Partial Derivatives Calc IV Lab Karen Donnelly All Rights Reserved LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2I1EhRictRiM2Iy1GLDYlUShyZXN0YXJ0RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEiO0YnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGNi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR1EsMC4yNzc3Nzc4ZW1GJw==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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2I1EhRictRiM2JUYrLUYjNiUtRiw2JVEld2l0aEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RMCZBcHBseUZ1bmN0aW9uO0YnL0Y6USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGRC8lKXN0cmV0Y2h5R0ZELyUqc3ltbWV0cmljR0ZELyUobGFyZ2VvcEdGRC8lLm1vdmFibGVsaW1pdHNHRkQvJSdhY2NlbnRHRkQvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZTLUkobWZlbmNlZEdGJDYkLUYjNiMtRiw2JVEqcGxvdHRvb2xzRidGNkY5RkBGK0Yr

1. Let f be a function of two variables defined around the point (a,b). Recall that the definition of the partial derivative with respect to x at (LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEiYUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIixGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRjEvJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMzMzMzMzM2VtRictRiw2JVEiYkYnRi9GMkY5) is given by the following limit (provided that the limit exists). 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. Similarly, the partial derivative with respect to y at (a,b) is 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. As an example of how Maple V can be used to compute the partial derivatives of the function: 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 at the point (x,y) we first define f and take the derivatives PkkiZkc2ImYqNiRJInhHRiRJInlHRiRGJDYkSSlvcGVyYXRvckdGJEkmYXJyb3dHRiRGJCwoKiY5JCIiJTklIiIjIiIkKiZGLiIiIkYwRjQhIiJGMEYxRiRGJEYk LUklZGlmZkclKnByb3RlY3RlZEc2JC1JImZHNiI2JEkieEdGKEkieUdGKEYq 2. Similarly, the following gives us the partial derivative with respect to y: LUklZGlmZkclKnByb3RlY3RlZEc2JC1JImZHNiI2JEkieEdGKEkieUdGKEYr 3. The second derivative with respect to x is computed as follows: LUklZGlmZkclKnByb3RlY3RlZEc2JS1JImZHNiI2JEkieEdGKEkieUdGKEYqRio= or LUklZGlmZkclKnByb3RlY3RlZEc2JS1JImZHNiI2JEkieEdGKEkieUdGKEYqRio= 4. We can compute the mixed order partial derivatives (which are the same if ). LUklZGlmZkclKnByb3RlY3RlZEc2JS1JImZHNiI2JEkieEdGKEkieUdGKEYrRio= LUklZGlmZkclKnByb3RlY3RlZEc2JS1JImZHNiI2JEkieEdGKEkieUdGKEYqRis= 5. We can compute higher order partial derivatives. To compute the fifth order derivative of f(x,y) three times with respect to x and then two with respect to y: LUklZGlmZkclKnByb3RlY3RlZEc2KC1JImZHNiI2JEkieEdGKEkieUdGKEYrRitGKkYqRio= Please note an important point: the diff command returns an expression, not a function. 6. Thus as is, to evaluate the derivative at a particular point, we use subs which substitutes a value into an expression and evauates the expression. PkkkZnh4RzYiLUklZGlmZkclKnByb3RlY3RlZEc2Ji1JImZHRiQ2JEkieEdGJEkieUdGJEYtRixGLA== LUklc3Vic0clKnByb3RlY3RlZEc2JS9JInhHNiIiIiQvSSJ5R0YoIiIjSSRmeHhHRig= If we wnt to make a true function from an expression we use the unapply command: PkkkZjJ4RzYiLUkodW5hcHBseUdGJDYlSSRmeHhHRiRJInhHRiRJInlHRiQ= This can now be evaluated as a true function, relieving us of using the subs command. LUkkZjJ4RzYiNiQiIiQiIiM=

The first order partial derivatives gives us the slopes of the surface in the x and y directions. See figures 12.29, 12.30 page 860. The partial derivative with respect to x at a particular point gives the slope of the tangent line to the curve of intersection of a plane parallel to the x- axis passing through that the surface at that point. The partial derivative with respect to y at a particular point gives the slope of the tangent line to the curve of intersection of a plane parallel to the y- axis passing through the surface at that point. 7. As an example of this, we complete exercise 43, page 912: The surface 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 (This is a quadric surface -- a hyperbolic paraboloid -- the "saddle".) PkkiZkc2ImYqNiRJInhHRiRJInlHRiRGJDYkSSlvcGVyYXRvckdGJEkmYXJyb3dHRiRGJCwmKiQ5JCIiIyIiKiokOSVGLyEiIkYkRiRGJA== QyQ+SSNQMUc2Ii1JJ3Bsb3QzZEc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJTYlLUkiZkdGJTYkSSJ4R0YlSSJ5R0YlL0YvOyEiIyIiIy9GMDshIiUiIiUhIiI= 8. We plot the plane of intersection, y = 3. QyQ+SSNQMkc2Ii1JL2ltcGxpY2l0cGxvdDNkR0YlNigvSSJ5R0YlIiIkL0kieEdGJTshIiUiIiUvRipGLi9JInpHRiU7ISM1IiM/L0kmc3R5bGVHRiVJLHBhdGNobm9ncmlkR0YlL0klYXhlc0dGJUkmYm94ZWRHRiUhIiI= and the "Point" where finding the slope: QyQ+SSJTRzYiLUknc3BoZXJlR0YlNiU3JSIiIiIiJCIiISQiIiMhIiIvSSZjb2xvckdGJUkkcmVkR0YlRi8= Display the surface and plane together, along with point on surface where finding the slope: LUkoZGlzcGxheUc2IjYjPCVJI1AxR0YkSSNQMkdGJEkiU0dGJA== 9. From the graph we see that the curve of intersection is a parabola. The equation can be obtained by substituting y =3 into the equation for the surface 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 to get 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. We can add this to the plot by using the spacecurve command. 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 We can calculate the slope of the surface in the x direction at the point (x,y,z) = (1,3,0) by evaluating the partial derivative with respect to x at this point: PkkjZnhHNiItSSVkaWZmRyUqcHJvdGVjdGVkRzYkLUkiZkdGJDYkSSJ4R0YkSSJ5R0YkRiw= LUklc3Vic0clKnByb3RlY3RlZEc2JC9JInhHNiIiIiJJI2Z4R0Yo Exercise: Complete Exercise 44, page 912: Plot the plane x = 1 intersecting the surface Use appropriate commands similar to those above.