß ) 0. (a) All (ß) so that 2 Á 0 Ê ( 2)( B) Á 0 Ê Á 2 and Á (b) All (ß) so that Á. (a) All (ßßz) (b) All (ßßz) ecept the interior of the clinder 2. (a) All (ßßz) so that z 0 (b) All (ßßz). (a) All (ßßz) ith z Á 0 (b) All (ßßz) ith z Á. (a) All (ßßz) ecept (ß0ß0) (b) All (ß ß z) ecept (!ß ß 0) or (ß 0ß 0) 5. lim È lim lim lim lim ; Ð Ä Ð00 Ä 0b È Ä 0b È2 Ä 0b È2 È Ä 0b 2 È2 along 0 lim lim lim lim È Ð Ä Ð00 Ä 0c È2 Ä 0c È2( ) Ä 0c È2 È2 along 0
% % % % % 6. lim lim ; lim lim lim Ð Ä Ð00 % Ä 0 % 0 Ð Ä Ð00 % Ä 0 Ä 0 2 % % a b along 0 along Section.2 Limits and Continuit 87 % a b % % % 7. lim lim lim Ð Ä Ð00 % Ä 0 Ä 0 % % % a b Ê different limits for different values of along () 8. lim lim lim lim ; if 0, the limit is ; but if Ð Ä Ð00 Ä 0 () Ä 0 Ä 0 0, the limit is along Á 0 9. lim lim Ð Ä Ð00 Ä 0 Ê different limits for different values of, Á along Á 0. lim lim Ð Ä Ð00 Ä 0 Ê different limits for different values of, Á along Á. lim lim Ð Ä Ð00 Ä 0 Ê different limits for different values of, Á 0 along Á 0 2. lim lim Ð Ä Ð00 Ä 0 Ê different limits for different values of, Á along Á. No, the limit depends onl on the values f(ß) has hen (ß) Á (! ß!). If f is continuous at (! ß!), then lim f(ß) must equal f(! ß!). If f is not continuous at Ð Ä Ð! ß! Ñ (! ß!), the limit could have an value different from, and need not even eist. c tan 5. lim Š and lim lim, b the Sandich Theorem Ð Ä Ð00 Ê Ð Ä Ð00 Ð Ä Ð00 2 Š 2 Š 6 6 6. If 0, lim lim lim ˆ 2 2 and Ð Ä Ð00 Ð Ä Ð00 Ð Ä Ð00 6 2 2 Š 6 2 Š 6 lim lim 2 2; if 0, lim lim Ð Ä Ð00 Ð Ä Ð00 Ð Ä Ð00 Ð Ä Ð00 2 cos È lim ˆ 2 2 and lim 2 Ê lim Ð Ä Ð00 6 Ð Ä Ð00 Ð Ä Ð00 2, b the Sandich Theorem 7. The limit is 0 since sin ˆ sin ˆ sin ˆ for 0, and sin ˆ Ÿ Ê Ÿ Ÿ Ê Ÿ Ÿ for Ÿ 0. Thus as (ß) Ä (!ß!), both and approach 0 Ê sin ˆ Ä 0, b the Sandich Theorem. 8. The limit is 0 since ¹ cos Š ¹ Ÿ Ê Ÿ cos Š Ÿ Ê Ÿ cos Š Ÿ for 0, and cos Š for Ÿ 0. Thus as (ß) Ä (!ß!), both and approach 0 Ê cos Š Ä 0, b the Sandich Theorem.
87 Chapter Partial Derivatives 2m 2 tan ) 9. (a) f(ß) m m tan ) sin 2 ). The value of f(ß) sin 2 ) varies ith ), hich is the line's angle of inclination. (b) Since f(ß) m sin 2 ) and since Ÿ sin 2) Ÿ for ever ), lim f(ß) varies from to Ð Ä Ð00 along m. 50. a b Ÿ È È Ÿ È È a b a b a b Ð Ä Ð00 a b Ð Ä Ð00 a b Ê ¹ ¹ Ÿ Ê a b Ÿ Ÿ a b Ê lim 0 b the Sandich Theorem, since lim 0; thus, define f(0ß0) 0 $ $ $ $ r cos ) (r cos )) ar sin ) b r acos ) cos ) sin ) b 5. lim lim lim 0 Ð Ä Ð00 r Ä 0 r cos ) r sin ) r Ä 0 $ $ $ $ $ $ $ $ r cos ) r sin ) racos ) sin ) b 52. lim cos Š lim cos lim cos cos 0 Ð Ä Ð00 Š r Ä 0 r cos ) r sin ) r Ä 0 r sin ) 5. lim lim lim sin sin ; the limit does not eist since sin is beteen Ð Ä Ð00 r Ä 0 r a ) b ) ) r Ä 0 0 and depending on ) 2 2r cos ) 2 cos ) 2 cos ) 5. lim lim lim ; the limit does not eist for co Ð Ä Ð00 r Ä 0 r r cos ) r Ä 0 r cos ) cos ) s ) 0 r cos ) r sin ) r acos ) sin ) b 55. lim tan lim tan lim tan ; Ð Ä Ð00 r Ä 0 r r Ä 0 r a r cos ) sin ) b cos ) sin ) if r Ä 0, then lim tan lim tan ; if r Ä 0, then r Ä! b r r Ä! b r a r cos ) sin ) b cos ) sin ) lim tan lim tan the limit is r Ä! c r r Ä! c Š r Ê r cos ) r sin ) 56. lim lim lim cos sin lim (cos 2 ) hich ranges beteen Ð Ä Ð00 r Ä 0 r a ) ) b ) r Ä 0 r Ä 0 and depending on ) Ê the limit does not eist r cos ) r cos ) sin ) r sin ) 57. lim ln Š lim ln Š % Ð Ä Ð00 r Ä 0 r lim ln a r cos ) sin ) b ln Ê define f(0ß0) ln r Ä 0 2 (2r cos )) ar sin ) b 58. lim lim lim 2r cos sin 0 define f(0 0) 0 Ð Ä Ð00 r Ä 0 r ) ) Ê ß r Ä 0 59. Let $ 0.. Then È $ Ê È 0. Ê 0.0 Ê 0 0.0 Ê f(ß) f(!ß!) 0.0 %. 60. Let $ 0.05. Then $ and $ Ê f(ß) f(0ß0) 0 Ÿ 0.05 %. 6. Let $ 0.005. Then $ and $ Ê f(ß) f(0ß0) 0 Ÿ 0.005 0.005 0.0 %.
Section. Partial Derivatives 875 62. Let $ 0.0. Since Ÿ cos Ÿ Ê Ÿ 2 cos Ÿ Ê Ÿ Ÿ Ê Ÿ Ÿ cos 2 cos 2cos 2cos Ÿ. Then $ and $ Ê f(ß) f(0ß0) 0 Ÿ 0.0 0.0 0.02 %. 6. Let $ È0.05. Then È z $ Ê f(ßßz) f(!ß0ß0) z 0 z Š È t Š È0.05 0.05 %. $ 6. Let $ 0.2. Then $, $, and z $ Ê f(ßßz) f(!ß0ß0) z 0 z z (0.2) 0.008 %. z z 65. Let $ 0.005. Then $, $, and z $ Ê f(ß ß z) f(!ß 0ß 0) ¹ 0¹ z z ¹ ¹ Ÿ z Ÿ z 0.005 0.005 0.005 0.05. % 66. Let $ tan (0.). Then $, $, and z $ Ê f(ßßz) f(!ß0ß0) tan tan tan z Ÿ tan tan tan z tan tan tan z tan $ tan $ tan $ 0.0 0.0 0.0 0.0 %. 67. lim f(ßßz) lim ( z)!! z! f(! ß! ßz!) Ê f is continuous at Ðz ß Ä Ð! ß! ßz! Ñ Ðz ß Ä Ð! ß! ßz! Ñ ever ( ß ßz )!!! 68. lim f(ßßz) lim a Ðz ß Ä Ð! ß! ßz! Ñ Ðz ß Ä Ð! ß! ßz! Ñ z b!! z! f(! ß! ßz!) Ê f is continuous at ever point ( ß ßz )!!!. PARTIAL DERIVATIVES ` ` ` `., 2. 2, 2 ` ` ` `. 2( 2),. 5, 5 2 6 ` ` ` ` 5. 2( ), 2( ) 6. 6(2 ), 9(2 ) 2 ` f ` È ` È ` $ É$ ˆ ` $ É$ ˆ 7., 8., ` f ` ` f ` ` () ` () ` () ` () 9. ( ), ( ) ` a b() (2) a b(0) (2) 2 a b a b ` a b a b 0., ( )() ( )() ( )() ( )() ` ( ) ( ) ` ( ) ( )., ` ` ` ` 2. ˆ, ˆ f f ` ˆ ` ˆ ` ˆ ` ˆ Ð Ñ ` Ð Ñ Ð Ñ ` Ð Ñ ` ` ` `. e ( ) e, e ( ) e
876 Chapter Partial Derivatives ` `. e sin ( ) e cos ( ), e cos ( ) ` f ` ` f ` ` ` ` ` 5. ( ), ( ) ` ` e ` ` ` ` 6. e () ln e ln, e () ln e e ln ` ` ` ` ` ` ` ` ` ` 7. 2 sin ( ) sin ( ) 2 sin ( ) cos ( ) ( ) 2 sin ( ) cos ( ), 2 sin ( ) sin ( ) 2 sin ( ) cos ( ) ( ) 6 sin ( ) cos ( ) ` ` ` ` ` 8. 2 cos a b cos a b 2 cos a b sin a b a b 6 cosa b sina b, ` ` ` ` ` 2 cos a b cos a b 2 cos a b sin a b a b cos a b sin a b ` c ` ln ` ` ` ` ln ` ln ` (ln ) 9., ln 20. f(ß) Ê and ` ` 2. g(), g()! _ n0 ` f ` ` () ` ( ) ( ) ` ( ) ` ( ) 22. f(ß) () n ` f `, Ê f(ß) Ê ( ) and z z 2. f, f 2, f z 2. f z, f z, f z È z z È z 25. f, f, f $Î $Î $Î z 26. f a z b, f a z b, f za z b z z Èz Èz z Èz 27. f, f, f z z È(z) z È(z) z z È(z) 28. f, f, f 2 2z 2z z 2z 29. f, f, f 0. f z (), f z ln () z ln () z ln () () z ln () z, ` (z)() z ` z ` ` ` ` ` `z f ln () z ln () ln () z a z b a z b a z b z. f 2e, f 2e, f 2ze z z z z 2. f ze, f ze, f e z. f sech ( 2 z), f 2 sech ( 2 z), f sech ( 2 z) z. f cosh a z b, f cosh a z b, f 2z cosh a z b
Section. Partial Derivatives 877! 5. `t 2 sin (2 t! ), ` sin (2 t! ) `g ` 2u `g ` 2u 6. v e ˆ 2ve, 2ve v e ˆ 2ve 2ue Ð2uÎvÑ Ð2uÎvÑ Ð2uÎvÑ Ð2uÎvÑ Ð2uÎvÑ Ð2uÎvÑ `u `u v `v `v v `h `h `h ` `9 `) 7. sin 9 cos ), cos 9 cos ), sin 9 sin ) `g `g `g `r `) `z 8. cos ), r sin ), $ p v v Vv v 2Vv $ Vv $ g Vv $ 2g $ 2g 2g g g 9. W V, W P, W, W, W `A `A q `A m `A `A m h `c ` h ` q `m q ` q q 0. m,,, c, ` f ` f ` f ` f ` ` ` ` `` ``.,, 0, 0, ` f ` f ` f ` f ` ` ` ` `` `` 2. cos, cos, sin, sin, cos sin `g `g ` g ` g ` g ` g ` ` ` ` `` ``. 2 cos, sin sin, 2 sin, cos, 2 cos `h `h ` h ` h ` h ` h ` ` ` ` `` ``. e, e, 0, e, e ` ` ` ` ` ` ` ` ` () ` () `` `` () 5.,,,, ` s ` ` s ` 6. ˆ ˆ, ˆ ˆ, ` ˆ ` ˆ ` ˆ ` ˆ ` s ` (2) 2 ` s (2) 2 a b a b ` a b a b,, ` s `` ` s `` a b( ) (2) a b a b ` 2 ` ` ` ` 2 ` 2 `` (2 ) `` (2 ) 7.,,, and ` ` ` ` ` ` `` `` 8. e ln, ln,, and ` $ % ` $ $ ` $ ` ` `` 9. 2, 2, 2 6 2, and ` `` $ 2 6 2 ` ` ` ` ` `` 50. sin cos, cos sin, cos cos, and ` `` cos cos 5. (a) first (b) first (c) first (d) first (e) first (f) first 52. (a) first three times (b) first three times (c) first tice (d) first tice
878 Chapter Partial Derivatives f( hß2) f(ß2) c (h) 26(h) d(26) h6a2hh b6 h h h 5. f (ß2) lim lim lim h Ä 0 h Ä 0 h Ä 0 h 6h lim lim ( 6h), h Ä 0 h h Ä 0 c (2h) (2h) d(26) f (ß2) lim lim lim h Ä 0 h Ä 0 h Ä 0 lim ( 2) 2 h Ä 0 f(ß2 h) f(ß2) (2 6 2h) (2 6) h h h f( 2 hß) f( 2ß) c2( 2h) ( 2h) d( 2) h 5. f ( 2ß) lim h lim h Ä 0 h Ä 0 (2hh) lim lim, h Ä 0 h h Ä 0 f( 2ß h) f( 2ß) c(h) 2(h ) d( 2) h f ( 2ß) lim h lim h Ä 0 h Ä 0 ah2h2h b h 2h lim lim lim ( 2h) h Ä 0 h h Ä 0 h h Ä 0 f( ß ßz h) f(, ßz )!!!!!! 55. f z(! ß! ßz!) lim h ; h Ä 0 f(ß2ß h) f(, 2ß) 2( h) 2(9) 2h 2h f z(ß2ß) lim h lim h lim h lim (2 2h) 2 h Ä 0 h Ä 0 h Ä 0 h Ä 0 f( ß hßz ) f(, ßz )!!!!!! 56. f (! ß! ßz!) lim h ; h Ä 0 f( h) ß ß f(,0) ß a2h 9hb0 f ( ß0ß) lim h lim h lim (2h 9) 9 h Ä 0 h Ä 0 h Ä 0 57. ˆ `z z $ `z `z $ `z ` z 2 ` 0 Ê az 2b ` z Ê at (ßß) e have ( 2) ` or `z ` 2 58. ˆ ` z ˆ ` ` 2 0 ˆ z 2 ` ` `z `z `z Ê `z Ê at (ßß) e have ( 2) `z or ` `z 6 ` A ` A a ` A `a `a bc sin A `b 59. a b c 2bc cos A Ê 2a (2bc sin A) Ê ; also 0 2b 2c cos A (2bc sin A) `A `A c cos A b Ê 2c cos A 2b (2bc sin A) Ê `b `b bc sin A a b (sin A) `a A a cos A `a `a a cos A ` 60. sin A sin B Ê sin A 0 Ê (sin A) ` a cos A 0 Ê `A sin A ; also ˆ `a `a b( csc B cot B) Ê b csc B cot B sin A sin A `B `B 6. Differentiating each equation implicitl gives v ln u ˆ v u u and 0 u ln v ˆ v or (ln u) v ˆ v u u ˆ u v (ln v) u 0 Ÿ v v u º 0 ln v º ln u u º u ln v º v Ê v v u v ln v (ln u)(ln v) 62. Differentiating each equation implicitl gives (2) u (2) u and 0 (2)u u or 2 º 0 º Ê u and 2 2 2 2 u u º 2 º 2 º 2 0 º 2 2 `s ` ` 2 2 2 `u `u `u (2) u (2)u (2) 0 u ; net s Ê 2 2 2 2 2 2 2 2 2 Š 2 Š ` f ` f ` f ` f ` f ` f ` ` `z ` ` `z ` ` `z 6. 2, 2, z Ê 2, 2, Ê 2 2 ( ) 0
Section. Partial Derivatives 879 ` f ` f ` f ` f ` f ` f ` ` `z ` ` `z ` ` `z 6. 6z, 6z, 6z a b, 6z, 6z, 2z Ê 6z 6z 2z 0 ` c ` c ` c ` c ` ` ` ` ` ` ` ` c2 c2 2 2 2 2 65. 2e sin 2, 2e cos 2, e cos 2, e cos 2 Ê e cos 2 e cos 2 0 ` f ` f ` f ` f ` ` ` a b ` a b ` ` a b a b 66.,,, Ê 0 ` f $Î $Î ` f $Î ` ` $Î ` f $Î $Î a z b, ` z a z b (2z) za z b ; ` $Î &Î f ` f $Î &Î ` a b a b ` a b a b ` $Î &Î ` ` ` ` z z z z a b a b Ê ` ` `z 67. a z b (2) a z b, a z b (2) z z, z z, $Î &Î $Î &Î a z b a z b a z b a z b $Î &Î $Î &Î a z b z a z b a z b a z ba z b 0 ` f ` f ` ` `z ` ` 68. e cos 5z, e cos 5z, 5e sin 5z; 9e cos 5z, 6e cos 5z, ` f ` f ` f ` f `z ` ` `z 25e cos 5z Ê 9e cos 5z 6e cos 5z 25e cos 5z 0 ` ` ` ` ` ` `t ` `t `t c 69. cos ( ct), c cos ( ct); sin ( ct), c sin ( ct) Ê c [ sin ( ct)] ` ` ` ` ` `t ` `t Ê ` ` `t c [ cos (2 2ct)] c ` 70. 2 sin (2 2ct), 2c sin (2 2ct); cos (2 2ct), c cos (2 2ct) ` ` ` ` ` `t ` `t ` ` 7. cos ( ct) 2 sin (2 2ct), c cos ( ct) 2c sin (2 2ct); sin ( ct) cos (2 2ct), c sin ( ct) c cos (2 2ct) ` `t Ê c [ sin ( ct) cos (2 2ct)] c ` ` c ` ` c ` ` ` ct `t ct ` (ct) `t (ct) `t (ct) ` 72., ;, Ê c c ` ` ` ` `t ` ` ` 8c sec (2 2ct) tan (2 2ct) ` ` ` c [8 sec (2 2ct) tan (2 2ct)] c t Ê t ` 7. 2 sec (2 2ct), 2c sec (2 2ct); 8 sec (2 2ct) tan (2 2ct), ` b ` b ` b ` `t ` ` ` ` ` 5c cos ( ct) c e ` c 5 cos ( ct) e c t bct ct Ê t c b d ` ct ct ct 7. 5 sin ( ct) e, 5c sin ( ct) ce ; 5 cos ( ct) e, ` `u ` ` f ` f ` `u ` ` f `t `u `t `u `t `u `u ` `u ` `u ` `u 75. (ac) Ê (ac) Š (ac) a c ; a Ê Š a a ` ` ` ` ` `u `t `u `u ` a Ê ac c Š a c
880 Chapter Partial Derivatives 76. If the first partial derivatives are continuous throughout an open region R, then b Theorem in this section of the tet, f(ß) f(! ß!) f (! ß!)? f (! ß!)? %? %?, here %, % Ä 0 as?,? Ä 0. Then as (ß) Ä (! ß!),? Ä 0 and? Ä 0 Ê lim f(ß) f(! ß!) Ê f is continuous at ever point Ð Ä Ð! ß! Ñ (! ß!) in R. 77. Yes, since f, f, f, and f are all continuous on R, use the same reasoning as in Eercise 76 ith f (ß) f (! ß!) f (! ß!)? f (! ß!)? %? %? and f (ß) f (! ß!) f (! ß!)? f (! ß!)? s%? s%?. Then lim f (ß) f (! ß!) Ð Ä Ð! ß! Ñ and lim f (ß) f (! ß!). Ð Ä ÐßÑ!!. THE CHAIN RULE ` ` d d d ` ` dt dt dt d Ê dt d dt. (a) 2, 2, sin t, cos t Ê 2 sin t 2 cos t 2 cos t sin t 2 sin t cos t (b) ( ) 0 0; cos t sin t 0 ` ` d d d ` ` dt dt dt 2. (a) 2, 2, sin t cos t, sin t cos t Ê (2)( sin t cos t) (2)( sin t cos t) 2(cos t sin t)(cos t sin t) 2(cos t sin t)(sin t cos t) a2 cos t 2 sin tba2 cos t 2 sin tb d dt 0; (cos t sin t) (cos t sin t) 2 cos t 2 sin t 2 Ê 0 d dt (b) (0) 0 ` ` ` ( ) d d dz ` z ` z `z z dt dt dt t d 2 2 cos t sin t cos t sin t d dt z z z t t Š z z a b Š Š t t t dt d dt. (a),,, 2 cos t sin t, 2 sin t cos t, Ê cos t sin t sin t cos t ; t Ê (b) () ` 2 ` 2 ` 2z d d dz Î ` z ` z `z z dt dt dt Ê d 2 sin t 2 cos t cî zt 2 cos t sin t 2 sin t cos t ˆ Î t cî t dt z z z cos tsin t6t ; ln z ln cos t sin t 6t ln ( 6t) Ê 6t a b a b dt 6t d 6 dt 9. (a),,, sin t, cos t, 2t (b) () ` ` ` d 2t d dz t d te 2e t e ` ` `z z dt t dt t dt dt t t z 5. (a) 2e, 2e,,,, e Ê (t) atan tbat b 2 at b t e t t e Ê d ˆ 2 dt t at ba2 tan t b(2t) t tan t d dt ˆ c t t tan t ; 2e ln z a2 tan tbat bt (b) () ()() ` ` ` d d dz tc d cos tc ` ` `z dt dt t dt dt t t cos (t ln t) t (ln t)[cos (t ln t)] e c t (ln t)[cos (t ln t)] cos (t ln t) e c t ; z sin t e csin (t ln t) Ê d t e c[cos (t ln t)] ln t t ˆ t e c dt t ( ln t) cos (t ln t) d 6. (a) cos, cos,,,, e Ê cos e (b) dt () ( 0)() 0
`z `z ` `z ` cos v e e ln e sin v `u ` `u ` `u u cos v u 7. (a) ae ln bˆ Š (sin v) (u cos v) ln (u sin v) (u cos v)(sin v) u u sin v ( cos v) ln (u sin v) cos v; `z `z ` `z ` u sin v e e u cos v `v ` `v ` `v ae ln bˆ u cos v Š (u cos v) ae ln b(tan v) (u cos v)(u cos v) [ (u cos v) ln (u sin v)](tan v) u sin v ( u sin v) ln (u sin v) sin v ; `z sin v z e ln (u cos v) ln (u sin v) Ê u ( cos v) ln (u sin v) (u cos v) ˆ ` u sin v `z u cos v ( cos v) ln (u sin v) cos v; also ( u sin v) ln (u sin v) (u cos v) ˆ `v u cos v ( u sin v) ln (u sin v) sin v `z `u `z ()(2) ˆ cos `v ˆ sin Section. The Chain Rule 88 u cos v (b) At ˆ 2 ß : cos ln ˆ 2 sin cos 2È2 ln È2 2È2 È2 (ln 2 2); ( )(2) sin ln ˆ 2 sin È2 ln È2 È2 2È2 ln 2 È2 u sin v `z Š Š c 8. (a) `u cos v sin v u 0; Š Š cos v sin v (u sin v)(cos v) (u cos v)(sin v) c Š Š `z u sin v u cos v (u sin v)(u sin v) (u cos v)(u cos v) `v ( u sin v) u cos v u Š Š `z ` z `u `v cot v sin v cos v ; z tan Š tan (cot v) Ê 0 and ˆ acsc vb sin v cos v `z `z 6 `u `v (b) At ˆ. ß : 0 and ` ` ` ` ` ` `z `u ` `u ` `u `z `u 9. (a) ( z)() ( z)() ( )(v) 2z v( ) ` ` ` ` ` ` `z `v ` `v ` `v `z `v (u v) (u v) 2uv v(2u) 2u uv; ( z)() ( z)( ) ( )(u) ( )u 2v (2u)u 2v 2u ; ` `u z z au v bau v uv bau v uv b u v 2u v Ê 2u uv and ` `v 2v 2u (b) At ˆ ` : 2 ˆ ˆ ` () and 2() 2 ˆ ß ` u ` v ` 2 v v 2 v v 2z v `u z z z 0. (a) Š ae sin u ue cos ubš ae cos u ue sin ubš ae b v ˆ 2ue sin u v v ae sin uue cos ub ue 2v sin uue 2v cos uue 2v v 2ue cos u v ue 2v sin uue 2v cos uue 2v e cos u ˆ 2ue v v a b 2 ue 2v sin uue 2v cos uue 2v u ˆ v a ue sin ub e ; ` 2 v 2 v 2z v `v z z z Š aue sin ubš aue cos ubš aue b v 2ue sin u v ue 2v sin uue 2v cos uue 2v v 2ue cos u ue 2v sin uue 2v cos uue 2v v ue cos u v 2ue ue 2v sin uue 2v cos uue 2v v ue 2; 2v ln u e sin u 2v u e cos u 2v u e 2v ln 2u e ` 2 ` `u u `v ` 2 ` `u `v ˆ aue sin ub ˆ a b ˆ a b a b a b ln 2 2 ln u 2v Ê and 2 (b) At ( ß!): and 2 `u `u `p `u `q `u ` r rp pq qrrppq ` `p ` `q ` `r ` qr (qr) (qr) (qr) `u `u `p `u `q `u ` r rp pq qrrppq 2p2r ` `p ` `q ` `r ` qr (qr) (qr) (qr) (qr) 2 2z) (2 2 2z) z `u `u ` ; `u ` `u `r (2z 2) (z ) `z `p `z `q `z `r `z rp pq qrrppq 2q2p q r (q r) (q r) (q r) (q r) (2z 2) (z ) ;. (a) 0;
882 Chapter Partial Derivatives u Ê 0,, and p q 2 `u `u (z ) ( ) z `u (z )(0) () q r 2z 2 z ` ` (z ) (z ) `z (z ) (z ) (b) At Š È ß 2 ß `u : ` u 0, `u, and 2 2 ` ` (2) `z (2) qr qr z ln ` Èp Èp È sin qr qr c `u e qr z qr z re sin p z ˆ z z z ` È (0) are sin pbš qe sin p (0) z ; p a b `u 2. (a) e (cos ) qr re sin p (0) qr e cos e cos qe sin p (0) z if a b a b ; qr e (0) are sin p b(2z ln ) aqe sin pbˆ a2zre sin p b(ln ) qr c ` u qr qr qr qe sin p `z È p z z z ˆ z az ln bab z zln z `u z z a b z Ÿ Ÿ Ê ` `u `u ` z z, and `z z ln from direct calculations ˆ ` u ˆ È u ˆ ˆ ˆ 2 ˆ ˆ ˆ ß ß u Î ` ÐÎÑ È ` ` ` ` z Î (2z) ln ln ; u e sin (sin ) if, (b) At : 2,, ln È 2 ln 2 dz `z d `z d dz `z du `z dv ` d dt ` dt ` dt dt `u dt `v dt ` dt.. 5. ` ` ` ` ` ` ` z ` ` ` ` ` ` ` z `u ` `u ` `u `z `u `v ` `v ` `v `z `v 6. ` ` `r ` `s ` `t ` ` `r ` `s ` `t ` `r ` `s ` `t ` ` `r ` `s ` `t `
Section. The Chain Rule 88 7. ` ` ` ` ` ` ` ` ` ` `u ` `u ` `u `v ` `v ` `v 8. ` ` `u ` `v ` ` `u ` `v ` `u ` `v ` ` `u ` `v ` 9. ` z ` z ` ` z ` ` z ` z ` ` z ` `t ` `t ` `t `s ` `s ` `s ` d `u ` d `u ` d `u `r du `r `s du `s `t du `t 20. 2. 22. ` ` ` ` ` ` ` z ` ` v `p ` `p ` `p `z `p `v `p
88 Chapter Partial Derivatives ` ` d ` d ` d d ` ` d ` d ` d d `r ` dr ` dr ` dr dr `s ` ds ` ds ` ds ds 2. since 0 since 0 25. Let F(ß) 2 0 Ê J(ß) 2. ` ` ` ` ` `s ` `s ` `s $ and F (ß) Ê d d Ê (ß) d F d F ( ) d F d F 2 26. Let F(ß) 0 Ê F (ß) and F (ß) 2 Ê d d Ê ( ß) 2 d F 2 d F 2 27. Let F(ß) 7 0 Ê F (ß) 2 and F (ß) 2 Ê d d 5 Ê (ß2) d F e cos d d F e sin d ( ln 2) (2 ln 2) 28. Let F(ß) e sin ln 2 0 Ê F (ß) e cos and F (ß) e sin Ê Ê!ß $ $ z ` z F ` z () ; ` z F z z ` F z z z ` ` Fz z z `z ` () 29. Let F(ßßz) z z 2 0 Ê F (ßßz), F (ßßz) z, F (ßßz) z Ê Ê ß ß Ê ß ß z z z 0. Let F(ßßz) 0 Ê F (ßßz), F (ßßz), F (ßßz) `z F Š z `z `z F Š z `z ` Fz ` ` Fz Š Š ` z z Ê Ê (26) ß ß 9; Ê (26) ß ß. Let F(ßßz) sin ( ) sin ( z) sin ( z) 0 Ê F (ßßz) cos ( ) cos ( z), F (ßßz) cos ( ) cos ( z), F (ßßz) cos ( z) cos ( z) Ê z `z ` cos ( ) cos ( z) `z `z F cos ( ) cos ( z) `z cos ( z) cos ( z) ` ` Fz cos ( z) cos ( z) ` Ê ( ß ß ) ; Ê ( ß ß ) z 2 z z z ` z F ˆ 2 e ` z ( ln 2 ln ) ; ` z F e z e ` z 5 ` F ez ` ln 2 ` F ez ` ( ln 2 ln ) z z ln 2 2. Let F(ßßz) e e 2 ln 2 ln 2 0 Ê F (ßßz) e, F (ßßz) e e, F (ßßz) e Ê Ê ß ß Ê ß ß ` ` ` ` ` ` `z `r ` `r ` `r `z `r. 2( z)() 2( z)[ sin (r s)] 2( z)[cos (r s)] 2( z)[ sin (r s) cos (r s)] 2[r s cos (r s) sin (r s)][ sin (r s) cos (r s)] F Fz
Section. The Chain Rule 885 ` Ê 2()(2) 2 `r rßs ` ` ` ` ` ` ` z 2v 2v v `. ˆ () ˆ (0) (u v) ˆ Ê () ˆ ˆ `v ` `v ` `v `z `v u z u u `v ußv2 8 ` 5. 2u v 2 ˆ 2 ( 2) ˆ () 2(u 2v ) ( 2) ` ` ` ` `v ` `v ` `v (u2v) u2v ` Ê 7 `v u0ßv0 `z `z ` `z ` `u ` `u ` `u $ $ $ $ 6. ( cos sin )(2u) ( cos cos )(v) cuv cos au v uv bsin uv d(2u) cau v bcos au v uv bau v b cos uv d(v) `z Ê 0 (cos 0cos 0)() 2 `u u0ßv `z dz ` 5 u 5 u `z 5 7. ˆ e e Ê (2) 2; `u d `u ae u ln vb `u u ln2 ß v (2) ˆ ˆ ˆ Ê () `z dz ` 5 5 `z 5 `v d `v v ae ln vb v `v u ln2 ß v (2) 8. ` z dz ` q È v È `u dq `u Š q Š v u Š È Š v tan u u c c atan ubau b ` z Ê 2 ` z dz `q c tan u c ; Š Š `u ußv2 atan ba b `v dq `v q 2Èv c tan u z Š Š Ê ` È c v tan u 2Èv (v ) ` v ußv2 `V `V dv `V di `V dr di dr `I `R dt `I dt `R dt dt dt di di dt dt 9. V IR Ê R and I; R I Ê 0.0 volts/sec (600 ohms) (0.0 amps)(0.5 ohms/sec) Ê 0.00005 amps/sec dv `V da `V db `V dc da db dc dt `a dt `b dt `c dt dt dt dt dv dt aßb2ßc ds `S da `S db `S dc Ê dt `a dt `b dt `c dt da db dc ds dt dt dt dt aßb2ßc 0. V abc Ê (bc) (ac) (ab) Ê (2 m)( m)( m/sec) ( m)( m)( m/sec) ( m)(2 m)( m/sec) m /sec and the volume is increasing; S 2ab 2ac 2bc 2(b c) 2(a c) 2(a b) Ê 2(5 m)( m/sec) 2( m)( m/sec) 2( m)( m/sec) 0 m /sec and the surface area is not changing; D Èa b c dd `D da `D db ` D dc Ê ˆ da db dc a b c dd Ê dt `a dt `b dt `c dt È a b c dt dt dt dt aßb2ßc 6 È m È Š [( m)( m/sec) (2 m)( m/sec) ( m)( m/sec)] m/sec 0 Ê the diagonals are decreasing in length ` ` ` ` ` ` ` ` ` ` ` ` ` `u ` `v ` ` ` `u `v ` `u ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ( ) ` () ` (0) u v u v `u `v, and `u `v ` `z `u `z `v `z ` `z `u `v ` `v ` ` ` `z. () (0) ( ), (0) ( ) () Ê 0 ` ` ` ` ` `r `r `r `) r `) ` cos ` `r ) ) ) ) ˆ ) r `) ) ) ) ` cos cos Ê f (sin )) ˆ ) ` ` r r ; then r f cos ) ` (sin )) r ˆ ) ` ` `) ` ` r `) (sin )) Ê f cos ) ` sin ` sin cos ` sin ` sin cos ` Ê f (cos ) ` sin ` `r a b ˆ ) ) `r r ` a b ˆ ) ) `r r ` ˆ ) ) ) ) ) ) `r r `) ` a b a bˆ ˆ 2 sin ) cos ) ˆ ` ` sin ) Š ˆ ` ) `r r `r `) r `) 2. (a) f f f cos ) f sin ) and f ( r sin )) f (r cos )) Ê f sin ) f cos ) (b) sin f sin cos f sin and f sin cos f cos (c) f cos and `r r `r `) r `) `r r `) af b asin bˆ ` ˆ 2 sin ) cos ) ˆ ` ` cos ) Š ˆ ` af b af b ˆ ` ˆ ` ) Ê $
886 Chapter Partial Derivatives ` ` `u ` `v ` ` ` `. Ê ˆ ` ` ` ˆ ` `u ` `v ` `u `v `u ` `u ` `v ` ` ` u ` ` v ` ` u ` ` v ` ` ` ` ` `u `u ` `v`u ` `u`v ` `v ` `u `u `v`u `u`v `v Š Š Š Š ` ` ` ` ` ` ` u ` ` v ` ` `u `u `v`u `v ` `u ` `v ` `u `v 2 ; Ê Š Š ` ` `u ` `v ` `u ` `v `u `u ` `v`u ` `u`v ` `v ` ` ` ` ` ` ` ` ` ` `u `u `v`u `u`v `v `u `u `v`u `v Š Š 2 ; thus ` ` `u `v uu vv uu vv a b a b a b( ) 0, since 0 ` ` ` `. f (u)() g (v)() f (u) g (v) Ê f (u)() g (v)() f (u) g (v); f (u)(i) g (v)( i) Ê f (u) ai bg (v) ai b f (u) g (v) Ê 0 df d d dz z dt ` dt ` dt `z dt df a b dt 5. f (ßßz) cos t, f (ßßz) sin t, and f (ßßz) t t 2 Ê (cos t)( sin t) (sin t)(cos t) t t 2 () t t 2; 0 Ê t t 2 0 Ê t 2 or t ; t 2 Ê cos ( 2), sin ( 2), z 2 for the point (cos ( 2) ßsin ( 2) ß2); t Ê cos, sin, z for the point (cos ßsin ß) d ` d ` d ` dz 2 2 2 6. a2e cos z b( sin t) a2 e cos zbˆ a e sin z b() dt ` dt ` dt `z dt t 2 2 2 e cos z t 2() ()() 2 2e cos z sin t e sin z; at the point on the curve z 0 Ê t z 0 d dt Ðln20 ß Ê 0 0 `T `T dt `T d `T d ` ` dt ` dt ` dt 7. (a) 8 and 8 Ê (8 )( sin t) (8 )(cos t) dt (8 cos t sin t)( sin t) (8 sin t cos t)(cos t) sin t cos t Ê dt 6 sin t cos t; dt 5 7 dt Ê Ê Ê Ê 0 sin t cos t 0 sin t cos t sin t cos t or sin t cos t t,,, on the interval 0 Ÿ t Ÿ 2; dt È2 È2 dt ¹ 6 sin cos 0 Ê T has a minimum at (ß) Š ß ; t dt È2 È2 dt Ê ß ß ¹ 6 sin cos 0 T has a maimum at ( ) Š ; t dt 5 5 È2 È2 dt ¹ 6 sin cos 0 Ê T has a minimum at (ß) Š ß ; t 5 dt 7 7 È2 È2 dt ¹ 6 sin cos 0 Ê T has a maimum at (ß) Š ß t 7 `T `T ` ` È2 È2 È2 È2 (b) T Ê 8, and 8 so the etreme values occur at the four points found in part (a): T Š ß T Š ß ˆ ˆ ˆ 6, the maimum and È È È È 2 2 2 2 T Š T Š ˆ ˆ ˆ ß ß 2, the minimum `T `T dt `T d `T d 8. (a) and Ê Š 2È2 sin t Š È2 cos t ` ` dt ` dt ` dt Š È2 sin t Š 2È2 sin t Š 2È2 cos t Š È 2 cos t sin t cos t sin t a sin tb dt dt dt dt È2 8 sin t Ê 6 sin t cost t; 0 Ê 8 sin t 0 Ê sin t Ê sin t Ê t, 5 7,, on the interval 0 Ÿ t Ÿ 2; dt ¹ 8 sin 2 ˆ 8 Ê T has a maimum at (ß) (2ß); dt t dt dt ¹ 8 sin 2 ˆ 8 Ê T has a minimum at (ß) ( 2ß); t
dt 5 dt Section.5 Directional Derivatives and Gradient Vectors 887 ¹ 8 sin 2 ˆ 8 Ê T has a maimum at (ß) ( 2ß); t 5 dt 7 dt ¹ 8 sin 2 ˆ 8 Ê T has a minimum at (ß) (2ß) t 7 `T ` `T ` (b) T 2 Ê and so the etreme values occur at the four points found in part (a): T(2ß) T( 2ß) 0, the maimum and T( 2ß) T(2ß), the minimum ' 9. G(uß) g(tß) dt here u f() Ê g(uß)f () g (tß) dt; thus a u dg `G du `G d d `u d ` d ' ' ' F() Èt% $ dt Ê F () Éa % b $ ` (2) Èt% $ dt 2È) $ dt 0 0 ` 0 ' 50. Using the result in Eercise 9, F() Èt$ dt Èt$ dt Ê F () ' ' Éa $ b ` Èt$ dt È' dt `.5 DIRECTIONAL DERIVATIVES AND GRADIENT VECTORS ' È t $ ' a u È % $ 2 t ` `., Ê f ij; f(2ß) Ê is the level curve 2 2 ` ` ` ` i j 2. Ê (ß ) ; Ê (ß) Ê f ; f(ß) ln 2 Ê ln 2 ln a b Ê 2 is the level curve `g `g `g ` ` `. 2 Ê ( ß0) 2; Ê g 2 ij; g( ß0) Ê is the level curve `g `g `g. Ê Š È2ß È2; ` ` ` `g Ê ` Š È2ß Ê g È2 ij; g Š È2 ß Ê or is the level curve z ` ` ` ` `z `z 5. 2 Ê (ß ß ) ; 2 Ê (ß ß ) 2; z ln Ê (ß ß ) ; thus f i2j
888 Chapter Partial Derivatives z ` z ` ` ` `z z Ê ` f (ßß) ; thus f 6 ` z i j 6. 6z Ê ( ßß ) ; 6z Ê (ßß) 6; 6z a b ` f 26 ` f 2 ` a z $Î b ` 27 ` a z $Î b ` 5 ` f z 2 26 2 2 `z ` ( 2 2) ; thus f z Ê z ß ß z 5 27 i5 j a b 5 7. Ê ( ß2ß2) ; Ê ( ß2ß2) ; È È ` È ` 6 ` ` 6 ` f 2 `z e sin z Ê ˆ È È ` z!ß!ß 6 ; thus f Š i j 8. e cos z Ê ˆ!ß!ß ; e cos z sin Ê ˆ 0ß0 ß ; v v i j È 5 5 9. u i j; f (ß) 2 Ê f (5ß5) 0; f (ß) 2 6 Ê f (5ß5) 20 Ê f 0i20 j Ê (Duf) P f u 0ˆ 20ˆ! v v i j È ( ) 5 5 5 5 0. u i j; f (ß) Ê f ( ß) ; f (ß) 2 Ê f ( ß) 2 Ê f i2 j Ê (D f) f u u P! 2 8 5 5 v 2i 5j 2 5 2È v È2 5 2È 2 2È 6 5 g ( ) g (D g) g 2È i j u P u!. u i j; g (ß) Ê g (ß) ; g (ß) Ê ß Ê Ê v i 2j 2 v È ( 2) È È c Š ˆ ˆ È Ê Š 2. u i j; h (ß) Ê h (ß) ; ˆ ˆ È 6 ˆ P! 2È 2È u Ê Š h ( ß ) Ê h ( ß ) Ê h Ê (D h) h i j u 2È v v i6j È 6 ( 2) 6 2 7 7 7. u i j ; f (ßßz) z Ê f (ßß2) ; f (ßßz) z Ê f (ßß2) ; f (ßßz) Ê f (ßß2) 0 Ê f i j Ê (D f) f u z z P v ij v È È È È. u i j ; f (ßßz) 2 Ê f (ßß) 2; f (ßßz) Ê f() ß ß ; f(z) ß ß 6z Ê f() ß ß 6 Ê f 2ij6 Ê (Df) f u z z P È È È 2Š Š 6Š 0 v v 2ij2 È2 ( 2) 2 2 u! u! 8 7 7 5. u i j ; g (ßßz) e cos z Ê g (0ß0ß0) ; g (ßßz) ze sin z Ê g (0ß0ß0) 0; g (ßßz) e sin z Ê g (0ß0ß0) 0 Ê g i Ê (D g) g u 2 z z P v i2j2 2 2 6. u i j ; h (ßßz) sin Ê h ˆ ß0ß ; v È 2 2 z z h (ß ß z) sin ze Ê h ˆ ß!ß ; h (ß ß z) e Ê h ˆ ß!ß 2 Ê h i j 2 z z z (Duh) P h u 2! Ê 7. f (2 ) i( 2) j Ê f( ß) ij Ê u i j; f increases u! f i j f È( ) È2 È2 È2 È2 È2 È2 most rapidl in the direction u i jand decreases most rapidl in the direction u i j; (D f) f u f È2 and (D f) È2 u P u!! P
Section.5 Directional Derivatives and Gradient Vectors 889 f f 8. f a2 e sin b i a e sin e cos b j Ê f(ß0) 2 j Ê u j ; f increases most rapidl in the direction u jand decreases most rapidl in the direction u j; (Duf) P! f u f 2 and (D f) 2 u P! f i 5 j f È ( 5) ( ) 5 5 È È È È È È 9. f i Š z j Ê f( ß ß ) i 5 j Ê u i j ; f increases most rapidl in the direction of u i j and decreases 5 most rapidl in the direction u È i È j È ; (Du f) P f u f È and! (Duf) P! È 20. g e e 2z Ê g ßln 2ß 2 2 Ê g 2 2 2 2 i j ˆ i j i j u i j ; g È2 2 2 2 2 2 u P! u P! g increases most rapidl in the direction u i j and decreases most rapidl in the direction u i j ; (D g) g u g and (D g) 2. f ˆ i Š j ˆ Ê f( ßß) 2 i 2 j 2 f Ê u i j ; z z f È È È u È i È j È and decreases most rapidl in the direction f increases most rapidl in the direction u i j ; (D f) f u f 2È and (D f) 2È È È È u P u!! P 2 2 h 2ij6 h È2 6 2 6 2 6 7 i 7 j 7 u 7 i 7 j 7 2 6 u 7 i 7 j 7 u P u u P!! 22. h Š iš j6 Ê h( ßß0) 2ij6 Ê u ; h increases most rapidl in the direction and decreases most rapidl in the direction ; (D h) h h 7 and (D h) 7 2. f 2i2 j Ê f Š È2ßÈ2 2È2 i2è2 j Ê Tangent line: 2È2 Š È2 2È2 Š È2 0 Ê È2 È2 2. f 2 ij Ê f Š È2ß 2È2 i j Ê Tangent line: 2È2 Š È2 ( ) 0 Ê 2È2 25. f i j Ê f(2ß2) 2i2j Ê Tangent line: 2( 2) 2( 2) 0 Ê
890 Chapter Partial Derivatives 26. f (2 ) i(2 ) j Ê f( ß2) i5j Ê Tangent line: ( ) 5( 2) 0 Ê 5 0 v 27. f i( 2) j Ê f(ß2) 2i7 j; a vector orthogonal to f is v 7i2 j Ê u 7 2 7 2 È5 È5 È5 È5 i j and u i j are the directions here the derivative is zero v 7i 2j È7 ( 2) a b a b v i j v È È È and 2 2 È2 È2 28. f i j Ê f( ß) ij; a vector orthogonal to f is v ij Ê u i j u i j are the directions here the derivative is zero 29. f (2 ) i( 8) j Ê f(ß 2) i j Ê f(ß 2) È( ) () È85 ; no, the maimum rate of change is È85 0. T 2 i(2 z) j Ê T(ßß) 2 ij Ê T(ßß) È( 2) È6 ; no, the minimum rate of change is È6. f f ( ß) i f ( ß) j and u i j Ê (Du f)(ß2) f (ß2) Š f (ß2) Š i j È È2 È2 È2 È2 2È2 Ê f (ß2) f (ß2) ; u j Ê (D f)(ß2) f (ß2)(0) f (ß2)( ) Ê f (ß2) u v i 2j i j u v È( ) ( 2) 2 (D f) f 6 7 È i 5 È j u P 5! u È5 È5 È5 Ê f (ß2) ; then f (ß2) Ê f (ß2) ; thus f(ß2) and Ê 2. (a) (Duf) P 2È Ê f 2È v ij ; u i j ; thus u f v È ( ) È È È f È È È Ê f f u Ê f 2È Š i j 2 i 2 j 2 v i j (b) v ij Ê u i j Ê (D f) f u 2 2 2(0) 2 2 u P! Š Š È v È È2 È2 È2 È2. The directional derivative is the scalar component. With f evaluated at P!, the scalar component of f in the direction of uis f u (D u f) P!.. D f f i (f if jf ) i f ; similarl, D f f j f and D f f f i z j z 5. If (ß) is a point on the line, then T(ß) (!) i(!) jis a vector parallel to the line Ê T N 0 Ê A( ) B( ) 0, as claimed.!! `(f) `(f) `(f) 6. (a) (f) i j ˆ i Š j ˆ Š i j f ` ` `z ` ` `z ` ` `z `(f g) `(f g) `(f g) `g `g `g ` ` `z ` ` ` ` `z `z (b) (f g) i j Š i Š j Š `g `g `g `g `g `g ` ` ` ` `z `z ` ` `z ` ` `z i i j j Š i j Š i j f g (c) (f g) f g (Substitute g for g in part (b) above)
Section.6 Tangent Planes and Differentials 89 `(fg) `(fg) `(fg) `g `g `g ` ` `z ` ` ` ` `z `z (d) (fg) i j Š g f i Š g f j Š g f ˆ g `g `g Š f Š g Š f ˆ g `g i i j j Š f ` ` ` ` `z `z `g `g `g ` ` `z ` ` `z fš i j gš i j f gg f f f f `Š `Š `Š `g `g f g f g f g g g g f g ` ` `z g g g ` ` ` ` `z `z (e) Š i j Š iœ jš `g `g `g `g `g `g g ig jg f if jf gš f ` i ` j `z Š ` i ` j `z g g g g ` ` `z ` ` `z Œ Œ g f f g g f f g g g g `g.6 TANTGENT PLANES AND DIFFERENTIALS. (a) f 2i2j2z Ê f(ßß) 2i2j2 Ê Tangent plane: 2( ) 2( ) 2(z ) 0 Ê z ; (b) Normal line: 2t, 2t, z 2t 2. (a) f 2i2j2z Ê f(ß5ß) 6i0j8 Ê Tangent plane: 6( ) 0( 5) 8(z ) 0 Ê 5 z 8; (b) Normal line: 6t, 5 0t, z 8t. (a) f 2i2 Ê f(2ß0ß2) i2 ÊTangent plane: ( 2) 2(z 2) 0 Ê 2z 0 Ê 2 z 2 0; (b) Normal line: 2 t, 0, z 2 2t. (a) f (2 2) i(2 2) j2z Ê f(ßß) j6 Ê Tangent plane: ( ) 6(z ) 0 Ê 2 z 7; (b) Normal line:, t, z 6t 5. (a) z z f a sin 2 ze bia zbjae b Ê f(0ßß2) 2i2 j Ê Tangent plane: 2( 0) 2( ) (z 2) 0 Ê 2 2 z 0; (b) Normal line: 2t, 2t, z 2 t 6. (a) f (2 ) i( 2) j Ê f(ßß) ij Ê Tangent plane: ( ) ( ) (z ) 0 Ê z ; (b) Normal line: t, t, z t 7. (a) f ijfor all points Ê f(0ßß0) ij Ê Tangent plane: ( 0) ( ) (z 0) 0 Ê z 0; (b) Normal line: t, t, z t 8. (a) f (2 2 ) i(2 2 ) j Ê f(2ßß8) 9i7 j Ê Tangent plane: 9( 2) 7( ) (z 8) 0 Ê 9 7 z 2; (b) Normal line: 2 9t, 7t, z 8 t 2 2 9. z f(ß) ln a b Ê f (ß) and f (ß) Ê f (ß0) 2 and f (ß0) 0 Ê from Eq. () the tangent plane at (ß0ß0) is 2( ) z 0 or 2 z 2 0
892 Chapter Partial Derivatives a b a b a b 0. z f(ß) e Ê f (ß) 2e and f (ß) 2e Ê f (0ß0) 0 and f (!ß!) 0 Ê from Eq. () the tangent plane at (0ß0ß) is z 0 or z È Î Î. z f( Bß) Ê f (ß) ( ) and f (ß) ( ) Ê f (ß2) and f (ß) Ê from Eq. () the tangent plane at (ß2ß) is ( ) ( 2) (z ) 0 Ê 2z 0 2. z f( Bß) Ê f (ß) 8 and f (ß) Ê f (ß) 8 and f ( ß) Ê from Eq. () the tangent plane at (ßß5) is 8( ) 2( ) (z 5) 0 or 8 2 z 5 0. f i2j2 Ê f(ßß) i2j2 and g i for all points; v f g i j Ê v 2 2 2j2 Ê Tangent line:, 2t, z 2t â 0 0â. f zizj Ê f(ßß) ij; g 2ij6z Ê g(ßß) 2ij6 ; i j Ê v f g Ê 2ij2 Ê Tangent line: 2t, t, z 2t â2 6â 5. f 2i2j2 Ê f ˆ ßß 2i2j2 and g j for all points; v f g i j Ê v 2 2 2 2i2 Ê Tangent line: 2t,, z 2t â0 0â 6. f i2 j Ê f ˆ ßß i2 j and g j for all points; v f g i j Ê v 2 i Ê Tangent line: t,, z t â0 0â 7. f a 6 bia6 bj2z Ê f(ßß) ij6 ; g 2i2j2z i j Ê g( ßß$) 2i2j6 ; v f g Ê v 6 90i 90 j Ê Tangent line: â 2 2 6 â 90t, 90t, z 8. f 2i2 j Ê f Š È2ßÈ2ß 2È2 i2è2 j; g 2i2 j Ê g Š È2ßÈ2ß i j 2È2 2È i 2 j; v f g Ê v 2È2 2È2 0 2È2i2È2 j Ê Tangent line: â2è2 2È2 â È22È2t, È22È2t, z z 2 z z z 69 69 69 9. f Š iš jš Ê f(ßß2) i j ; v i6j2 6 2 u i j Ê f 9 u and df ( f 9 u) ds ˆ (0.) 0.0008 v È 6 ( 2) 7 7 7 8 8 2 i 2 j 2 v È2 2 ( 2) È i f and df ( f ) ds (0.) 0.0577 È j È u È u È v 20. f ae cos zbiaze sin zbjae sin z b Ê f(0ß0ß0) i; u Ê
Section.6 Tangent Planes and Differentials 89 Ä 2. g ( cos z) i( sin z) j( sin z cos z) Ê g(2ßß0) 2 ij ; A P P 2i2j2 v 2i2j2 v È( 2) 2 2 È È È Ê u i j Ê g u 0 and dg ( g u) ds (0)(0.2) 0 22. h c sin ( ) z dic sin ( ) dj2z Ê h( ßß) ( sin ) i( sin ) j 2 2 ; P Ä v ij i v P ij here P (!ß!ß!) Ê u i j! Ê h u È and dh ( h u) ds È(0.) 0.72 È! v È È È È È È 2. (a) The unit tangent vector at Š ß in the direction of motion is u i j; È È u T (sin 2) i(2 cos 2) j Ê T Š ß Š sin È iš cos È j Ê D T Š ß T u È sin È cos È 0.95 C/ft d dt ` dt ` dt È È dt `T d `T (b) r(t) (sin 2t) i(cos 2t) j Ê v(t) (2 cos 2t) i(2 sin 2t) jand v 2; v v T v Š T v (D T) v, here u ; at Š ß e have u i j from part (a) u dt Ê Š sin È cos È 2 È sin È cos È.87 C/sec dt È v v 2. (a) T ( z) izj Ê T(8ß6ß) 56i2j8 ; r(t) 2t itjt Ê the particle is v at the point P( )ß6ß) hen t 2; v(t) tij2t Ê v(2) 8ij Ê u v 8 76 È i Ê ß ß 89 È j 89 È u u 89 È 89 È89 dt `T d `T d dt 76 dt ` dt ` dt v u v Ê dt v Š È u t2 È89 D T(8 6 ) T [56 8 2 8 ( )] C/m (b) T ( T ) at t 2, D T (2) 89 76 C/sec 25. (a) f(!ß0), f (ß) 2 Ê f (0ß0) 0, f (ß) 2 Ê f (0ß0) 0 Ê L(ß) 0( 0) 0( 0) (b) f(ß), f (ß) 2, f (ß) 2 Ê L(ß) 2( ) 2( ) 2 2 26. (a) f(!ß0), f (ß) 2( 2) Ê f (0ß0), f (ß) 2( 2) Ê f (0ß0) Ê L(ß) ( 0) ( 0) (b) f(ß2) 25, f (ß2) 0, f (ß2) 0 Ê L(ß) 25 0( ) 0( 2) 0 0 5 27. (a) f(0ß0) 5, f (ß) for all (ß), f (ß) for all (ß) Ê L(ß) 5 ( 0) ( 0) 5 (b) f(ß), f (ß), f (ß) Ê L(ß) ( ) ( ) 5 % $ $ 28. (a) f(ß), f (ß) Ê f (ß), f (ß) Ê f (ß) Ê L(ß) ( ) ( ) 6 (b) f(0ß0) 0, f (!ß0) 0, f (0ß0) 0 Ê L(ß) 0 29. (a) f(0ß0), f (ß) e cos Ê f (0ß0), f (ß) e sin Ê f (0ß0) 0 Ê L(ß) ( 0) 0( 0) (b) f ˆ 0ß 0, f ˆ 0ß 0, f ˆ 0ß Ê L(ß) 0 0( 0) ˆ 2 2 0. (a) f(0ß0), f (ß) e Ê f (!ß!), f (ß) 2e Ê f (0ß0) 2 Ê L(ß) (0) 2(0) 2 $ $ $ $ $ $ (b) f(2) ß e, f(2) ß e, f(2) ß 2e Ê L() ß e e() 2e(2) $ $ $ e 2e 2e
89 Chapter Partial Derivatives. f(2ß), f (ß) 2 Ê f (2ß), f (ß) Ê f (2ß) 6 Ê L(ß) ( 2) 6( ) 7 6; f (ß) 2, f (ß) 0, f (ß) Ê M ; thus E(ß) Ÿ ˆ () a 2 b Ÿ ˆ (0. 0.) 0.06 2. f(2ß2), f (ß) Ê f (2ß2) 7, f (ß) Ê f (2ß2) 0 Ê L(ß) 7( 2) 0( 2) 7 ; f (ß), f (ß), f (ß) Ê M ; thus E(ß) Ÿ ˆ () a 2 2 b Ÿ ˆ (0. 0.) 0.02. f(0ß0), f (ß) cos Ê f (0ß0), f (ß) sin Ê f (0ß0) Ê L(ß) ( 0) ( 0) ; f (ß) 0, f (ß) cos, f (ß) sin Ê Q ; thus E( ) ˆ () a b ˆ ß Ÿ Ÿ (0.2 0.2) 0.08. f( ß) 6, f (ß) sin ( ) Ê f (ß2), f (ß) 2 cos ( ) Ê f (ß2) 5 Ê L(ß) 6 ( ) 5( 2) 5 8; f (ß) cos ( ), f (ß) 2, f (ß) 2 sin ( ); Ÿ 0. Ê 0.9 Ÿ Ÿ. and 2 Ÿ 0. Ê.9 Ÿ Ÿ 2.; thus the ma of f (ß) on R is 2., the ma of f (ß) on R is 2.2, and the ma of f (ß) on R is 2(2.) sin (0.9 ) Ÿ. Ê M.; thus E(ß) Ÿ ˆ (.) a 2 b Ÿ (2.5)(0. 0.) 0.086 0 Þ 0 Þ 0 Þ ˆ a b 5. f(!ß0), f (ß) e cos Ê f (0ß0), f (ß) e sin Ê f (0ß0) 0 Ê L(ß) ( 0) 0( 0) ; f (ß) e cos, f (ß) e cos, f (ß) e sin ; Ÿ 0. Ê 0. Ÿ Ÿ 0. and Ÿ 0. Ê 0. Ÿ Ÿ 0.; thus the ma of f (ß) on R is e cos (0.) Ÿ., the ma of f (ß) on R is e cos (0.) Ÿ., and the ma of f (ß) on R is e sin (0.) Ÿ 0.2 Ê M.; thus E(ß) Ÿ (.) Ÿ (0.555)(0. 0.) 0.0222 (0.98) 6. f(ß) 0, f (ß) Ê f (ß), f (ß) Ê f (ß) Ê L(ß) 0 ( ) ( ) 2; f (ß), f (ß), f (ß) 0; Ÿ 0.2 Ê 0.98 Ÿ Ÿ.2 so the ma of f (ß) on R is Ÿ.0; Ÿ 0.2 Ê 0.98 Ÿ Ÿ.2 so the ma of f (ß) on R is (0.98) Ÿ.0 Ê M.0; thus E(ß) Ÿ ˆ (.0) a b Ÿ (0.52)(0.2 0.2) 0.082 7. (a) f( ß ß ), f (ß ß ) z 2, f (ß ß ) z 2, f (ß ß ) 2 Ð ß Ð ß z Ð ß Ê L(ßßz) 2( ) 2( ) 2(z ) 2 2 2z (b) f(00) ß ß 0, f(00) ß ß 0, f(00) ß ß, f(00) ß ß Ê L(z) ß ß 00() (0) (z0) z z (c) f(000) ß ß 0, f(000) ß ß 0, f(000) ß ß 0, f(000) ß ß 0 Ê L(z) ß ß 0 z 8. (a) f(ßß), f (ßß) 2 2, f (ßß) 2 2, f (ßß) 2z 2 ÐßßÑ ÐßßÑ z Ê L(ßßz) 2( ) 2( ) 2(z ) 2 2 2z (b) f(0ßß0), f (0ßß0) 0, f (!ßß0) 2, f (0ßß0) 0 Ê L(ßßz) 0( 0) 2( ) 0(z 0) z 2 (c) f(ß0ß0), f (ß0ß0) 2, f (ß0ß0) 0, f (ß0ß0) 0 Ê L(ßßz) 2( ) 0( 0) 0(z 0) 2 z ÐßßÑ 9. (a) f(ß0ß0), f (ß0ß0) ¹, f (ß0ß0) ¹ 0, È z È z 00 00 Ð ß Ð ß f ( ß 0 ß 0) ¹ 0 Ê L( ß ß z) ( ) 0( 0) 0(z 0) z z z 00 È Ð ß
(b) f(0) ß ß 2, f(0) ß ß, f(0) ß ß, f(0) ß ß 0 Section.6 Tangent Planes and Differentials 895 È È z 2 È2 L( z) È 2 È ( ) È ( ) 0(z 0) È È 2 2 2 2 2 2 2 2 z 2 2 z Ê ß ß (c) f(22) ß ß, f(22) ß ß, f(22) ß ß, f(22) ß ß Ê L(z) ß ß () (2) (z2) 0. (a) f ˆ, f ˆ cos 0, f ˆ cos ß ß ß ß ß ß 0, 2 z ˆ ßß z ˆ ßß f ˆ sin L( z) 0 ˆ ß ß ¹ Ê ß ß 0( ) (z ) 2 z z z ˆ ßß (b) f(2ß0ß) 0, f (2ß0ß) 0, f (2ß0ß) 2, f (2ß0ß) 0 Ê L(ßßz) 0 0( 2) 2( 0) 0(z ) 2 z. (a) f(0ß0ß0) 2, f (0ß0ß0) e, f (0ß0ß0) sin ( z) 0, Ð!ß!ß!Ñ f z(0ß0ß0) sin ( z) Ð!ß!ß!Ñ 0 Ê L(ßßz) 2 ( 0) 0( 0) 0(z 0) 2 (b) f ˆ 0 0, f ˆ 0 0, f ˆ 0 0, f ˆ ß ß ß ß ß ß 0ß ß0 z Ê L(ßßz) (0) ˆ 2 (z0) z (c) f ˆ 0, f ˆ 0, f ˆ ß ß ß ß 0ß ß, f ˆ 0ß ß z Ê L(ßßz) (0) ˆ ˆ z z Ð!ß!ß!Ñ 2. (a) f(ß0ß0) 0, f (ß0ß0) ¹ 0, f (ß0ß0) ¹ 0, z z (z) (z) Ðß!ß!Ñ Ðß!ß!Ñ f(00) ß ß ¹ 0 Ê L(z) ß ß 0 z (z) Ðß!ß!Ñ (b) f(ßß0) 0, f (ßß0) 0, f (ßß0) 0, f (ßß0) Ê L(ßßz) 0 0( ) 0( ) (z 0) z z z z (c) f() ß ß, f() ß ß, f() ß ß, f() ß ß Ê L(z) ß ß () () (z). f(ßßz) z z 2 at P!(ßß2) Ê f(ßß2) 2; f z, f z, fz Ê L(ßßz) 2 2( ) 6( ) 2(z 2) 2 6 2z 6; f 0, f 0, fzz 0, f 0, fz Ê M ; thus, E(ßßz) Ÿ ˆ ()(0.0 0.0 0.02) 0.002! z zz z ˆ. f(ßßz) z z at P (ßß2) Ê f(ßß2) 5; f 2, f z, f z Ê L(ßßz) 5 ( ) ( ) 2(z 2) 2z 5; f 2, f 0, f, f, f 0, fz Ê M 2; thus E(ßßz) Ÿ (2)(0.0 0.0 0.08) 0.0 5. f(ßßz) 2z z at P!(ßß0) Ê f(ßß0) ; f z, f 2z, fz 2 Ê L(ßßz) ( ) ( ) (z 0) z ; f 0, f 0, fzz 0, f, fz, fz 2 Ê M ; thus E(ßßz) Ÿ ˆ ()(0.0 0.0 0.0) 0.005 6. f(ßßz) È2 cos sin ( z) at P ˆ 0ß0 ß Ê f ˆ 0ß0ß ; f È! 2 sin sin ( z), f È2 cos cos ( z), fz È2 cos cos ( z) Ê L(ßßz) 0( 0) ( 0) ˆ z z ; f È2 cos sin ( z), f È2 cos sin ( z), f È zz 2 cos sin ( z), f È2 sin cos ( z), fz È2 sin cos ( z), fz È2 cos sin ( z). The absolute value of each of these second partial derivatives is bounded above b È2 Ê M È2; thus E(ßßz) Ÿ ˆ Š È 2 (0.0 0.0 0.0) 0.00066.
896 Chapter Partial Derivatives 7. T (ß) e e and T (ß) ae e b Ê dt T (ß) d T (ß) d ae e bd ae e b d Ê dt 2.5 d.0 d. If d Ÿ 0. and d Ÿ 0.02, then the Ðßln 2Ñ maimum possible error in the computed value of T is (2.5)(0.) (.0)(0.02) 0. in magnitude. dv 2 rh dr r dh 2 dr 8. Vr 2rh and Vh r Ê dv V r dr V h dh Ê V r h r dr h dh; no r 00 Ÿ and dh 00 dv 00 ˆ dr 2 (00) ˆ dh (00) 2 dr 00 dh Ÿ Ê Ÿ Ÿ 00 Ÿ 2() Ê % h V r h r h 9. Vr 2rh and Vh r Ê dv V r dr V h dh Ê dv 2rh dr r dh Ê dv Ð 52 20 dr 25 dh; $ $ dr Ÿ 0. cm and dh Ÿ 0. cm Ê dv Ÿ (20)(0.) (25 )(0.).5 cm ; V(5ß2) 00 cm.5 00 Ê maimum percentage error is 00.8% R R R R R R R R R R 50. (a) Ê dr dr dr Ê dr Š dr Š dr R R Ð00 00Ñ (00) (00) ß (b) dr R Š dr Š dr Ê dr R dr dr Ê R ill be more sensitive to a variation in R since R R (00) (00) R R (c) From part (a), dr Š dr Š dr so that R changing from 20 to 20. ohms Ê dr 0. ohm and R changing from 25 to 2.9 ohms Ê dr 0. ohms; Ê R ohms 00 R R R 9 ˆ 00 ˆ 00 Ð20ß25Ñ (20) (25) R Ð20ß25Ñ 0.0 ˆ 00 9 9 9 dr Ê dr (0.) ( 0.) 0.0 ohms Ê percentage change is 00 00 0.% 5. A Ê da d d; if then a -unit change in gives a greater change in da than a -unit change in. Thus, pa more attention to hich is the smaller of the to dimensions. 52. (a) f (ß) 2( ) Ê f (ß0) 2 and f (ß) Ê f (ß0) Ê df 2 d d Ê df is more sensitive to changes in (b) df 0 Ê 2 d d 0 Ê d 2 0 Ê d d 5. (a) r Ê 2r dr 2 d 2 d Ê dr d d Ê dr ˆ a 0.0b ˆ a 0.0b d r r Ð$ß%Ñ 5 5 Š Š 0.07 dr 0.0 5 0.0 r 00 5 00 0.28%; d ) d d ˆ ˆ 0.0 0.0 d d d ) Ð$ß%Ñ ˆ 25 a 0.0b ˆ 25 a 0.0b 25 5 Ê Ê Ê maimum change in d ) occurs hen d and d have opposite signs (d 0.0 and d 0.0 or vice 0.07 versa) Ê d) 0.0028; ) tan ˆ 0.92725528 Ê d) 0.0028 00 00 5 ) 0.92725528 0.0% (b) the radius r is more sensitive to changes in, and the angle ) is more sensitive to changes in 5. (a) V r h Ê dv 2rh dr r dh Ê at r and h 5 e have dv 0 dr dh Ê the volume is about 0 times more sensitive to a change in r (b) dv 0 Ê 0 2rh dr r dh 2h dr r dh 0 dr dh Ê dr 0 dh; choose dh.5 Ê dr 0.5 Ê h 6.5 in. and r 0.85 in. is one solution for? V dv 0 55. f(a ß b ß c ß d) a b º ad bc f d, f c, f b, f a df d da c db b dc a dd; since c d º Ê Ê a b c d a is much greater than b, c, and d, the function f is most sensitive to a change in d.
z 56. u e, u e sin z, u cos z Ê du e d ae sin z b d ( cos z) dz Section.7 Etreme Values and Saddle Points 897 Ê du d 7 d 0 dz d 7 d Ê magnitude of the maimum possible error ˆ 2ln ß ß 2 Ÿ (0.2) 7(0.6).8 Î Î Î K h h M h h h h h 2KM Î 2M 2KM Î 2K 2KM Î 2KM h h h h h h Î ˆ 2KM 2M 2K 2KM dk dm dh h h h h dq Ð22000.05 ß ß Þ Ñ Î (2)(2)(20) (2)(20) (2)(2) (2)(2)(20) 0.05 0.05 0.05 (0.05) 57. Q ˆ 2KM ˆ 2M, Q ˆ 2KM ˆ 2K, and Q ˆ 2KM ˆ 2KM Ê dq ˆ ˆ dk ˆ ˆ dm ˆ ˆ dh Ê dk dm dh (0.025)(800 dk 80 dm 2,000 dh) Ê Q is most sensitive to changes in h 58. A ab sin C Ê Aa b sin C, Ab a sin C, Ac ab cos C Ê da ˆ b sin C da ˆ a sin C db ˆ ab cos C dc; dc 2 0.09 radians, da 0.5 ft, db 0.5 ft; at a 50 ft, b 200 ft, and C 60, e see that the change is approimatel da (200)(sin 60 ) 0.5 (50)(sin 60 ) 0.5 (200)(50)(cos 60 ) 0.09 8 ft 59. z f(ß) Ê g(ßßz) f(ß) z 0 Ê g (ßßz) f (ß), g (ßßz) f (ß) and g z(ßßz) Ê g(ßßf(ß)) f(ß), g(ßßf(ß)) f(ß) and g(ßßf(ß)) Ê the tangent!!!!!!!!!!!! z!!!! plane at the point P! is f (! ß!)(!) f (! ß!)(!) [z f(! ß!)] 0 or z f ( ß )( ) f ( ß )( ) f( ß )!!!!!!!! v 60. f 2i2j 2(cos t t sin t) i2(sin t t cos t) j and v (t cos t) i(t sin t) j Ê u (t cos t) i (t sin t) j (t cos t) (t sin t) (cos t) i(sin t) jsince t 0 Ê (D f) f u È! 2(cos t t sin t)(cos t) 2(sin t t cos t)(sin t) 2 v 6. f 2i2j2z (2 cos t) i(2 sin t) j2t and v ( sin t) i(cos t) j Ê u ( sin t) i(cos t) j sin t cos t È(sin t) (cos t) È2 È2 È2 Š iš j Ê (D f) f u! sin t cos t 2t (2 cos t) Š (2 sin t) Š (2t) Š Ê (D f) ˆ, (D f)(0) 0 and (Duf) ˆ È2 È2 È2 È2 u 2È2 2 È 2 62. r Èt È Î Î i t j (t ) Ê v t i t j ; t Ê,, z Ê P (ßß) and v() i j ; f(ßßz) z 0 Ê f 2i2j Ê f(ßß) 2i2 j; therefore v ( f) Ê the curve is normal to the surface 6. r Èt È Î Î i t j(2t ) Ê v t i t j2 ; t Ê,, z Ê P (ßß) and v() i j2 ; f(z) ß ß z 0 Ê f 2i2 j Ê f() ß ß 2i2 j; no vab faßßb 0, thus the curve is tangent to the surface hen t.7 EXTREME VALUES AND SADDLE POINTS. f (ß) 2 0 and f (ß) 2 0 Ê and Ê critical point is ( ß); f ( ß) 2, f ( ß) 2, f ( ß) Ê f f f 0 and f 0 Ê local minimum of f( ß) 5 u u P P u v v!!
898 Chapter Partial Derivatives 2. f (ß) 2 6 0 and f (ß) 6 0 Ê 5 and 8 Ê critical point is (5ß8); f (5ß8) 2, f (5ß8) 6, f (5ß8) Ê f f f 0 and f 0 Ê local minimum of f(5ß8) 6 2 2. f (ß) 2 0 0 and f (ß) 2 0 Ê and Ê critical point is ˆ ß ; f ˆ 2 2 2 ß 0, f ˆ ß, f ˆ ß 2 Ê ff f 6 0 and f 0 Ê local maimum of fˆ 2 ß 0 2 2. f (ß) 2 0 0 and f (ß) 2 0 Ê 9 and 9 Ê critical point is ˆ 9ß 9 ; f ˆ 2 2 2 9ß 9 0, f ˆ 9ß 9, f ˆ 9ß 9 2 Ê ff f 6 0 and f 0 Ê local maimum of fˆ ß 2 28 9 9 9 5. f (ß) 2 0 and f (ß) 2 0 Ê 2 and Ê critical point is ( 2ß); f ( 2ß) 2, f ( 2ß) 0, f ( 2ß) Ê f f f 0 Ê saddle point 6. f (ß) 2 0 and f (ß) 2 2 0 Ê 2 and 2 Ê critical point is ( 2ß2); f ( 2ß2) 0, f ( 2ß2) 2, f ( 2ß2) Ê f f f 0 Ê saddle point 6 69 6 69 7. f (ß) 5 0 and f (ß) 5 6 0 Ê 5 and 5 Ê critical point is ˆ 5ß 25 ; f ˆ 6 69, f ˆ 6 69 0, f ˆ 6 69 ß ß ß 5 Ê f f f 25 0 Ê saddle point 5 25 5 25 5 25 8. f (ß) 2 2 0 and f (ß) 2 0 Ê and Ê critical point is ˆ ß 2 ; f ˆ 2, f ˆ, f ˆ ß ß ß 2 Ê f f f 0 and f 0 Ê local maimum of 2 2 2 ˆ ß 7 f 9. f (ß) 2 0 and f (ß) 2 6 0 Ê 2 and Ê critical point is (2ß); f (2ß) 2, f (2ß) 2, f (2ß) Ê f f f 2 0 Ê saddle point 0. f (ß) 6 6 2 0 and f (ß) 6 0 Ê and Ê critical point is ˆ 2ß ; f ˆ 6, f ˆ, f ˆ ß ß ß 6 Ê ff f 8 0 and f 0 Ê local minimum of fˆ 2ß. f (ß) 5 0 and f (ß) 8 2 0 Ê 2 and Ê critical point is (2ß); f (2ß), f (2ß) 8, f (2ß) Ê f f f 2 0 and f 0 Ê local minimum of f(2ß) 6 2. f (ß) 8 6 20 0 and f (ß) 6 0 26 0 Ê and 2 Ê critical point is (ß2); f (ß2) 8, f (ß2) 0, f (ß2) 6 Ê f f f 0 and f 0 Ê local minimum of f(ß2) 6. f (ß) 2 2 0 and f (ß) 2 0 Ê and 2 Ê critical point is (ß2); f (ß2) 2, f (ß2) 2, f (ß2) 0 Ê f f f 0 Ê saddle point. f (ß) 2 2 2 0 and f (ß) 2 2 0 Ê and 0 Ê critical point is (ß0); f (ß0) 2, f (ß0), f (ß0) 2 Ê f f f 0 and f 0 Ê local minimum of f(ß0) 0
Section.7 Etreme Values and Saddle Points 899 5. f (ß) 2 2 0 and f (ß) 2 0 Ê 0 and 0 Ê critical point is (0ß0); f (0ß0) 2, f (0ß0) 0, f (0ß0) 2 Ê f f f 0 Ê saddle point 6. f (ß) 2 2 0 and f (ß) 2 2 2 0 Ê 0 and Ê critical point is (0ß); f (0ß), f (0ß) 2, f (0ß) 2 Ê f f f 0 and f 0 Ê local maimum of f(0ß) 2 2 7. f (ß) 2 0 and f (ß) 2 0 Ê 0 and 0, or and Ê critical points are (0ß0) and ˆ 2 2 ß ; for (0ß0): f (0ß0) 6 0, f (0ß0) 6 0, f (0ß0) 2 Ð00 Ð00 2 2 2 2 2 2 2 2 2 2 70 ˆ 27 Ê f f f 0 Ê saddle point; for ˆ ß : f ˆ ß, f ˆ ß, f ˆ ß 2 Ê f f f 2 0 and f 0 Ê local maimum of f ß 8. f (ß) 0 and f (ß) 0 Ê 0 and 0, or and Ê critical points Ð00 Ð00 are (0ß 0) and ( ß ); for (!ß!): f (0ß 0) 6 0, f (0ß 0) 6 0, f (0ß 0) Ê f f f 9 0 Ê saddle point; for ( ß): f ( ß) 6, f ( ß) 6, f ( ß) Ê ff f 27 0 and f 0 Ê local maimum of f( ß) 9. f (ß) 2 6 6 0 and f (ß) 6 6 0 Ê 0 and 0, or and Ê critical Ð00 points are (0ß0) and (ß ); for (!ß!): f (0ß0) 2 2 2, f (0ß0) 6, f (0ß0) 6 Ê f f f 6 0 and f 0 Ê local minimum of f(0ß0) 0; for (ß): f (ß) 0, f (ß) 6, ß Ê Ê f ( ) 6 f f f 6 0 saddle point 20. f (ß) 6 6 0 Ê ; f (ß) 6 6 6 0 Ê 2 6 0 Ê 6(2 ) 0 Ê 0 or 2 Ê (0ß0) and (2ß2) are the critical points; f (ß) 6, f (ß) 6 2, f (ß) 6; for (0ß0): ß ß Ê Ê ß f (0ß0) 6, f (0ß0) 6, f (0ß0) 6 Ê f f f 72 0 Ê saddle point; for (2ß2): f (2ß2) 6, f (2 2) 8, f (2 2) 6 f f f 72 0 and f 0 local maimum of f(2 2) 8 2. f (ß) 27 0 and f (ß) 0 Ê 0 and 0, or 9 and Ê critical points are (0ß0) and ˆ ß ; for (!ß!): f (0ß0) 5 0, f (0ß0) 2 0, f (0ß0) Ê f f f 9 Ð00 Ð00 ˆ ˆ ˆ 8 ˆ 9 9 9 9 ˆ 6 9 8 6 0 Ê saddle point; for ß : f ß 2, f ß, f ß Ê f f f 8 0 and f 0 Ê local minimum of f ß % 22. f (ß) 2 6 0 Ê ; f (ß) 6 0 Ê a b 6 0 Ê 6 2 0 $ Ê 2 a8 b 0 Ê 0 or Ê (!ß 0) and ˆ ß are the critical points; f (ß ) 8, Ê saddle point; for ˆ ß : f ˆ ß 2, f ˆ ß 6, f ˆ ß 2 2 2 2 6 Ê ff f 08 0 and f 0 Ê local maimum of f ˆ 2ß f (ß) 6, and f (ß) 6; for (0ß0): f (0ß0) 0, f (0ß0) 0, f (0ß0) 6 Ê f f f 6 0 2. f (ß) 6 0 Ê 0 or 2; f (ß) 6 0 Ê 0 or 2 Ê the critical points are (0ß0), (0ß2), ( 2ß0), and ( 2ß 2); for (!ß!): f (0ß0) 6 6 6, f (0ß0) 6 6 6, Ð00 Ð00 ß ß ß ß Ê Ê f (0ß0) 0 Ê f f f 6 0 Ê saddle point; for (0ß2): f (0ß2) 6, f (0ß2) 6, f (0ß2) 0 Ê f f f 6 0 and f 0 Ê local minimum of f(0ß2) 2; for ( 2ß0): f ( 2ß0) 6, f ( 2ß0) 6, f ( 2ß0) 0 Ê f f f 6 0 and f 0 Ê local maimum of f( 2ß0) ; for ( 2 2): f ( 2 2) 6, f ( 2 2) 6, f ( 2 2) 0 f f f 6 0 saddle point
900 Chapter Partial Derivatives 2. f (ß) 6 8 0 Ê 6( ) 0 Ê 0 or ; f (ß) 6 6 2 0 Ê 6( 2)( ) 0 Ê 2 or Ê the critical points are (0ß2), (0ß), (ß2), and (ß); f (ß) 2 8, f (ß) 2 6, and f (ß) 0; for (!ß2): f (0ß2) 8, f (0ß2) 8, f (0ß2) 0 Ê f f f 2 0 and f 0 Ê local maimum of f(0ß2) 20; for (0ß): f (0ß) 8, f (0ß) 8, f (0ß) 0 Ê f f f 2 0 Ê saddle point; for (ß2): f (ß2) 8, f (ß2) 8, f (ß2) 0 Ê f f f 2 0 Ê saddle point; for (ß): f (ß) 8, f (ß) 8, f (ß) 0 Ê f f f 2 0 and f 0 Ê local minimum of f(ß) $ $ 25. f (ß) 0 and f (ß) 0 Ê Ê a b 0 Ê 0,, Ê the critical points are (0ß0), (ß), and ( ß ); for (!ß!): f (0ß0) 2 0, f (0ß0) 2 0, Ð00 Ð00 Ê Ê ß ß ß f (0ß0) Ê f f f 6 0 Ê saddle point; for (ß): f (ß) 2, f (ß) 2, f (ß) f f f 28 0 and f 0 local maimum of f( ) 2; for ( ): f ( ) 2, f ( ß) 2, f ( ß) Ê f f f 28 0 and f 0 Ê local maimum of f( ß) 2 $ $ $ 26. f (ß) 0 and f (ß) 0 Ê Ê 0 Ê a b 0 Ê 0,, Ê the critical points are (0ß0), (ß), and ( ß); f (ß) 2, f (ß) 2, and f (ß) ; ß ß ß Ê Ê for (!ß0): f (0ß0) 0, f (0ß0) 0, f (0ß0) Ê f f f 6 0 Ê saddle point; for (ß): f ( ) 2, f ( ) 2, f ( ) f f f 28 0 and f 0 local minimum of f( ß) 2; for ( ß): f ( ß) 2, f ( ß) 2, f ( ß) Ê f f f 28 0 and f 0 Ê local minimum of f( ß) 2 27. f (ß) 0 and f (ß) 0 Ê 0 and 0 Ê the critical point is (!ß0); 2 2 a b a b 2 2 2 2 8 $ $ $ a b a b a b Ê ff f 0 and f 0 Ê local maimum of f(0ß0) f, f, f ; f (!ß!) 2, f (0ß0) 2, f (0ß0) 0 2 2 $ $ 28. f ( ß ) 0 and f ( ß ) 0 Ê and Ê the critical point is ( ß ); f, f, f ; f (ß) 2, f (ß) 2, f (ß) Ê f f f 0 and f 2 Ê local minimum of f(ß) 29. f (ß) cos 0 and f (ß) sin 0 Ê n, n an integer, and 0 Ê the critical points are (nß 0), n an integer (Note: cos and sin cannot both be 0 for the same, so sin must be 0 and 0); f sin, f 0, f cos ; f (nß0) 0, f (nß0) 0, f (nß0) if n is even and f (nß0) if n is odd Ê f f f 0 Ê saddle point. 2 2 2 0. f (ß) 2e cos 0 and f (ß) e sin 0 Ê no solution since e Á 0 for an and the functions cos and sin cannot equal 0 for the same Ê no critical points Ê no etrema and no saddle points. (i) On OA, f(ß) f(0ß) on 0 Ÿ Ÿ 2; f (0ß) 2 0 Ê 2; f(0ß 0) and f(!ß ) (ii) On AB, f(ß) f(ß2) 2 on 0 Ÿ Ÿ ; f (ß2) 0 Ê ; f(0ß2) and f( ß ) 5 (iii) On OB, f(ß) f(ß2) 6 2 on 0 Ÿ Ÿ ; endpoint values have been found above; f (ß2) 2 2 0 Ê and 2, but (ß) is not an interior point of OB
Section.7 Etreme Values and Saddle Points 90 (iv) For interior points of the triangular region, f (ß) 0 and f (ß) 2 0 Ê and 2, but (ß2) is not an interior point of the region. Therefore, the absolute maimum is at (0ß 0) and the absolute minimum is 5 at (ß ). 2. (i) On OA, D(ß) D(0ß) on 0 Ÿ Ÿ ; D(0ß) 2 0 Ê 0; D(!ß!) and D(!ß %) 7 (ii) On AB, D(ß) D(ß) 7 on 0 Ÿ Ÿ ; D (ß) 2 0 Ê 2 and (2ß) is an interior point of AB; D( ß %) and D( %ß %) D(!ß %) 7 (iii) On OB, D(ß) D(ß) on 0 Ÿ Ÿ ; D (ß) 2 0 Ê 0 and 0, hich is not an interior point of OB; endpoint values have been found above (iv) For interior points of the triangular region, f (ß) 2 0 and f (ß) 2 0 Ê 0 and 0, hich is not an interior point of the region. Therefore, the absolute maimum is 7 at (!ß %) and (%ß %), and the absolute minimum is at (0ß 0).. (i) On OA, f(ß ) f(!ß ) on 0 Ÿ Ÿ 2; f (0ß) 2 0 Ê 0 and 0; f(0ß0) 0 and f(0 ß ) (ii) On OB, f(ß) f(ß0) on 0 Ÿ Ÿ ; f (ß0) 2 0 Ê 0 and 0; f(0ß0) 0 and f(ß0) (iii) On AB, f(ß) f(ß2 2) 5 8 on 0 Ÿ Ÿ ; f (ß22) 08 0 Ê 5 2 and ; f ˆ 2 ß 5 5 5 5; endpoint values have been found above. (iv) For interior points of the triangular region, f (ß) 2 0 and f (ß) 2 0 Ê 0 and 0, but (!ß0) is not an interior point of the region. Therefore the absolute maimum is at (0ß 2) and the absolute minimum is 0 at (0ß 0).. (i) On AB, T(ß) T(!ß) on Ÿ Ÿ ; T(0ß) 2 0 Ê 0 and 0; T(0ß0) 0, T(!ß ) 9, and T(!ß ) 9 (ii) On BC, T(ß) T(ß) 9 on 0 Ÿ Ÿ 5; T (ß) 2 0 Ê and ; T ˆ 27 ß and T(5 ß ) 9 (iii) On CD, T(ß) T(5ß) 5 5 on 5 Ÿ Ÿ ;T (5ß) 2 5 0 Ê and 5;T ˆ 5 5 ß 5, T( &ß) and T(5ß) 9 (iv) On AD, T(ß) T(ß) 9 9 on 0 Ÿ Ÿ 5; T (ß) 2 9 0 Ê and ; T ˆ 9 5 ß, T( ) 9 and T( ) (v) For interior points of the rectangular region, T (ß) 2 6 0 and T (ß) 2 0 Ê and 2 Ê (ß2) is an interior critical point ith T(ß2) 2. Therefore the absolute maimum is 9 at (5ß) and the absolute minimum is 2 at (ß2). 9
902 Chapter Partial Derivatives 5. (i) On OC, T(ß) T(ß0) 6 2 on 0 Ÿ Ÿ 5; T(ß0) 26 0 Ê and 0; T(ß0) 7, T(0ß0) 2, and T(5ß0) (ii) On CB, T(ß) T(5ß) 5 on 5 Ÿ Ÿ 0; T (5ß) 2 5 0 Ê and 5; T ˆ 5 5 ß 7 and T(5ß) 9 (iii) On AB, T(ß) T(ß) 9 on 9 0 Ÿ Ÿ 5; T(ß) 29 0 Ê and ; T ˆ 9ß 7 and T(!ß) (iv) On AO, T(ß) T(!ß) 2 on Ÿ Ÿ 0; T (0ß) 2 0 Ê 0 and 0, but (0ß0) is not an interior point of AO (v) For interior points of the rectangular region, T (ß) 2 6 0 and T (ß) 2 0 Ê and 2, an interior critical point ith T( %ß 2) 0. Therefore the absolute maimum is at (!ß ) and the absolute minimum is 0 at (ß 2). 6. (i) On OA, f(ß) f(!ß) 2 on 0 Ÿ Ÿ ; f (0ß) 8 0 Ê 0 and 0, but (0ß0) is not an interior point of OA; f(!ß 0) 0 and f(!ß ) 2 (ii) On AB, f(ß) f(ß) 8 2 $ 2 on 0 Ÿ Ÿ; f (ß) 8 96 0 Ê and È2 È2, or and, but Š ß is not in È 2 the interior of AB; f Š 6È È 2 ß 2 2 and f(ß) 8 (iii) On BC, f(ß) f( ß) 8 2 2 on 0 Ÿ Ÿ ; f (ß) 8 8 0 Ê and, but (ß ) is not an interior point of BC; f( ß 0) 2 and f( ß ) 8 $ (iv) On OC, f(ß) f(ß0) 2 on 0 Ÿ Ÿ ; f (ß0) 96 0 Ê 0 and 0, but (0ß0) is not an interior point of OC; f(!ß 0) 0 and f( ß 0) 2 (v) For interior points of the rectangular region, f (ß) 8 96 0 and f (ß) 8 8 0 Ê 0 and 0, or and, but (0ß0) is not an interior point of the region; f ˆ ß 2. Therefore the absolute maimum is 2 at ˆ ß and the absolute minimum is 2 at (ß0). 7. (i) On AB, f(ß) f(ß) cos on Ÿ Ÿ ; f (ß) sin 0 Ê 0 and ; f( ß0), f ˆ ß È2, and f ˆ ß È2 (ii) On CD, f(ß) f( $ß) cos on Ÿ Ÿ ; f (ß) sin 0 Ê 0 and ; f(ß0), f ˆ ß È2 and f ˆ ß È2 ˆ È 2 a b (iii) On BC, f(ß) f ß on Ÿ Ÿ ; f ˆ ß È2(2 ) 0 Ê 2 and ; f ˆ 2ß 2È2, f ˆ ß È2, and fˆ ß È2 (iv) On AD, f(ß) f ˆ È ß 2 on Ÿ Ÿ; f ˆ a b ß È 2(2 ) 0 Ê 2 and ; È È2 È2 ß ß a b Ê f ˆ 2ß 2 2, f ˆ ß, and f ˆ ß (v) For interior points of the region, f ( ) ( 2) cos 0 and f ( ) sin 0 2 and 0, hich is an interior critical point ith f(2ß0). Therefore the absolute maimum is at
È 2 (2ß0) and the absolute minimum is at ˆ ß, ˆ ß, ˆ ß, and ˆ ß. 8. (i) On OA, f(ß) f(!ß) 2 on 0 Ÿ Ÿ ; f (0ß) 2 Ê no interior critical points; f(0ß0) and f(0ß) (ii) On OB, f(ß) f(ß0) on 0 Ÿ Ÿ ; f (ß0) Ê no interior critical points; f(ß0) 5 (iii) On AB, f(ß) f(ß ) 8 6 on 0 Ÿ Ÿ ; f (ß) 66 0 Ê 8 5 and ; f ˆ 5 ß 5, f(0ß), and f( ß0) 5 8 8 8 8 Section.7 Etreme Values and Saddle Points 90 (iv) For interior points of the triangular region, f (ß) 8 0 and f (ß) 8 2 0 Ê and hich is an interior critical point ith f ˆ ß 2. Therefore the absolute maimum is 5 at (ß 0) and the absolute minimum is at (!ß0). ' b a 9. Let F(aßb) 6 d here a Ÿ b. The boundar of the domain of F is the line a b in the a b ab-plane, and F(aßa) 0, so F is identicall 0 on the boundar of its domain. For interior critical points e `F `F have: `a a6aa b 0 Ê a, 2 and `b a6 b b b 0 Ê b, 2. Since a Ÿb, there is onl 2 one interior critical point ( ß2) and F( ß2) a 6 b d gives the area under the parabola 6 that is above the -ais. Therefore, a and b 2. ' ' c 0. Let F(aßb) 2 2 b d here a Ÿ b. The boundar of the domain of F is the line a b and a b a Î$ on this line F is identicall 0. For interior critical points e have: a 2 2a a b 0 Ê a, 6 `F and 2 2b b Î$ `b a b 0 Ê b, 6. Since a Ÿ b, there is onl one critical point ( 6ß) and F( 6ß) a 2 2 b d gives the area under the curve a 2 2 b Î$ that is above the -ais. ' c6 Therefore, a 6 and b.. T ( ß ) 2 0 and T ( ß ) 0 Ê and 0 ith T ˆ ß 0 ; on the boundar Š ß È 9 Š ß È 9 ß ß Ê Š ß È È : T(ß) 2 for Ÿ Ÿ Ê T (ß) 2 0 Ê and È ; T, T, T( 0) 2, and T( 0) 0 the hottest is 2 at and Š ; the coldest is at ˆ ß ß0. 2 2 2. f (ß) 2 0 and f (ß) 0 Ê and 2; f ˆ ß2 8, ˆ ß2 2 f ˆ 2 ¹, f ˆ 2 f f f 0 and f 0 a local minimum of f ˆ ß ß Ê Ê ß2 ˆ ß2 2 2 ln 2 ln 2. (a) f (ß) 2 0 and f (ß) 2 0 Ê 0 and 0; f (0ß0) 2, f (0ß0) 2, f (0ß0) Ê f f f 2 0 Ê saddle point at (0ß0) (b) f (ß) 2 2 0 and f (ß) 2 0 Ê and 2; f (ß2) 2, f (ß2) 2, f (ß 2) 0 Ê f f f 0 and f 0 Ê local minimum at (ß ) Ð ß 2 Ñ (c) f (ß) 9 9 0 and f (ß) 2 0 Ê and 2; f (ß2) 8 8, f (ß2) 2, f (ß2) 0 Ê f f f 6 0 and f 0 Ê local minimum at (ß); f ( ß 2) 8, f ( ß 2) 2, f ( ß 2) 0 Ê f f f 6 0 Ê saddle point at ( ß 2) `F `a Î$
90 Chapter Partial Derivatives. (a) Minimum at (0ß0) since f(ß) 0 for all other (ß) (b) Maimum of at (!ß!) since f(ß ) for all other (ß ) (c) Neither since f(ß) 0 for 0 and f(ß) 0 for 0 (d) Neither since f(ß) 0 for 0 and f(ß) 0 for 0 (e) Neither since f(ß) 0 for 0 and 0, but f(ß) 0 for 0 and 0 (f) Minimum at (0ß0) since f(ß) 0 for all other (ß) 5. If 0, then f(ß) Ê f (ß) 2 0 and f (ß) 2 0 Ê 0 and 0 Ê (!ß0) is the onl 2 critical point. If Á 0, f (ß) 2 0 Ê ; f (ß) 2 0 Ê 2 ˆ 2 0 Ê 0 Ê ˆ 0 Ê 0 or 2 Ê ˆ 2 (0) 0 or ; in an case (0ß0) is a critical point. 6. (See Eercise 5 above): f (ß) 2, f (ß) 2, and f (ß) Ê f f f ; f ill have a saddle point at (0ß0) if 0 Ê 2 or 2; f ill have a local minimum at (0ß0) if 0 Ê 2 2; the test is inconclusive if 0 Ê 2. 7. No; for eample f(ß) has a saddle point at (aßb) (0ß0) here f f 0. 8. If f (aßb) and f (aßb) differ in sign, then f (aßb) f (aßb) 0 so f f f 0. The surface must therefore have a saddle point at (aß b) b the second derivative test. 9. We ant the point on z 0 here the tangent plane is parallel to the plane 2 z 0. To find a normal vector to z 0 let z 0. Then 2i2 j is normal to z 0 at (ß). The vector is parallel to i2j hich is normal to the plane 2 z 0 if 6i6j i2j or and. Thus the point is ˆ ß ß0 or ˆ 55 ß ß. 6 6 6 9 6 6 50. We ant the point on z 0 here the tangent plane is parallel to the plane 2 z 0. Let z 0, then 2i2 j is normal to z 0 at (ß). The vector is parallel to i2 j hich is normal to the plane if and. Thus the point ˆ ßß 0 or ˆ 5 ß ß is the point on the surface z 0 nearest the plane 2 z 0. 5. No, because the domain 0 and 0 is unbounded since and can be as large as e please. Absolute etrema are guaranteed for continuous functions defined over closed and bounded domains in the plane. Since the domain is unbounded, the continuous function f(ß) need not have an absolute maimum (although, in this case, it does have an absolute minimum value of f(0ß0) 0). 52. (a) (i) On 0, f(ß) f(0ß) for 0 Ÿ Ÿ ; f (0ß) 2 0 Ê and 0; f ˆ 0 ß, f(0ß0), and f(0ß) (ii) On, f(ß) f(ß) for 0 Ÿ Ÿ ; f (ß) 2 0 Ê and, but ˆ ß is outside the domain; f(0ß) and f( ß) (iii) On, f(ß) f( ß) for 0 Ÿ Ÿ ; f (ß) 2 0 Ê and, but ˆ ß is outside the domain; f(ß0) and f( ß) (iv) On 0, f(ß) f(ß0) for 0 Ÿ Ÿ ; f (ß0) 2 0 Ê and 0; f ˆ 0 ß ; f(0 ß 0), and f( ß 0) (v) On the interior of the square, f (ß) 2 2 0 and f (ß) 2 2 0 Ê 2 2 Ê (). Then f(ß) 2 () () is the absolute minimum value hen 2 2.
Section.7 Etreme Values and Saddle Points 905 (b) The absolute maimum is f( ß ). df d d d d dt ` dt ` dt dt dt 5. (a) 2 sin t 2 cos t 0 Ê cos t sin t Ê (i) On the semicircle, 0, e have t and È 2 Ê f Š È2ßÈ2 2È2. At the endpoints, f( 2ß0) 2 and f( ß!) 2. Therefore the absolute minimum is f( 2ß0) 2 hen t ; the absolute maimum is f Š È2ßÈ2 2È2 hen t. (ii) On the quartercircle, 0 and 0, the endpoints give f(!ß2) 2 and f( ß0) 2. Therefore the absolute minimum is f(2ß0) 2 and f(!ß2) 2 hen t 0, respectivel; the absolute maimum is f Š È2ßÈ2 2È2 hen t. dg `g d `g d d d dt ` dt ` dt dt dt (b) sin t cos t 0 Ê cos t sin t Ê. (i) On the semicircle, 0, e obtain È 2 at t and È2, È 2 at t. Then g Š È2ß È2 2 and g Š È2ß È2 2. At the endpoints, g( 2ß 0) g( ß 0) 0. Therefore the absolute minimum is g Š È2ßÈ2 2 hen t ; the absolute maimum is gš È2 ß È2 2 hen t. (ii) On the quartercircle, 0 and 0, the endpoints give g(!ß2) 0 and g( ß0) 0. Therefore the absolute minimum is g(2ß0) 0 and g(!ß2) 0 hen t 0, respectivel; the absolute maimum is g Š È2 ß È2 2 hen t. dh `h d `h d d d dt ` dt ` dt dt dt (c) 2 (8 cos t)( 2 sin t) ( sin t)(2 cos t) 8 cos t sin t 0 Ê t 0,, ielding the points (2ß0), (0ß2) for 0 Ÿ t Ÿ. (i) On the semicircle, 0 e have h(2ß0) 8, h(0ß2), and h( 2ß0) 8. Therefore, the absolute minimum is h(!ß 2) hen t ; the absolute maimum is h(2ß 0) 8 and h( 2ß 0) 8 hen t 0, respectivel. (ii) On the quartercircle, 0 and 0 the absolute minimum is h(0ß2) hen t ; the absolute maimum is h(2ß0) 8 hen t 0. df d d d d dt ` dt ` dt dt dt È2 È2 9 5. (a) 2 6 sin t 6 cos t 0 Ê sin t cos t Ê t for 0 Ÿ t Ÿ. (i) On the semi-ellipse,, 0, f(ß) 2 6 cos t 6 sin t 6 Š 6 Š 6È2 at t. At the endpoints, f( ß0) 6 and f(ß0) 6. The absolute minimum is f( ß0) 6 hen È2 t ; the absolute maimum is f Š ßÈ2 6È 2 hen t. (ii) On the quarter ellipse, at the endpoints f(0ß2) 6 and f(ß0) 6. The absolute minimum is f(ß0) 6 È dg `g d `g d d d dt ` dt ` dt dt dt Ê t, for 0 Ÿ t Ÿ. È È È È 2 and f(0ß2) 6 hen t 0, respectivel; the absolute maimum is f Š ßÈ2 6È2 hen t. (b) (2 sin t)( sin t) ( cos t)(2 cos t) 6 acos t sin tb 6 cos 2t 0 2 (i) On the semi-ellipse, g(ß) 6 sin t cos t. Then g Š ßÈ 2 hen t, and 2 g Š È ß 2 hen t. At the endpoints, g( ß 0) g( $ß 0) 0. The absolute minimum is 2 g Š È 2 ß 2 hen t ; the absolute maimum is g Š ßÈ 2 hen t. (ii) On the quarter ellipse, at the endpoints g(!ß2) 0 and g( $ß0) 0. The absolute minimum is g(ß0) 0 È 2 and g(0ß2) 0 at t 0, respectivel; the absolute maimum is g Š ßÈ2 hen t.
906 Chapter Partial Derivatives dh `h d `h d d d dt ` dt ` dt dt dt (c) 2 6 (6 cos t)( sin t) (2 sin t)(2 cos t) 6 sin t cos t 0 Ê t 0,, for 0 Ÿ t Ÿ, ielding the points (ß0), (0ß2), and ( ß0). (i) On the semi-ellipse, 0 so that h(ß0) 9, h(0ß2) 2, and h( ß0) 9. The absolute minimum is h(ß 0) 9 and h( ß 0) 9 hen t 0, respectivel; the absolute maimum is h(!ß 2) 2 hen t. (ii) On the quarter ellipse, the absolute minimum is h(ß0) 9 hen t 0; the absolute maimum is h(!ß 2) 2 hen t. df d d d d dt ` dt ` dt dt dt df dt ˆ ß ˆ ß 55. (i) 2t and t Ê (t )(2) (2t)() t 2 0 Ê t Ê and ith f. The absolute minimum is f hen t ; there is no absolute maimum. (ii) For the endpoints: t Ê 2 and 0 ith f( 2ß0) 0; t 0 Ê 0 and ith f(!ß ) 0. The absolute minimum is f ˆ ß hen t ; the absolute maimum is f(0ß ) 0 and f( ß 0) 0 hen t, 0 respectivel. (iii) There are no interior critical points. For the endpoints: t 0 Ê 0 and ith f(0ß) 0; t Ê 2 and 2 ith f(2ß2). The absolute minimum is f(0ß) 0 hen t 0; the absolute maimum is f(2ß2) hen t. df d d d d dt ` dt ` dt dt dt df 2 Ê dt Ê 5 Ê 5 5 f ˆ 2 6. The absolute minimum is f ˆ 2 5 5 5 25 5 5 5 5 hen t 5 ; there is no absolute 56. (a) 2 2 (i) t and 2 2t (2t)() 2(2 2t)( 2) 0t 8 0 t and ith maimum along the line. (ii) For the endpoints: t 0 Ê 0 and 2 ith f(0ß2) ; t Ê and 0 ith f(ß0). The absolute minimum is f ˆ 2 5ß 5 5 at the interior critical point hen t 5 ; the absolute maimum is f(0ß2) at the endpoint hen t 0. (b) dg g g d 2 d dt ` d dt ` 2 d dt dt ` ` a b a b dt dg (i) t and 2 2t Ê 5t 8t Ê dt a5t 8t b [( 2t)() ( 2)(2 2t)( 2)] 2 a5t 8t b ( 0t 8) 0 Ê t Ê and ith g ˆ 2 ß 5. The absolute 5 5 5 5 5 ˆ 5 maimum is g ˆ 2 5 5ß 5 hen t 5 ; there is no absolute minimum along the line since and can be as large as e please. (ii) For the endpoints: t 0 Ê 0 and 2 ith g(0ß2) ; t Ê and 0 ith g(ß0). The absolute minimum is g(0ß2) hen t 0; the absolute maimum is g ˆ 2 ß 5 hen t. 5 5 5 (2)( ) ( ) 20 (2) (0) ˆ 20 9 20 9 ; 7 57. m and b (2) Ê 2 2 2 0 0 0 9 2 D 2 0 (0)(5) (6) (0) (8) 5 5 ; 58. m and b 5 (0) Ê 2 0 0 2 0 2 0 0 2 6 D 0 5 8 6
Section.7 Etreme Values and Saddle Points 907 ()(5) (8) () (5) 2 2 6 ; 7 2 6 6 59. m and b 5 () Ê 0 0 0 0 2 2 2 2 6 D 5 5 8 (5)(5) (0) 5 (5) () 5 5 5 5 ; 5 5 60. m and b 5 (5) Ê 0 0 0 2 2 2 2 9 6 D 5 5 0 (62)(.2) 6(92.8) (62) 6(500) 6. m 0.22 and b 6 c.2 (0.22)(62) d.59 Ê 0.22.59 2 5.27 6.2 2 8 5.68 2 02.2 2 6.25 576 50 0 7.2 900 26. 5 6 8.20 296 295.2 6 2 8.7 76 65.82 D 62.2 500 92.8 (0.0086)(9) (0.065852) (0.0086) (0.000002) 62. m 5,55 and b (9 5,55(0.0086)).26 Ê F 5,55.26 D ˆ F ˆ ˆ D D D F 0.00 5 0.00000 0.05 2 0.0005 22 0.00000025 0.0 0.0002 0.0000000576 0.006 0. 0002 0.00000005 0.00092 D 0.0086 9 0.000002 0.065852
908 Chapter Partial Derivatives 6. (b) m (20)(7,785) 0(5,70,292) (20) 0(,0,89) 0.027 and b 0 [7,785 (0.027)(20)] 76.8 Ê 0.027K 76.8 K K K 76 76 2 75 77 5625 2,825 55 772 2,025 27,660 29 775 7,96 88,725 5 27 777 7, 8,567 6 5 780 2,20 62,780 7 25 78 80,625 757,775 8 50 786 25,009 898,58 9 575 789 0,625,028,675 0 626 79 9,876,2,66 D 20 7,785,0,89 5,70,292 (c) K 6 Ê (0.027)(6) Ê (0.027)(6) 76.8 780 (2)(0) 6() (2) 6(287) 6. m.0 and b 6[0 (.0)(2)] 0.755 Ê.0 0.755 9 9 2 2 2 6 6 2 2 6 5 5 25 20 6 5 25 5 7 9 8 99 8 2 9 08 9 8 0 6 80 0 6 69 208 96 82 2 5 9 5 6 6 2 9 69 27 5 0 5 00 50 6 6 5 256 20 D 2 0 287 65-70. Eample CAS commands: Maple: f := (,) -> ^2+^-**; 0, := -5,5; 0, := -5,5; plotd( f(,), =0.., =0.., aes=boed, shading=zhue, title=65(a) (Section.7) ); plotd( f(,), =0.., =0.., grid=[0,0], aes=boed, shading=zhue, stle=patchcontour, title=65(b) (Section.7) ); f := D[](f); (c) f := D[2](f); crit_pts := solve( {f(,)=0,f(,)=0}, {,} ); f := D[](f); (d) f := D[2](f); f := D[2](f); discr := unappl( f(,)*f(,)-f(,)^2, (,) ); for CP in {crit_pts} do (e) eval( [,,f(,),discr(,)], CP );
Section.8 Lagrange Multipliers 909 end do; (0,0) is a saddle point ( 9/, /2) is a local minimum Mathematica: (assigned functions and bounds ill var) Clear[,,f] 2 f[_,_]:= min= 5; ma= 5; min= 5; ma= 5; PlotD[f[,], {, min, ma}, {, min, ma}, AesLabel Ä {,, z}] ContourPlot[f[,], {, min, ma}, {, min, ma}, ContourShading Ä False, Contours Ä 0] f= D[f[,], ]; f= D[f[,], ]; critical=solve[{f==0, f==0},{, }] f= D[f, ]; f= D[f, ]; f= D[f, ]; discriminant= f f f 2 {{, }, f[, ], discriminant, f} /.critical.8 LAGRANGE MULTIPLIERS. f i j and g 2i j so that f - g Ê i j -(2i j) Ê 2 - and - È 2 Ê 8 - Ê - or 0. CASE : If 0, then 0. But (0ß0) is not on the ellipse so Á 0. È CASE 2: Á 0 Ê - 2 Ê È2 Ê Š È2 2 Ê. È2 È2 2 2 Therefore f taes on its etreme values at Š ß and Š ß. The etreme values of f on the ellipse are È 2. 2. f i j and g 2i2 j so that f - g Ê i j -(2i2 j) Ê 2 - and 2- Ê - Ê 0 or - 2. CASE : If 0, then 0. But (0ß0) is not on the circle 0 0 so Á 0. CASE 2: Á 0 Ê - Ê 2 ˆ Ê a b 0 0 Ê È5 Ê È5. 2 Therefore f taes on its etreme values at Š È5ß È5 and Š È5ßÈ5. The etreme values of f on the circle are 5 and 5. - -. f 2i2 j and g i j so that f - g Ê 2i2 j -( i j) Ê and Ê ˆ - ˆ - 0 Ê - 2 Ê and Ê f taes on its etreme value at (ß) on the line. The etreme value is f( ß $ ) 9 9 9.. f 2i j and g ij so that f - g Ê 2i j -( ij) Ê 2 - and - Ê 2 Ê 0 or 2. CASE : If 0, then Ê. CASE 2: If Á 0, then 2 so that Ê 2 Ê Ê 2. Therefore f taes on its etreme values at (!ß ) and (ß ). The etreme values of f are f(0ß ) 0 and f( ß ).
90 Chapter Partial Derivatives 5. We optimize f(ß), the square of the distance to the origin, subject to the constraint g(ß) 5 0. Thus f 2i2 j and g i2 j so that f - g Ê 2i2j -a i2 jb Ê 2 - and 2 2-. CASE : If 0, then 0. But (0ß0) does not satisf the constraint 5 so Á 0. CASE 2: If Á 0, then 2 2- Ê Ê 2 ˆ 2 - - - Ê -. Then 5 Ê ˆ ˆ 2 - - 5 $ Ê - Ê - Ê and 8 Ê and È 27 2. Therefore Š È $ß 2 are the points on the curve 5 nearest the origin (since 5 has points increasingl far aa as gets close to 0, no points are farthest aa). 6. We optimize f(ß), the square of the distance to the origin subject to the constraint g(ß) 2 0. Thus f 2i2 j and g 2i j so that f - g Ê 2 2 - and 2-2 Ê -, since 0 Ê 0 (but g(0ß0) Á 0). Thus Á 0 and 2 2 ˆ 2 Ê 2 Ê a2 b 2 0 Ê (since 0) Ê È2. Therefore Š È2ß are the points on the curve 2 nearest the origin (since 2 has points increasingl far aa as gets close to 0, no points are farthest aa). 7. (a) f ijand g i j so that f - g Ê ij -(i j) Ê - and - Ê - and - Ê - 6 Ê -. Use - since 0 and 0. Then and Ê the minimum value is 8 at the point (ß). No, 6, 0, 0 is a branch of a hperbola in the first quadrant ith the -and -aes as asmptotes. The equations c give a famil of parallel lines ith m. As these lines move aa from the origin, the number c increases. Thus the minimum value of c occurs here c is tangent to the hperbola's branch. (b) f i jand g ijso that f - g Ê i j -( ij) Ê - 6 Ê 8 Ê 8 Ê f( )ß)) 6 is the maimum value. The equations c ( 0 and 0 or 0 and 0 to get a maimum value) give a famil of hperbolas in the first and third quadrants ith the - and - aes as asmptotes. The maimum value of c occurs here the hperbola c is tangent to the line 6. 8. Let f(ß) be the square of the distance from the origin. Then f 2i2 jand g (2 ) i(2 ) j so that f - g Ê 2 -(2 ) and 2 -(2 ) Ê - 2 2 Ê 2 Š (2 ) Ê (2 ) (2 ) Ê Ê. È CASE : Ê () 0 Ê and. È È CASE 2: Ê ( ) ( ) 0 Ê and. Thus f Š ß È È f Š ß and f(ß) 2 f( ß). È È È È Therefore the points (ß) and ( ß) are the farthest aa; Š ß and Š ß are the closest points to the origin. 9. V r h Ê 6 r h Ê 6 r h Ê g(rßh) r h 6; S 2rh 2r Ê S (2h r) i2r j and g 2rhir j so that S - g Ê (2rh r) i2rj -a2rhir jb Ê 2rh r 2rh - and 2 2 2r -r Ê r 0 or - r. But r 0 gives no phsical can, so r Á 0 Ê - r Ê 2h r 2rh ˆ 2 Ê 2r h Ê 6 r (2r) Ê r 2 Ê h ; thus r 2 cm and h cm give the onl etreme r $ surface area of 2 cm. Since r cm and h cm Ê V 6 cm and S 0 cm, hich is a larger surface area, then 2 cm must be the minimum surface area. 2 2 2
Section.8 Lagrange Multipliers 9 0. For a clinder of radius r and height h e ant to maimize the surface area S 2rh subject to the constraint -h h ˆ h ˆ h r a r r È2 g(rßh) r ˆ h h a 0. Thus S 2hi2r jand g 2r i jso that S - g Ê 2h 2-r and 2r Ê - and 2r Ê r h Ê h 2r Ê r a Ê 2r a Ê r Ê h aè a 2 Ê S 2 aè Š Š 2 2a. È 2 2 6 9 8 9 2 2 2 2 2 8 9 8 9 9 ˆ 6 9 È È È È 2 È È. A (2)(2) subject to g(ß) 0; A i jand g i jso that A - g Ê i j -ˆ i j Ê ˆ - and ˆ - Ê - and ˆ ˆ Ê Ê Ê 8 Ê 2 2. We use 2 2 since represents distance. Then Š 2 2, so the length is 2 2 and the idth is 2 2. 2 2 a b a b 2. P subject to g(ß) 0; P i j and g i j so that P - g 2 2a 2a b b Š b a b b a a b a a% Ê ˆ and ˆ 2 Ê and ˆ 2 a - - - Š Ê Š Ê Ê % a b b 2a Èa b a Èa b Èa b Èa b Èa b Ê aa b b a Ê, since 0 Ê Š Ê idth 2 2b a b and height 2 Ê perimeter is P Èa b. f 2i2 j and g (2 2) i(2 ) j so that f - g 2i2 j -[(2 2) i(2 ) j] - 2- Ê 2 -(2 2) and 2 -(2 ) Ê and, - Á Ê 2 Ê 2 (2) (2) - - 0 Ê 0 and 0, or 2 and. Therefore f(0ß0) 0 is the minimum value and f(2ß) 20 is the maimum value. (Note that - gives 2 2 2 or! 2, hich is impossible.). f ij and g 2i2 j so that f - g Ê 2- and 2- Ê - and 2 ˆ 2 2 ˆ 6 6 2 6 È0 È0 È0 È0 Ê Ê Ê 0 6 Ê Ê and, or and 2 6 2 20. Therefore f Š ß 6 2È0 6 2.25 is the maimum value, and È0 È0 È0 È0 6 2 f Š ß 2È0 6 0.25 is the minimum value. È0 È0-5. T (8 ) i( 2) j and g(ß) 25 0 Ê g 2i2 j so that T g 2 Ê (8 ) i( 2) j -(2i2 j) Ê 8 2- and 2 2- Ê -, - Á Ê 8 ˆ 2-2- Ê 0, or - 0, or - 5. CASE : 0 Ê 0; but (0ß0) is not on 25 so Á 0. È CASE 2: - 0 Ê 2 Ê (2) 25 Ê 5 and 2. 2 CASE : - 5 Ê Ê ˆ 25 Ê 2È5 Ê 2È 5 and È 5, or 2È5 and È5. Therefore T Š È5ß2È5 0 T Š È5ß2È5 is the minimum value and T Š 2È5ßÈ5 25 T Š 2È5ßÈ5 is the maimum value. (Note: - Ê 0 from the equation 2 2-; but e found Á 0 in CASE.) $ 6. The surface area is given b S r 2rh subject to the constraint V(rßh) r r h 8000. Thus S (8r 2h) i2r j and V ar 2rhbir j so that S - V (8r 2h) i2rj -car 2rhbir jd Ê 8r2h -ar 2rh b and 2r -r Ê r 0 or 2 r -. But r Á 0 2 2 so 2 r - Ê - r Ê r h r a 2r rh b Ê h 0 Ê the tan is a sphere (there is no clindrical part) and 6 r 8000 Ê r 0 ˆ. $ Î$
92 Chapter Partial Derivatives 7. Let f(ßßz) ( ) ( ) (z ) be the square of the distance from (ßß). Then f 2( ) i2( ) j2(z ) and g i2j so that f - g Ê 2( ) i2( ) j2(z ) -( i2j ) Ê 2( ) -, 2( ) 2 -, 2(z ) - Ê 2( ) 2[2( )] and 2(z ) [2( )] Ê Ê z 2 ˆ or z ; thus 5 5 2 ˆ 0 Ê 2 Ê and z. Therefore the point ˆ ß2 ß is closest (since no point on the plane is farthest from the point (ßß)). 8. Let f(ßßz) ( ) ( ) (z ) be the square of the distance from (ßß). Then f 2( ) i2( ) j2(z ) and g 2i2j2z so that f - g Ê -, - - - - - - - 2 2 2 2 2 2 2 È È È È È È È and z -z Ê,, and z for - Á Ê ˆ ˆ ˆ - Ê Ê,, z or,, z. The largest value of f 2 2 2 È È È occurs here 0, 0, and z 0 or at the point Š ß ß on the sphere. 9. Let f(ßßz) z be the square of the distance from the origin. Then f 2i2j2z and g 2i2j2z so that f - g Ê 2i2j2z -(2i2j2z ) Ê 2 2 -, 2 2 -, and 2z 2z - Ê 0 or -. CASE : - Ê 2 2 Ê 0; 2z 2z Ê z 0 Ê 0 Ê and z 0. CASE 2: 0 Ê z, hich has no solution. Therefore the points on the unit circle, are the points on the surface z closest to the originþ The minimum distance is. 20. Let f(ßßz) z be the square of the distance to the origin. Then f 2i2j2z and g i j so that f - g Ê 2i2j2z -(i j) Ê 2 -, 2 -, and 2z - - - Ê Ê 2 -Š Ê 0 or - 2. CASE : 0 Ê 0 Ê z 0 Ê z. CASE 2: - 2 Ê and z Ê ( ) 0 Ê 2 0, so no solution. CASE : - 2 Ê and z Ê ( ) 0 Ê 0, again. Therefore (0ß0ß) is the point on the surface closest to the origin since this point gives the onl etreme value and there is no maimum distance from the surface to the origin. 2. Let f(ßßz) z be the square of the distance to the origin. Then f 2i2j2z and g ij2z so that f - g Ê 2i2j2z -( ij2z ) Ê 2 -, 2 -, and 2z 2z - Ê - or z 0. CASE : - Ê 2 and 2 Ê 0 and 0 Ê z 0 Ê z 2 and 0. 8 8 CASE 2: z 0 Ê 0 Ê. Then 2 - Ê -, and - Ê Š % Ê 6 Ê 2. Thus, 2 and 2, or = 2 and 2. Therefore e get four points: (ß 2ß 0), ( 2ß 2ß 0), (0ß 0ß 2) and (!ß 0ß 2). But the points (!ß 0ß 2) and (!ß!ß 2) are closest to the origin since the are 2 units aa and the others are 2È2 units aa. 22. Let f(ßßz) z be the square of the distance to the origin. Then f 2i2j2z and g zizj so that f - g Ê 2 -z, 2 -z, and 2z - Ê 2 -z and 2 -z Ê Ê Ê z Ê a ba b Ê Ê the points are (ßß ), (ßß), ( ß ß ), and ( ß, ). 2. f i2j5 and g 2i2j2z so that f - g Ê i2j5 -(2i2j2z ) Ê 2 -, 5 2 2 -, and 5 2z - Ê, 2, and z 5 Ê ( 2) (5) 0 Ê. - - -
Section.8 Lagrange Multipliers 9 Thus,, 2, z 5 or, 2, z 5. Therefore f(ß2ß5) 0 is the maimum value and f( ß2ß5) 0 is the minimum value. 2. f i2j and g 2i2j2z so that f - g Ê i2j -(2i2j2z ) Ê 2 -, 5 2 2 -, and 2z - Ê, 2, and z Ê (2) () 25 Ê. - - - 5 0 5 5 0 5 5 0 5 È È È È È È È È È Thus,,, z or,, z. Therefore f Š ß ß 5È 5 0 5 is the maimum value and f Š ß, 5È is the minimum value. È È È 25. f(ßßz) z and g(ßßz) z 9 0 Ê f 2i2j2z and g ijso that f - g Ê 2i2j2z -( ij) Ê 2 -, 2 -, and 2z - Ê z Ê 9 0 Ê,, and z. 26. f(ßßz) z and g(ßßz) z 6 0 Ê f zizj and g ij2z so that f - g Ê zizj -( ij2z ) Ê z -, z -, and 2z- Ê z z Ê z 0 or. But z 0 so that Ê 2z - and z -. Then 2z(z) Ê 0 or 2z. But 0 so that 2 2 2z Ê 2z Ê 2z 2z z 6 Ê z. We use z since z 0. Then and 2 2 096 5 5 È5 25È5 hich ields f Š ß ß. È È5 È5 5 5 27. V 6z and g(ßßz) z 0 Ê V 6zi6zj6 and g 2i2j2z so that V - g Ê z -, z -, and -z Ê z - and z - Ê Ê z 2 2 2 Ê Ê since 0 Ê the dimensions of the bo are b b for maimum È È È È volume. (Note that there is no minimum volume since the bo could be made arbitraril thin.) z a b c 28. V z ith ßßz all positive and ; thus V z and g(ßßz) bc ac abz abc 0 Ê V zizj and g bciacjab so that V - g Ê z -bc, z -ac, and -ab Ê z -bc, z -ac, and z -abz Ê - Á 0. Also, -bc -ac -abz Ê b a, c bz, and b c c c az Ê and z. Then Ê ˆ b ˆ c a a a a b z a b a c a Ê a Ê Ê ˆ b ˆ a b and z ˆ c ˆ a c Ê V z ˆ a ˆ b ˆ c abc a a 27 is the maimum volume. (Note that there is no minimum volume since the bo could be made arbitraril thin.) 29. T 6iz j( 6) and g 8i2j8z so that T - g Ê 6iz j( 6) -(8i2j8z ) Ê 6 8 -, z 2 -, and 6 8z - Ê - 2 or 0. CASE : - 2 Ê z 2(2) Ê z. Then z 6 6z Ê z Ê. Then ˆ ˆ 6 Ê. CASE 2: 0 Ê Ê 6 8z Š Ê z Ê (0) a b6 0 2z 2z - Ê 2 8 0 Ê ( )( 2) 0 Ê or 2. No Ê z () È Ê z 0 and 2 Ê z ( 2) ( 2) Ê z. The temperatures are T ˆ 2 ß ß 62, T(0ßß0) 600, T Š 0ß2ß È Š 600 2È, and T Š 0 2 È Š 600 2È 6.6. Therefore ˆ ß ß ß ß are the hottest points on the space probe. 0. T 00z i00z j800z and g 2i2j2z so that T - g Ê 00z i00z j800z -(2i2j2z ) Ê 00z 2 -, 00z 2 -, and 800z 2z -. È 2 Solving this sstem ields the points a!ß ß 0 b, a ß0ß 0 b, and Š ß ß. The corresponding
9 Chapter Partial Derivatives È 2 temperatures are T a!ß ß 0b 0, T a ß 0ß 0b 0, and T Š ß ß 50. Therefore 50 is the È2 È2 maimum temperature at Š ß ß and Š ß ß ; 50 is the minimum temperature at È2 È2 Š ß ß and Š ß ß.. U ( 2) i j and g 2 ij so that U - g Ê ( 2) i j -(2 ij) Ê 2 - and - Ê 2 2 Ê 2 2 Ê 2 (2 2) 0 Ê 8 and. Therefore U(8ß) $28 is the maimum value of U under the constraint. 2. M (6 z) i2j and g 2i2j2z so that M - g Ê (6 z) i2j -(2i2j2z ) Ê 6 z 2 -, 2 2 -, 2z - Ê - or 0. CASE : - Ê 6 z 2 and 2z Ê 6 z 2( 2z) Ê z 2 and. Then ( ) 2 6 0 Ê. CASE 2: 0, 6 z 2 -, and 2z - Ê - Ê 6 z 2 ˆ 2z 2z Ê 6z z Ê a6z z b0 z 6 Ê z 6 or z. No z 6 Ê 0 Ê 0; z È Ê 27 Ê. Therefore e have the points Š Èß0ß, (0ß0ß6), and aß ß2 b. Then M Š Èß0ß 27È 60 06.8, M Š Èß0ß 60 27È.2, M(0ß0ß6) 60, and M( ßß2) 2 M( ßß2). Therefore, the eaest field is at aß ß2 b.. Let g (ßßz) 2 0 and g (ßßz) z 0 Ê g 2 ij, g j, and f 2i2j2z so that f - g. g Ê 2i2j2z -(2 ij). ( j) Ê 2i2j2z 2 - i(. - ) j. Ê 2 2 -, 2. -, and 2z. Ê -. Then 2 2z Ê 2z2 so that 2 0 Ê 2( 2z 2) 0 Ê z 0. This equation coupled ith z 0 implies z and. 2 2 2 2 Then so that ˆ ß ß is the point that gives the maimum value f ˆ ß ß ˆ 2 ˆ ˆ.. Let g (ßßz) 2 z 6 0 and g (ßßz) 9z 9 0 Ê g i2j, g ij9, and f 2i2j2z so that f - g. g Ê 2i2j2z -( i2j ). ( ij9 ) Ê 2 -., 2 2-., and 2z - 9.. Then 0 2 z 6 (-. ) (2-.) ˆ 9 27 -. 6 Ê 7-7. 6; 0 9z 9 Ê (-. ) ˆ 9 -. ˆ 27 8 -. 9 Ê - 9. 8. Solving these to equations for - and. gives 20 78 -. 8 2-. 2-9. 9 59 59 Ê 59 59 59 8 2 9 2,77 69 59 ß 59 ß 59 59 59 maimum - and.,, and z. The minimum value is f ˆ. (Note that there is no value of f subject to the constraints because at least one of the variables,, or z can be made arbitrar and assume a value as large as e please.) 5. Let f(ßßz) z be the square of the distance from the origin. We ant to minimize f(ßßz) subject to the constraints g (ßßz) 2z 2 0 and g (ßßz) 6 0. Thus f 2i2j2z, g j2, and g ij so that f - g. g Ê 2., 2 -., and 2z 2 -. Then 0 2z 2 ˆ -. 5. 2 2 Ê 2 Ê 5 2; 0 6 ˆ -. - -. -. 6. -. - Ê -. 6 Ê -. 2. Solving these to equations for - and. gives - and. Ê 2,, and z. The point (2ßß) on the line of intersection is closest to the origin. (There is no maimum distance from the origin since points on the line can be arbitraril far aa.) 6. The maimum value is f ˆ 2 ß ß from Eercise above.
Section.8 Lagrange Multipliers 95 7. Let g (ßßz) z 0 and g (ßßz) z 0 0 Ê g, g 2i2j2z, and f 2zi zj so that f - g. g Ê 2zi zj -( ). (2i2j2z ) Ê 2z 2., z 2., and 2z. - Ê z. Ê 0 or z. Ê. since z. CASE : 0 and z Ê 9 0 (from g ) Ê ielding the points a0ß ß b. CASE 2:. Ê z 2 Ê 2 (since z ) Ê 2 0 0 (from g ) Ê 9 0 Ê È Ê 2 Š È Ê È6 ielding the points Š È6ß È ß. No f a!ß ßb and f Š È6ß Èß 6 Š È 6È. Therefore the maimum of f is 6È at Š È6ßÈß, and the minimum of f is 6È at Š È6ßÈ ß. 8. (a) Let g (ßßz) z 0 0 and g (ßßz) z 0 Ê g ij, g ij, and zizj so that - g. g Ê zizj -( ij). ( ij) Ê z -., z -., and -. Ê z z Ê z 0 or. CASE : z 0 Ê 0 and 0 Ê no solution. CASE 2: Ê 2 z 0 0 and 2 z 0 Ê z 20 Ê 0 and 0 Ê (0)(0)(20) 2000 i j (b) n 2i2 jis parallel to the line of intersection Ê the line is 2t 0, â â 2t 0, z 20. Since z 20, e see that z ( 2t 0)(2t 0)(20) at 00 b(20) hich has its maimum hen t 0 Ê 0, 0, and z 20. 9. Let g ( Bßßz) 0 and g (ßßz) z 0. Then f ij2z, g ij, and g 2i2j2z so that f - g. g Ê ij2z -( ij). (2i2j2z ) Ê - 2., - 2., and 2z 2z. Ê z 0 or.. CASE : z 0 Ê 0 Ê 2 0 (since ) Ê È2 and È2 ielding the points Š È 2ß È2 ß!. CASE 2:. Ê -2 and -2 Ê 2( ) Ê 2 2(2) since Ê 0 Ê 0 Ê z 0 Ê z 2 ielding the points a!ß!ß 2 b. No, f a!ß!ß 2b and f Š È2ß È2ß! 2. Therefore the maimum value of f is at a!ß!ß 2 b and the minimum value of f is 2 at Š È 2ß È2 ß!. 0. Let f(ßßz) z be the square of the distance from the origin. We ant to minimize f(ßßz) subject to the constraints g (ßßz) 2 z 5 0 and g (ßßz) z 0. Thus f 2i2j2z, g 2j, and g 8i8j2z so that f - g. g Ê 2i2j2z -(2j ). (8i8j2z ) Ê 2 8., 2 2-8., and 2z - 2z. Ê 0 or.. CASE : 0 Ê (0) z 0 Ê z 2 Ê 2 (2) 5 0 Ê, or 2 ( 2) 5 0 5 Ê ielding the points ˆ!ß ß 5 5 6 and ˆ!ß 6ß. CASE 2:. Ê - Ê - 0 Ê 2z (0) 2z ˆ 5 Ê z 0 Ê 2 (0) 5 Ê and (0) ˆ 5 Ê no solution. 5 5 5 25 6 6 9 6 Then f ˆ!ß ß and f ˆ!ß ß 25 ˆ Ê the point ˆ!ß ß is closest to the origin.. f ij and g i j so that f - g Ê ij -(i j) Ê - and - Ê Ê 6 Ê Ê (ß ) and ( %ß ) are candidates for the location of etreme values. But as Ä _, Ä _ and f(ß) Ä _; as Ä _, Ä 0 and f(ß) Ä _. Therefore no maimum or minimum value eists subject to the constraint.
96 Chapter Partial Derivatives! 2. Let f(aßbßc) (A B C z ) C (B C ) (A B C ) (A C ). We ant to minimize f. Then f A(AßBßC) A 2B C, f B(AßBßC) 2A B C, and f C(AßBßC) A B 8C 2. Set each partial derivative equal to 0 and solve the sstem to get A, B, and C or the critical point of f is ˆ ß ß.. (a) Maimize f(aßbßc) a b c subject to a b c r. Thus f 2ab c i2a bc j2a b c and g 2ai2bj2c so that f - g Ê 2ab c 2a -, 2a bc 2b -, and 2a b c 2c- Ê 2a b c 2a - 2b - 2c - Ê - 0 or a b c. CASE : - 0 Ê a b c 0. CASE 2: a b c Ê f(abc) ß ß aaa and a r Ê f(abc) ß ß Š is the maimum value. (b) The point Š Èa Èb È ß ß c is on the sphere if a b c r. Moreover, b part (a), abc f Š ÈaßÈbßÈc r $ Î$ r abc Ÿ Š Ê (abc) Ÿ, as claimed. r $! n n i i n n n n i. Let f( ß ßá ß ) a a a á a and g( ß ßá ß ) á. Then e a a a a n n n i 2- - - - n n Î n n n n Î!! Œ!! ˆ a i!! i i n i i i - - i Œ i i i i i i i i ant f - g Ê a -(2 ), a -(2 ), á, a -(2 ), - Á 0 Ê Ê á Ê - a Ê 2 - a Ê f( ß ßá ß ) a a a a is the maimum value. 5-50. Eample CAS commands: Maple: f := (,,z) -> *+*z; g := (,,z) -> ^2+^2-2; g2 := (,,z) -> ^2+z^2-2; h := unappl( f(,,z)-lambda[]*g(,,z)-lambda[2]*g2(,,z), (,,z,lambda[],lambda[2]) ); h := diff( h(,,z,lambda[],lambda[2]), ); h := diff( h(,,z,lambda[],lambda[2]), ); hz := diff( h(,,z,lambda[],lambda[2]), z ); hl := diff( h(,,z,lambda[],lambda[2]), lambda[] ); hl2 := diff( h(,,z,lambda[],lambda[2]), lambda[2] ); ss := { h=0, h=0, hz=0, hl=0, hl2=0 }; q := solve( ss, {,,z,lambda[],lambda[2]} ); q2 := map(allvalues,{q}); for p in q2 do eval( [,,z,f(,,z)], p ); ``=evalf(eval( [,,z,f(,,z)], p )); end do; Mathematica: (assigned functions ill var) Clear[,, z, lambda, lambda2] f[_,_,z_]:= z 2 2 g[_,_,z_]:= 2 2 2 g2[_,_,z_]:= z 2 h = f[,, z] lambda g[,, z] lambda2 g2[,, z]; h= D[h, ]; h= D[h, ]; hz= D[h,z]; hl=d[h, lambda]; hl2= D[h, lambda2]; critical=solve[{h==0, h==0, hz==0, hl==0, hl2==0, g[,,z]==0, g2[,,z]==0}, (a) (b) (c) (d)
Section.9 Partial Derivatives ith Constrained Variables 97 {,, z, lambda, lambda2}]//n {{,, z}, f[,, z]}/.critical.9 PARTIAL DERIVATIVES WITH CONSTRAINED VARIABLES. z and z : Î (ßz) Ñ ` ` (a) Œ z Ä Ä Ê Š Ï z z Ò z ` ` ` ` `z `z ; `z ` 0 and 2 2 ` ` ` ` ` ` `z ` ` ` ` ` ` ` ` ` 2 2 Ê 0 2 2 Ê Ê Š (2) ˆ (2)() (2z)(0) 2 2 0 ` ` ` ` z Î Ñ (b) Œ ( z) ; 0 and 2 2 z ˆ ` ` ` ` ` ` `z ` `z ` Ä ß Ä Ê Ï z z Ò ` `z ` `z ` `z `z `z `z `z `z `z ` ` Ê 2 Ê Ê ˆ ` (2)(0) (2) Š (2z)() 2z `z ` z `z 2 Î (ßz) Ñ (c) Œ ; 0 and 2 2 z ˆ ` Ä Ä Ê ` ` ` ` `z `z ` Ï z z Ò ` ` Ê 2 Ê Ê ˆ ` (2) ˆ (2)(0) (2z)() 2z `z `z 2 ` z ` ` ` `z ` `z ` `z `z `z `z `z `z `z 2. z sin t and t: Î Ñ Î Ñ ` ` ` ` ` ` `z ` `t ` `z (a) Ä Ð ÓÄ Ê Š ; 0, 0, and Ï z z z Ò ` ` ` ` ` `z ` `t ` ` ` z ß Ït Ò `t ` ` ` Ê Š (2)(0) ()() ( )(0) (cos t)() cos t cos ( ) t ß Î tñ Î Ñ ` ` ` ` ` ` `z ` `t `z `t (b) z Ä Ð ÓÄ Ê Š ; 0 and 0 Ï z z t Ò ` ` ` ` ` `z ` `t ` ` ` zt ß Ï t t Ò ` `t ` ` ` ` ` ` Ê Ê Š (2)( ) ()() ( )(0) (cos t)(0) 2at b 2 2t zt ß Î Ñ Î Ñ (c) Ð Ó ˆ ` ` ` ` ` ` `z ` `t ` ` Ä Ä Ê ; 0 and 0 Ï z z z Ò `z ß ` `z ` `z `z `z `t `z `z `z Ït Ò Ê ˆ ` (2)(0) ()(0) ( )() (cos t)(0) `z ß Î tñ Î Ñ (d) z Ð Ó ˆ ` ` ` ` ` ` `z ` `t ` `t Ä Ä Ê ; 0 and 0 Ï z z t Ò `z t ß ` `z ` `z `z `z `t `z `z `z Ï t t Ò Ê ˆ ` (2)(0) ()(0) ( )() (cos t)(0) `z t ß Î Ñ Î Ñ t (e) z Ð Ó ˆ ` ` ` ` ` ` `z ` `t ` `z Ä Ä Ê ; 0 and 0 Ï z z t Ò `t z ß ` `t ` `t `z `t `t `t `t `t Ï t t Ò Ê ˆ ` (2)(0) ()() ( )(0) (cos t)() cos t `t z ß
98 Chapter Partial Derivatives Î tñ Î Ñ (f) z Ð Ó ˆ ` ` ` ` ` ` `z ` `t ` `z Ä Ä Ê ; 0 and 0 Ï z z t Ò `t z ß ` `t ` `t `z `t `t `t `t `t Ï t t Ò Ê ˆ ` (2)() ()(0) ( )(0) (cos t)() cos t 2 cos t 2(t ) `t z ß. U f(pßvßt) and PV nrt P Î P P Ñ (a) Œ V V U (0) V Ä ˆ `U `U `P `U `V `U `T `U Ä Ê ˆ `U ˆ `U ˆ V ÏT PV `P V `P `P `V `P `T `P `P `V `T nr Ò nr `U `U V ˆ ˆ `P `T nr nrt P V V Î Ñ (b) Œ V V U T Ä Ä Ê ˆ V Ï T T Ò ˆ ˆ ˆ (0) ˆ `U ˆ nr `U `P V `T `U `U `P `U `V `U `T `U nr `U `U `T `P `T `V `T `T `T `P V `V `T. z and sin z z sin 0 Î Ñ (a) Œ ; 0 and ˆ ` ` ` ` ` ` `z ` Ä Ä Ê ` ` ` ` ` ` ` ` z Ïz z(ß) Ò `z `z `z z cos `z ( cos z) (sin ) z cos 0 Ê. At (0ß ß), ` ` ` cos zsin ` ` ` (2)() (2)(0) (2z)( ) 2 Ð 0 ß ßÑ (0 ß ß ) Ê ˆ Î (ßz) Ñ (b) Œ (2) (2)(0) (2z)() z ˆ ` Ä Ä Ê ` ` ` ` ` `z ` `z ` ` ` ` ` ` ` z z z z z Ï z z Ò ` ` ` ` `z `z `z `z ` ` cos z sin ` 0 `z `z z cos `z ( )() ` `z 2(0) C (!ß, Ñ 2 2 (2) 2z. No (sin z) cos z sin (z cos ) 0 and 0 Ê cos z sin (z cos ) 0 Ê. At (!ß ß ), Ê ˆ ˆ $ 5. z z and z 6 Î Ñ (a) Œ Ä Ä Ê Š Ïz z( ) Ò ß ` ` ` ` ` ` `z ` ` ` ` ` `z ` `z `z ` `z ` ` ` ` ` `z `z ` z ` ` 0 Ê 2 (2z) ` 0 Ê ` z. At (ßßßz) (ß2ßß), ` Ê Š ` ¹ (2 ß ß ßc) a2 b(0) a2 z b() a z b 2 z a z b. No (2) 2 (2z) 0 and c(2)(2) () ( ) dc ( ) d() 5 Î (ßz) Ñ (b) Œ z Ä Ä Ê Š Ï z z Ò z ` ` ` ` ` ` `z ` ` ` ` ` `z ` ` ` ` `z ` ` ` ` `z ` ` ` ` ` 0 Ê (2) ` 2 0 Ê `. At (ßßßz) (ß2ßß), ` 2 Ê Š ` ¹ z ( 2 ß ß ßc) (2)(2)() ˆ (2)(2) () ( ) 5 a2 b a2 z b() a z b(0) a2 b 2 z. No (2) 2 (2z) 0 and `u `v ` `u `v `v ` 6. uv Ê v u ; u v and 0 Ê 0 2u 2v Ê ˆ u u Ê ` ` ` ` ` ` v ` `u u `u v u `u `u v ` u v u Š Š Ê. At (ußv) Š È2ß, ` v ` v ` ` v u ` Š È2
Section.0 Talor's Formula for To Variables 99 `u ` Ê Š 7. Œ r Œ r cos ) Ä cos ; r 2 2 2r and 0 2 2r ) r sin ) Ê ˆ ` ` ) Ê Ê ) `r ` Ê Ê ˆ r È ` r ` ` `r `r `r ` ` ` ` 8. If,, and z are independent, then ˆ ` ß ` ` ` ` ` `z ` `t ` z ` ` ` ` `z ` `t ` ˆ `t `t `t `t ` ` ` ` ` ` z ß ` t ß ` ` ` ` ` `z ` `t `z `z ` ` ` ` `z ` `t ` ` ` `z ` z 2 0 0 ˆ ` ` ` ` 2 ˆ t ß 2 2. (2)() ( 2)(0) ()(0) () 2. Thus 2z t 25 Ê 0 0 Ê Ê ˆ ` 2. On the other hand, if,, and t are independent, then ˆ (2)() ( 2)(0) ()(0) 2. Thus, 2z t 25 Ê Ê Ê ` ` `z ` ` ` ` ` `z ` ` ` ` 9. If is a differentiable function of and z, then f(ßßz) 0 Ê 0 Ê 0 ` / ` ` / `z ` z / `z `z / ` Ê Š. Similarl, if is a differentiable function of and z, Š and if z is a differentiable function of and, ˆ `z / ` ` `. Then Š Š ˆ `z / ` Š ˆ / `z / ` Š. / `z / ` / ` ` / ` ` `z ` z `z df `u df `z df `u df `z `z ` du ` du ` du ` du ` ` 0. z z f(u) and u Ê ; also 0 so that ˆ df ˆ df du du `g ` `g ` `g `z ` `g `g `z ` ` ` ` `z ` ` ` `z `. If and are independent, then g(ßßz) 0 Ê 0 and 0 Ê 0 Ê `z `g/ ` ` `g/ `z Š, as claimed. 2. Let and be independent. Then f(ßßzß) 0, g(ßßzß) 0 and 0 ` ` `z ` `z ` ` ` ` ` `z ` ` ` ` `z ` ` ` `g ` `g ` `g `z `g ` `g `g `z `g ` ` ` ` ` `z ` ` ` ` `z ` ` ` Ê 0 and `z ` `z ` ` ` ` `g `z `g ` `g `z ` ` ` ` ˆ `z ` ` Ê ` ` ` ` `» `g `g» `z `» `g `g» `z f f ` 0 impl ` ` `g `g `g `g ` ` ` ` ` ` ` ` `g `g `g `g `z ` `z ` `z ` ` `z Lieise, f(ßßzß) 0, g(ßßzß) 0 and 0 Ê, as claimed. ` ` ` `z ` ` ` ` ` ` `z ` ` ` `z ` `g `g `z `g ` ` `z ` ` ` 0 and (similarl) ` `z ` ` ` 0 impl `z ` `z ` ` `» `g `g» ` ` `z ` `g ` ` Š `z g ` g Ê ` `z ` `z ` ` ` ` `z ` ` `» `g `g» `z ` f `g `g f `g `g `z ` `z ` z ` ` ` `z `g `g `g `g `z ` `z ` `z ` ` `z, as claimed..0 TAYLOR'S FORMULA FOR TWO VARIABLES c a b. f(ß) e Ê f e, f e, f 0, f e, f e Ê f(ß) f(0ß0) f (0ß0) f (0ß0) f (0ß0) 2f (0ß0) f (0ß0) d 0 0 0 2 0 quadratic approimation; f 0, f 0, f e, f e
920 Chapter Partial Derivatives $ $ 6 $ $ 6 a b Ê f(ß ) quadratic c f (!ß!) f (0ß 0) f (!ß!) f (0ß 0) d 0 0 0, cubic approimation Ê f(ß ) f(0ß 0) f (0ß 0) f (!ß 0) c f (!ß!) 2f (!ß!) f (0ß 0) d c d a b Ê f(ß) quadratic $ f (0ß0) f (!ß0) f (0ß0) $ 6 c f (0ß0) d $ $ a b 6 c 0 ( ) 0d $ a b 6 a b, cubic approimation 2. f(ß) e cos Ê f e cos, f e sin, f e cos, f e sin, f e cos 0 2 0 ( ), quadratic approimation; f e cos, f e sin, f e cos, f e sin. f(ß) sin Ê f cos, f sin, f sin, f cos, f 0 c 0 0 0 a 0 2 0b, quadratic approimation; $ $ 6 c $ $ Ê f(ß) f(0ß0) f (0ß0) f (!ß0) f (0ß0) 2f (0ß0) f (0ß0) d f cos, f sin, f 0, f 0 Ê f(ß) quadratic f (0ß0) f (!ß0) f (0ß0) f (0ß0) d 6 a 0 0 0 0 b, cubic approimation. f(ß) sin cos Ê f cos cos, f sin sin, f sin cos, f cos sin, c 0 0 a 0 2 0 0b, quadratic approimation; Ê f(ß) quadratic $ f (0ß0) f (!ß0) f (0ß0) $ 6 c f (0ß0) d $ $ $ 6 c ( ) 0 ( ) 0d 6 a b, cubic approimation f sin cos Ê f(ß) f(0ß0) f (0ß0) f (0ß0) f (0ß0) 2f (0ß0) f (0ß0) d f cos cos, f sin sin, f cos cos, f sin sin e e e ( ) Ê f(ß) f(0ß0) f (0ß0) f (0ß0) c f (0ß0) 2f (0ß0) f (0ß0) d 0 0 c 0 2 ( ) d a2 b, quadratic approimation; e e 2e ( ) ( ) $ Ê f(ß) quadratic $ f (0ß0) f (!ß0) f (0ß0) $ 6 c f (0ß0) d $ $ 2 a2 b 6 c 0 ( ) 2d $ a2 b 6 a 2 b, cubic approimation 5. f(ß) e ln ( ) Ê f e ln ( ), f, f e ln ( ), f, f f e ln ( ), f, f, f 2 2 2 (2 ) (2 ) (2 ) 0 2 c ( ) 2 ( 2) ( ) d 2 a b (2 ) (2 ), quadratic approimation; 6 8 2 (2 ) $ (2 ) $ (2 ) $ (2 ) $ Ê f(ß) quadratic $ f (0ß0) f (!ß0) f (0ß0) $ 6 c f (0ß0) d $ $ (2 ) (2 ) 6 a 6 8 2 b $ (2 ) (2 ) a 8 2 6 b $ (2 ) (2 ) (2 ), cubic approimation 6. f(ß) ln (2 ) Ê f, f, f, f, f Ê f( ß ) f(0 ß 0) f (0 ß 0) f (0 ß 0) f (0 ß 0) 2f (0 ß 0) f (0 ß c 0) d f, f, f, f a b a b a b 7. f(ß) sin a b Ê f 2 cos a b, f 2 cos a b, f 2 cos a b sin a b, f sin, f 2 cos sin
Section.0 Talor's Formula for To Variables 92 Ê f(ß) f(0ß0) f (0ß0) f (0ß0) c f (0ß0) 2f (0ß0) f (0ß0) d 0 0 0 a 2 2 0 2b, quadratic approimation; $ $ f sin 8 cos, f 2 sin 8 cos Ê f(ß) quadratic $ f (0ß0) f (!ß0) f (0ß0) $ 6 c f (0ß0) d $ $ f 2 sin a b 8 cos a b, f sin a b 8 cos a b, a b a b a b a b 6 a 0 0 0 0 b, cubic approimation Ê f(ß) f(0ß0) f (0ß0) f (0ß0) c f (0ß0) 2f (0ß0) f (0ß0) d 0 0 c 0 2 0 0d, quadratic approimation; $ a b a b a b a b $ a b a b a b a b Ê f(ß) quadratic $ f (0ß0) f (!ß0) f (0ß0) $ 6 c f (0ß0) d $ $ 8. f(ß) cos a b Ê f 2 sin a b, f 2 sin a b, f 2 sin a b cos a b, f cos a b, f 2 sin a b cos a b f 2 cos 8 sin, f cos 8 sin, f cos 8 sin, f 2 cos 8 sin 6 a 0 0 0 0 b, cubic approimation ( ) 2 ( ) $ c d a b a b 6 ( ) % $ $ 6 c d $ $ 9. f(ß) Ê f f, f f f Ê f(ß) f(0ß0) f (0ß0) f (0ß0) f (0ß0) 2f (0ß0) f (0ß0) 2 2 2 2 ( ) 2 ( ) ( ), quadratic approimation; f f f f Ê f(ß) quadratic f (0ß0) f (!ß0) f (0ß0) f (0ß0) () () 6 a 6 6 6 6 b $ $ $ ( ) ( ) a b ( ) ( ) ( ), cubic approimation 2( ) ( ) ( ) ( ) 0. f(ß) Ê f, f, f, f $ 2( ) ( ), f ( ) $ a b 6( ) $ [ ( ) 6( )( )]( ) ( ) % ( ) % [ ( ) 6( )( )]( ), f 6( ) $ ( ) % ( ) % $ $ 6 c $ $ Ê f(ß) f(0ß0) f (0ß0) f (0ß0) c f (0ß0) 2f (0ß0) f (0ß0) d 22 2, quadratic approimation; f, f, f Ê f(ß) quadratic f (0ß0) f (!ß0) f (0ß0) f (0ß0) d 6 a 6 2 2 6 b $ $, cubic approimation. f(ß) cos cos Ê f sin cos, f cos sin, f cos cos, f sin sin, c c d f cos cos Ê f(ß) f(0ß0) f (0ß0) f (0ß0) f (0ß0) 2f (0ß0) f (0ß0) d 0 0 ( ) 2 0 ( ), quadratic approimation. Since all partial derivatives of f are products of sines and cosines, the absolute value of these derivatives is less than or equal $ $ $ $ to Ê E(ß) Ÿ 6 c (0.) (0.) (0.) 0.) d Ÿ 0.00. Ê f(ß) f(0ß0) f (0ß0) f (0ß0) c f (0ß0) 2f (0ß0) f (0ß0) d 0Þ 0Þ 2. f(ß) e sin Ê f e sin, f e cos, f e sin, f e cos, f e sin 0 0 a 0 2 0b, quadratic approimation. No, f e sin, f e cos, f e sin, and f e cos. Since Ÿ 0., e sin Ÿ e sin 0. 0. and e cos Ÿ e cos 0... Therefore,
922 Chapter Partial Derivatives $ $ $ $ E(ß) Ÿ 6 c (0.)(0.) (.)(0.) (0.)(0.) (.)(0.) d Ÿ 0.0008. CHAPTER PRACTICE EXERCISES. Domain: All points in the -plane Range: z 0 Level curves are ellipses ith major ais along the -ais and minor ais along the -ais. 2. Domain: All points in the -plane Range: 0 z _ Level curves are the straight lines ln z ith slope, and z 0.. Domain: All (ß) such that Á 0 and Á 0 Range: z Á 0 Level curves are hperbolas ith the - and -aes as asmptotes.. Domain: All (ß) so that 0 Range: z 0 Level curves are the parabolas c, c 0. 5. Domain: All points (ßßz) in space Range: All real numbers Level surfaces are paraboloids of revolution ith the z-ais as ais.
Chapter Practice Eercises 92 6. Domain: All points (ßßz) in space Range: Nonnegative real numbers Level surfaces are ellipsoids ith center (0ß0ß0). 7. Domain: All (ßßz) such that (ßßz) Á (0ß!ß0) Range: Positive real numbers Level surfaces are spheres ith center (0ß0ß0) and radius r 0. 8. Domain: All points (ßßz) in space Range: (0ß ] Level surfaces are spheres ith center (0ß0ß0) and radius r 0. ln2 9. lim e cos e cos (2)( ) 2 Ð Ä Ðßln2Ñ 2 2 0 0. lim 2 Ð Ä Ð00 cos 0cos 0. lim lim lim Ð Ä Ð Ð Ä Ð ( )( ) Ð Ä Ð Á Á $ $ () a b 2. lim lim lim Ð Ä Ð Ð Ä Ð a b Ð Ä Ð. lim ln z ln ( ) e ln e P ÄÐßßeÑ. lim tan ( z) tan ( ( ) ( )) tan ( ) P ÄÐßßÑ 5. Let, Á. Then lim lim hich gives different limits for Ð Ä Ð00 a ß b Ä Ð00 Á different values of Ê the limit does not eist. () 6. Let, Á 0. Then lim lim Ð Ä Ð00 ( Ä Ð00 () hich gives different limits for Á 0
92 Chapter Partial Derivatives different values of Ê the limit does not eist. 7. Let. Then lim hich gives different limits for different values Ð Ä Ð00 of Ê the limit does not eist so f(0ß0) cannot be defined in a a that maes f continuous at the origin. sin ( ) sin, 0 8. Along the -ais, 0 and lim lim, so the limit fails to eist Ð Ä Ð00 Ä 0, 0 Ê f is not continuous at (0ß0). `g `r `g `) 9. cos ) sin ), r sin ) r cos ) Š ` f 2 ` ˆ 20. Š, ` f 2 Š ` Š ˆ ` f ` f ` f `R R `R R `R R 2.,, $ $ 22. h (ßßz) 2 cos (2 z), h (ßßz) cos (2 z), h (ßßz) cos (2 z) z ` RT ` nt ` nr ` nrt `n V `R V `T V `V V 2.,,, 2. f (rßjßtß) É, f (rßjßtß) É, f (rßjßtß) ˆ Š Š r T T 2r j j rj T rj È 2ÈT É É, f (rßjßtß) ˆ É ˆ $Î É T T T rj T rj T rj rj `g `g ` g ` g 2 ` g ` g ` ` ` ` $ `` `` 25., Ê 0,, 26. g (ß) e cos, g (ß) sin Ê g (ß) e sin, g (ß) 0, g (ß) g (ß) cos 2 ` f 22 ` f ` f ` f ` ` ` a b ` `` `` 27. 5, Ê 0, 0, 28. f (ß), f (ß) 2 sin 7e Ê f (ß) 0, f (ß) 2 cos 7e, f (ß) f (ß) ` ` d d ` ` dt dt t d t dt ˆ t t 29. cos ( ), cos ( ), e, Ê [ cos ( )]e [ cos ( )] ; t 0 Ê and 0 Ê d 0 [ ( )] ˆ dt 0 t 0 ` ` ` d Î d dz ` ` `z dt dt t dt d Î dt t Ê d dt (2 0)(2) (0 0) 5 t 0. e, e sin z, cos z sin z, t,, Ê e t ae sin zbˆ ( cos z sin z) ; t Ê 2, 0, and z ` ` ` ` ` ` ` ` `r `s `r `s ` `. 2 cos (2 ), cos (2 ),, cos s, s, r Ê r [2 cos (2 )]() [ cos (2 )](s); r and s 0 Ê and 0
` Ê ` `r (2 cos 2) (cos 2)(0) 2; `s [2 cos (2 )](cos s) [ cos (2 )](r) Ðß0Ñ ` Ê `s (2 cos 2)(cos 0) (cos 2)( ) 2 Ðß0Ñ ` 2. ` ` d ˆ u a2e cos v b; u v 0 Ê 2 Ê ˆ 2 (2) 2 ; `u d `u `u Ð00 5 5 5 ` d ` ` 2 `v d `v ˆ u 2e sin v Ê `v ˆ 5 a b 5 (0) 0 Ð00 d d dz ` ` `z dt dt dt df dt df dt t. z, z,, sin t, cos t, 2 sin 2t Chapter Practice Eercises 925 Ê ( z)(sin t) ( z)(cos t) 2( )(sin 2t); t Ê cos, sin, and z cos 2 Ê (sin cos 2)(sin ) (cos cos 2)(cos ) 2(sin cos )(sin 2) ` d `s d ` d `s d d ` ` d d ` ds ` ds ` ds ` ds ds ` ` ds ds. (5) and () Ê 5 5 5 0 d F cos d F 2 cos 5. F(ß) sin Ê F cos and F 2 cos Ê cos d 2 cos d Ð0 2 Ê at (ß ) (!ß ) e have ¹ d F 2 e b d F 2 eb 6. F(ß) 2 e 2 Ê F 2 e and F 2 e Ê d 2 ln 2 2 d Ð0ln2 0 2 Ê at (ß) (!ßln 2) e have ¹ (ln 2 ) È ˆ ß È2 f È2 È2 È2 È2 f Ê f increases most rapidl in the direction and decreases most È2 È2 È2 È2 u i j u P u P! c! v i j Ê (D f) f v È 7 5 5 u P! ˆ ˆ ˆ ˆ 5 5 0 7. f ( sin cos ) i(cos sin ) j Ê f i j Ê f Ɉ ˆ 2 ; u i j u i j rapidl in the direction ; (D f) f and (D f) ; u i j u 8. f 2ec 2 2 e c2 i j Ê f i j Ê f È2 ( 2) 2È f 2; u i j 0 Ð È2 È2 f È2 È2 Ê f increases most rapidl in the direction u i j and decreases most rapidl in the direction u i j; (D f) f 2È2 and (D f) 2È v i j 2 ; u i j È2 È u P! c u P! 2 v È È2 È2 Ê (D f) P f u (2) Š ( 2) Š 0 u! È2 È2 2 6 2 6z 2 6z 2 6z 9. f Š iš jš Ê f 2ij6 ; ÐcßcßÑ f 2ij6 2 6 2 6 f È2 6 7 7 7 7 7 7 2 6 u 7 i 7 j 7 u P! u P! v 2 6 v 7 7 7 (Du f) P! (Duf) P 7! u i j Ê f increases most rapidl in the direction u i j and decreases most rapidl in the direction ; (D f) f 7, (D f) 7; u i j Ê f 2 000 ß f È5 È5 2 2 È 5 È5 È5 È5 v ij u P u v È i j È È È 0. f (2 ) i( 2) j( 2z) Ê f 2 j; u j Ê f increases most Ð rapidl in the direction u j and decreases most rapidl in the direction u j ; (D f) f È5 and (D f) È5 ; u P!! Ê (Du f) P! f u (0) Š (2) Š () Š È È È È È
926 Chapter Partial Derivatives. r (cos t) i(sin t) jt Ê v(t) ( sin t) i( cos t) j Ê vˆ j È2 È2 Ê u j ; f(ßßz) z Ê f zizj ; t ields the point on the heli ( ß0 ß) È2 È2 È2 Ê f j Ê f u ( j) Š j Ðc0 ß 2. f(ßßz) z Ê f zizj ; at (ßß) e get f ij Ê the maimum value of Df f È u Ð ß. (a) Let f aib j at (ß2). The direction toard (2ß2) is determined b v (2 ) i(2 2) j i u so that f u 2 Ê a 2. The direction toard (ß) is determined b v ( ) i( 2) j j u so that f u 2 Ê b 2 Ê b 2. Therefore f 2i2 j; f a, 2b f a, 2b 2. 5 5 (b) The direction toard (ß6) is determined b v ( ) i(6 2) j i j Ê u i j Ê f u. 5. (a) True (b) False (c) True (d) True $ 5. f 2ij2z Ê f Ð0ßcßcÑ j2, f Ð000 ß j, f Ð0ßcßÑ j2 6. f 2j2z Ê f Ð220 ß j, f Ð2ßc2ß0Ñ j, f Ð202 ß, f Ð20 ß ßc2Ñ 7. f 2ij5 Ê f Ð2ßcßÑ ij5 Ê Tangent Plane: ( 2) ( ) 5(z ) 0 Ê 5z ; Normal Line: 2 t, t, z 5t 8. f 2i2 j Ê f Ð2 ß 2i2 j Ê Tangent Plane: 2( ) 2( ) (z 2) 0 Ê 2 2 z 6 0; Normal Line: 2t, 2t, z 2 t `z 2 `z `z 2 `z 9. Ê 0 and Ê ¹ 2; thus the tangent plane is ` ` Ð00 ß ` ` 2( ) (z 0) 0 or 2 z 2 0 00 Ð ß
` ` ˆ ß ß ` ` 2 ˆ ß ß 2 plane is ( ) ( ) ˆ z 0 or 2z 0 Chapter Practice Eercises 927 `z ` z `z ` z 50. 2 a b Ê and 2 a b Ê ¹ ; thus the tangent 5. f ( cos ) ij Ê f ij Ê the tangent Ð line is ( ) ( ) 0 Ê ; the normal line is ( ) Ê 52. f i j Ê f i2 j Ê the tangent Ð 2 line is ( ) 2( 2) 0 Ê ; the normal line is 2 2( ) Ê 2 5. Let f(ßßz) 2 2z and g(ßßz). Then f 2i2j2 a b 2i2j2 ß ß 2 i j and g j Ê f g 2 2 2 2i2 Ê the line is 2t,, z 2t â0 0â 5. Let f(ßßz) z 2 and g(ßßz). Then f i2j a b i2 jand ß 2 ß 2 i j g j Ê f g 2 i Ê the line is t,, z t â0 0â 55. f ˆ, f ˆ cos cos, f ˆ ß ß ÐÎßÎÑ ß sin sin ÐÎß ÎÑ Ê L(ß) ˆ ˆ ; f (ß) sin cos, f (ß) sin cos, and f (ß) cos sin. Thus an upper bound for E depends on the bound M used for f, f, and f. È È È 2 2 2 With M e have E(ß) Ÿ Š ˆ Ÿ (0.2) Ÿ 0.02; ith M, E(ß) Ÿ () ˆ (0.2) 0.02. 56. f(ß) 0, f (ß), f (ß) 6 5 Ê L(ß) ( ) 5( ) 5 ; Ð Ð f (ß) 0, f (ß) 6, and f (ß) Ê maimum of f, f, and f is 6 Ê M 6 Ê E(ß) Ÿ (6) a b (6)(0. 0.2) 0.27 57. f(ß0ß0) 0, f (ß0ß0) z 0, f (ß0ß0) 2z, f (ß0ß0) 2 00 Ð00 ß z 00 Ð ß Ð ß Ê L(ßßz) 0( ) ( 0) (z 0) z; f(ßß0), f (ßß0), f (ßß0), f z( ßß!) Ê L(ßßz) ( ) ( ) (z 0) z 58. f ˆ 0, f ˆ 0 È2 sin sin ( z) 0, f ˆ ß!ß!ß ß ¹!ß0ß È2 cos cos ( z) ¹, 00 ß ß 00 ß ß ˆ ˆ f ˆ 0 È2 cos cos ( z) L( z) ( 0) ˆ z!ß ß ¹ Ê ß ß z ; z ˆ 00 ß ß È2 È2 È2 È2 ß ß ß ß ß ß z ß ß L(z) È 2 È 2 2 2 (z 0) 2 È 2 2 2 z f ˆ 0, f ˆ 0, f ˆ 0, f ˆ 0 Ê ß ß
928 Chapter Partial Derivatives Ð55280 Þ 59. V r h Ê dv 2rh dr r dh Ê dv 2(.5)(5280) dr (.5) dh 5,80 dr 2.25 dh. You should be more careful ith the diameter since it has a greater effect on dv. 60. df (2 ) d ( 2) d Ê df d Ê f is more sensitive to changes in ; in fact, near the point (ß 2) a change in does not change f. 2 Ð V 6. di dv dr di 2 Ê dv dr Ê di 0.0 (80)(.000) 0.08, R R Ð2ß00 00 00 dvßdr20 Ñ 20 2 00 2 di 0.08 00 I 0.2 or increases b 0.08 amps; % change in V (00) ˆ.7%; % change in R ˆ (00) 20%; I 0.2 Ê estimated % change in I 00 00 5.8% Ê more sensitive to voltage change. 62. A ab Ê da b da a db Ê da 6 da 0 db; da 0. and db 0. 0 6 Ð Ê da 26(0.) 2.6 and A (0)(6) 60 Ê da 00 2.6 00.625% A 60 6. (a) uv Ê d v du u dv; percentage change in u Ÿ 2% Ê du Ÿ 0.02, and percentage change in v Ÿ % d v du u dv du dv d Ê dv Ÿ 0.0; Ê ¹ 00¹ du dv 00 00 Ÿ du 00 dv 00 uv u v u v u v Ÿ 2% % 5% dz du dv du dv du dv (b) z u v Ê Ÿ (since u 0, v 0) z uv uv uv u v Ê dz 00 Ÿ du dv 00 00 d ¹ 00¹ z u v 7 ( 0.25)(7) ( 0.725)(7) 7.80Þ25 h0 Þ725 7.8 Þ25 h0 Þ725 h 7.80 Þ25 h Þ725 2.975 5.075 7.8Þ25 h0 Þ725 7.80 Þ25 h Þ725 Ð70ß80Ñ Ê 6. C Ê C and C Ê dc d dh; thus hen 70 and h 80 e have dc (0.00000225) d (0.000009) dh g error in eight has more effect 65. f (ß) 2 2 0 and f (ß) 2 2 0 Ê 2 and 2 Ê ( 2ß2) is the critical point; f ( 2ß2) 2, f ( ß2) 2, f ( ß2) Ê f f f 0 and f 0 Ê local minimum value of f( ß 2) 8 66. f (ß) 0 0 and f (ß) 0 Ê 0 and Ê (0ß) is the critical point; f (0ß) 0, f (0ß), f (0ß) Ê f f f 56 0 Ê saddle point ith f(0ß) 2 % $ 67. f (ß) 6 0 and f (ß) 6 0 Ê 2 and 6 a b 0 Ê a 8 b 0 Ê 0 and 0, or and Ê the critical points are (0ß 0) and ˆ ß. For (!ß!): f (!ß!) 2 0, f (!ß!) 2 0, f (!ß 0) Ê f f f 9 0 Ê saddle point ith Ð00 Ð00 ˆ ß f(0ß0) 0. For ß : f 6, f 6, f Ê f f f 27 0 and f 0 Ê local maimum value of f ˆ % $ 68. f (ß) 0 and f (ß) 0 Ê and 0 Ê a b 0 Ê the critical points are (0ß 0) and (ß ). For (!ß!): f (!ß!) 6 0, f (!ß!) 6 0, f (!ß 0) Ð00 Ð00 ff f 27 0 and f 0 local minimum value of f( ) Ê f f f 9 0 Ê saddle point ith f(0ß0) 5. For (ß): f (ß) 6, f (ß) 6, f (ß) Ê Ê ß 69. f (ß) 6 0 and f (ß) 6 0 Ê ( 2) 0 and ( 2) 0 Ê 0 or 2 and 0 or 2 Ê the critical points are (0ß 0), (0ß 2), ( 2ß 0), and ( 2ß 2). For (!ß!): f (!ß!) 6 6 Ð00 Ð00 6, f (!ß!) 6 6 6, f (!ß 0) 0 Ê f f f 6 0 Ê saddle point ith f(0ß 0) 0. For (0ß 2): f (!ß2) 6, f (0 ß ) 6, f (!ß2) 0 Ê f f f 6 0 and f 0 Ê local minimum value of
Chapter Practice Eercises 929 f(!ß 2). For ( ß 0): f ( 2ß 0) 6, f ( ß 0) 6, f ( 2ß 0) 0 Ê f f f 6 0 and f 0 Ê local maimum value of f( 2ß0). For ( 2ß2): f ( 2ß2) 6, f ( 2ß2) 6, f ( 2ß2) 0 Ê f f f 6 0 Ê saddle point ith f( 2ß2) 0. $ 70. f (ß) 6 0 Ê a b 0 Ê 0, 2, 2; f (ß) 6 6 0 Ê. Therefore the critical points are (0ß ), (2ß ), and ( 2ß ). For (!ß ): f (!ß ) 2 6 6, f (!ß ) 6, f (!ß ) 0 Ð0 ß ß ß Ê Ê Ê f f f 96 0 Ê saddle point ith f(0ß). For (2ß): f (2ß) 2, f (2ß) 6, f (2ß) 0 Ê f f f 92 0 and f 0 Ê local minimum value of f(2ß) 9. For ( ß): f ( 2 ) 2, f ( ) 6, f ( 2 ) 0 f f f 92 0 and f 0 local minimum value of f( 2ß) 9. 7. (i) On OA, f(ß) f(0ß) for 0 Ÿ Ÿ Ê f (!ß) 2 0 Ê. But ˆ!ß is not in the region. Endpoints: f(0ß0) 0 and f(0ß) 28. (ii) On AB, f(ß) f(ß ) 0 28 for 0 Ÿ Ÿ Ê f (ß) 20 0 Ê 5,. But (5ß) is not in the region. Endpoints: f(ß 0) and f(!ß ) 28. (iii) On OB, f(ß) f(ß0) for 0 Ÿ Ÿ Ê f (ß0) 2 Ê and 0 Ê ˆ ß0 is a critical point ith f ˆ 9 ß!. Endpoints: f(0ß 0) 0 and f( %ß 0). (iv) For the interior of the triangular region, f (ß) 2 0 and f (ß) 2 0 Ê and. But (ß) is not in the region. Therefore the absolute maimum is 28 at (0ß) and the 9 absolute minimum is at ˆ ß!. 72. (i) On OA, f(ß) f(0ß) for 0 Ÿ Ÿ 2 Ê f (!ß) 2 0 Ê 2 and 0. But (0ß2) is not in the interior of OA. Endpoints: f(0ß0) and f(0ß2) 5. (ii) On AB, f(ß) f(ß2) 2 5 for 0 Ÿ Ÿ Ê f (ß2) 2 2 0 Ê and 2 Ê (ß2) is an interior critical point of AB ith f(ß2). Endpoints: f(ß2) and f(!ß2) 5. (iii) On BC, f(ß) f(ß) 9 for 0 Ÿ Ÿ 2 Ê f (ß) 2 0 Ê and. But (ß2) is not in the interior of BC. Endpoints: f(ß0) 9 and f( %ß2). (iv) On OC, f(ß) f(ß0) 2 for 0 Ÿ Ÿ Ê f (ß0) 2 2 0 Ê and 0 Ê (ß0) is an interior critical point of OC ith f(ß0) 0. Endpoints: f(0ß0) and f(ß0) 9. (v) For the interior of the rectangular region, f (ß) 2 2 0 and f (ß) 2 0 Ê and 2. But (ß2) is not in the interior of the region. Therefore the absolute maimum is at (ß 2) and the absolute minimum is 0 at (ß 0).
90 Chapter Partial Derivatives 7. (i) On AB, f(ß) f( 2ß) for 2 Ÿ Ÿ 2 Ê f ( 2ß) 2 Ê and 2 Ê ˆ 2 ß is an interior critical point in AB ith f ˆ 2 7 ß. Endpoints: f( 2ß2) 2 and f(2ß2) 2. (ii) On BC, f(ß) f(ß2) 2 for 2 Ÿ Ÿ 2 Ê f (ß2) 0 Ê no critical points in the interior of BC. Endpoints: f( 2ß2) 2 and f(2ß2) 2. (iii) On CD, f(ß) f(2ß) 5 for 2 Ÿ Ÿ 2 Ê 5 f (2ß) 2 5 0 Ê and 2. But ˆ 5 ß is not in the region. Endpoints: f(2ß2) 8 and f(2ß2) 2. (iv) On AD, f(ß) f(ß2) 0 for 2 Ÿ Ÿ2 Ê f (ß2) Ê no critical points in the interior of AD. Endpoints: f( 2ß2) 2 and f(2ß2) 8. (v) For the interior of the square, f (ß) 2 0 and f (ß) 2 0 Ê 2 and Ê (ß2) is an interior critical point of the square ith f(ß2) 2. Therefore the absolute maimum 7 is 8 at (2ß2) and the absolute minimum is at ˆ ß. 7. (i) On OA, f(ß) f(0ß) 2 for 0 Ÿ Ÿ 2 Ê f (!ß) 2 2 0 Ê and 0 Ê (!ß ) is an interior critical point of OA ith f(0ß). Endpoints: f(0ß0) 0 and f(0ß2) 0. (ii) On AB, f(ß) f(ß2) 2 for 0 Ÿ Ÿ 2 Ê f (ß2) 2 2 0 Ê and 2 Ê (ß2) is an interior critical point of AB ith f(ß2). Endpoints: f(0ß2) 0 and f(2ß2) 0. (iii) On BC, f(ß) f(2ß) 2 for 0 Ÿ Ÿ 2 Ê f (2ß) 2 2 0 Ê and 2 Ê (2ß) is an interior critical point of BC ith f(2ß). Endpoints: f(2ß0) 0 and f(2ß2) 0. (iv) On OC, f(ß) f(ß0) 2 for 0 Ÿ Ÿ 2 Ê f (ß0) 2 2 0 Ê and 0 Ê (ß0) is an interior critical point of OC ith f(ß0). Endpoints: f(0ß0) 0 and f(0ß2) 0. (v) For the interior of the rectangular region, f (ß) 2 2 0 and f (ß) 2 2 0 Ê and Ê (ß) is an interior critical point of the square ith f(ß) 2. Therefore the absolute maimum is 2 at (ß) and the absolute minimum is 0 at the four corners (0ß0), (0ß2), (2ß2), and (2ß0). 75. (i) On AB, f(ß) f(ß 2) 2 for 2 Ÿ Ÿ 2 Ê f (ß 2) 2 0 Ê no critical points in the interior of AB. Endpoints: f( 2ß0) 8 and f(2ß) 0. (ii) On BC, f(ß) f(2ß) for 0 Ÿ Ÿ Ê f (2ß) 2 0 Ê 2 and 2 Ê (2ß2) is an interior critical point of BC ith f(2ß2). Endpoints: f(2ß0) 0 and f(2ß) 0. (iii) On AC, f(ß) f(ß0) 2 for 2 Ÿ Ÿ 2 Ê f (ß0) 2 2 Ê and 0 Ê (ß0) is an interior critical point of AC ith f(ß0). Endpoints: f( 2ß0) 8 and f(2ß0) 0. (iv) For the interior of the triangular region, f (ß) 2 2 0 and f (ß) 2 0 Ê and 2 Ê (ß2) is an interior critical point of the region ith f(ß2). Therefore the absolute maimum is 8 at ( 2ß0) and the absolute minimum is at (ß0).
% 76. (i) On AB, f(ß) f(ß) 2 6 for $ 2 Ÿ Ÿ 2 Ê f (ß) 88 0 Ê 0 and 0, or and, or and Ê (0ß 0), (ß ), ( ß ) are all interior points of AB ith f(0ß0) 6, f(ß) 8, and f( ß) 8. Endpoints: f( 2ß2) 0 and f(2ß2) 0. % (ii) On BC, f(ß) f(2ß) 8 for 2 Ÿ Ÿ 2 $ $È Ê f (2ß) 8 0 Ê 2 and 2 Ê Š 2ß 2 is an interior critical point of BC ith È$ $ f Š 2 È $ ß 2 6 È2. Endpoints: f(2ß2) 2 and f(2ß2) 0. Chapter Practice Eercises 9 % $ (iii) On AC, f(ß) f(ß2) 8 for 2 Ÿ Ÿ2 Ê f (ß2) 8 0 Ê $È 2 and 2 $ Ê Š È $ 2ß2 is an interior critical point of AC ith f Š È $ 2ß2 6 È. Endpoints: f( 2ß2) 0 and f(2ß2) 2. $ $ (iv) For the interior of the triangular region, f (ß) 0 and f (ß) 0 Ê 0 and 0, or and or and. But neither of the points (0ß0) and (ß), or ( ß) are interior to the region. Therefore the absolute maimum is 8 at (ß) and ( ß), and the absolute minimum is 2 at (2ß2). $ 77. (i) On AB, f(ß) f( ß) 2 for Ÿ Ÿ Ê f ( ß) 6 0 Ê 0 and, or 2 and Ê ( ß0) is an interior critical point of AB ith f( ß0) 2; ( ß2) is outside the boundar. Endpoints: f( ß) 2 and f( ß) 0. (ii) $ On BC, f(ß) f(ß) 2 for Ÿ Ÿ Ê f (ß) 6 0 Ê 0 and, or 2 and Ê (0ß) is an interior critical point of BC ith f(!ß ) 2; ( 2ß ) is outside the boundar. Endpoints: f( ß ) 0 and f( ß ) 2. $ (iii) On CD, f(ß) f( ß) for Ÿ Ÿ Ê f (ß) 6 0 Ê 0 and, or 2 and Ê (ß 0) is an interior critical point of CD ith f( ß0) ; (ß2) is outside the boundar. Endpoints: f(ß) 2 and f( ß) 0. $ (iv) On AD, f(ß) f(ß) for Ÿ Ÿ Ê f (ß) 6 0 Ê 0 and, or 2 and Ê (0ß) is an interior point of AD ith f(0ß) ; ( ß) is outside the boundar. Endpoints: f( ß) 2 and f( ß) 0. (v) For the interior of the square, f (ß) 6 0 and f (ß) 6 0 Ê 0 or 2, and 0 or 2 Ê (0ß 0) is an interior critical point of the square region ith f(!ß0) 0; the points (0ß2), ( 2ß0), and ( 2ß2) are outside the region. Therefore the absolute maimum is at (ß0) and the absolute minimum is at (0ß).