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Chapter14Sec6ShowCode

Chapter14Sec6ShowCode - Initialization Cells(Code Needed...

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Initialization Cells (Code Needed for Most Plot3D Statements) In[160]:= ticks @ min_, max_ D : = Join @ Table @8 i,If @ i 0, ,Style @ i,12 DD , 8 0.01,0 << , 8 i,Ceiling @ min D ,Floor @ max D , 2 <D , H * Numbered ticks * L Table @8 j, , 8 0.01,0 << , 8 j,Round @ min D ,Round @ max - 1 D ,1 <DD H * Un - numbered ticks * L H * 2D Options * L SetOptions @ Plot, ImageSize fi 350, AxesOrigin fi 8 0,0 < , AspectRatio fi 1, AxesStyle fi Directive @ Medium, Bold D , Ticks fi ticks D ; H * 3D Options * L SetOptions @ Plot3D, ImageSize fi 600, BoxRatios fi 1,Boxed fi False, AxesOrigin fi 8 0,0,0 < , AxesStyle fi Directive @ Medium, Bold D , Ticks fi ticks, Mesh fi None, ViewPoint fi 8 2,0.9,1 < , ViewVertical fi 8 0,0,1 < D ; SetOptions @ ContourPlot3D, ImageSize fi 600, BoxRatios fi 1,Boxed fi False, AxesOrigin fi 8 0,0,0 < , AxesStyle fi Directive @ Medium, Bold D , Ticks fi ticks, Mesh fi None, ViewPoint fi 8 2,0.9,1 < , ViewVertical fi 8 0,0,1 < D ; SetOptions @ ParametricPlot3D, ImageSize fi 600, BoxRatios fi 1,Boxed fi False, AxesOrigin fi 8 0,0,0 < , AxesStyle fi Directive @ Medium, Bold D , Ticks fi ticks, Mesh fi None,

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ViewPoint fi 8 2,0.9,1 < , ViewVertical fi 8 0,0,1 < D ; SetOptions @ RegionPlot3D, ImageSize fi 600, BoxRatios fi 1,Boxed fi False, AxesOrigin fi 8 0,0,0 < , AxesStyle fi Directive @ Medium, Bold D , Ticks fi ticks, Mesh fi None, ViewPoint fi 8 2,0.9,1 < , ViewVertical fi 8 0,0,1 < D ; Chapter 14 : Partial Derivatives 2 Chapter14Sec6.nb
Section 14.6: Directional Derivatives and the Gradient Vector Directional Derivatives ª Definition Recall that if z = f H x , y L , then f x H x 0 , y 0 L = lim h fi 0 f H x 0 + h , y 0 L - f H x 0 , y 0 L h and f y H x 0 , y 0 L = lim h fi 0 f H x 0 , y 0 + h L - f H x 0 , y 0 L h are the rates of change of z in the x- and y- directions, that is along the unit vectors i and j . The directional derivative of f at the point H x 0 , y 0 L in the direction of a unit vector u = X a , b \ is D u f H x 0 , y 0 L = lim h fi 0 f H x 0 + h , y 0 + h L - f H x 0 , y 0 L h provided the limit exits. This represents the rate of change in z = f H x , y L in the direction of the unit vector u ; that is, the slope of the tangent line to the curve of intersection of the surface and the vertical plane containing u . Note that when u = X 1, 0 \ we obtain f x and when u = X 0, 1 \ we obtain f y . Thus, the concept of directional derivatives generalizes the idea of partial derivatives. Chapter14Sec6.nb 3

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ª Example Consider the curve is f H x , y L = - H x - 1 L 2 + H y - 1 L 2 + 4. In[166]:= myF @ x_, y_ D = - H x - 1 L ^2 - H y - 1 L ^2 + 4; myFx @ x_, y_ D = D @ myF @ x,y D , x D ; myFy @ x_, y_ D = D @ myF @ x,y D , y D ; Manipulate @ Show @ Plot3D @ H * Plot the surface * L myF @ x,y D , 8 x, - 1,3 < , 8 y, - 1,3 < , PlotRange fi 8 - 4,4 < , PlotStyle fi Opacity @ 0.4 D D , ContourPlot3D @ H * Plot the plane * L 0 - rSin @ Θ D x + rCos @ Θ D y, 8 x, - 8,8 < , 8
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