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265L09 - LESSON 9 Partial Derivative and Tangent Planes...

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LESSON 9 Partial Derivative and Tangent Planes READ: Sections 15.3, 15.4 NOTES: The role of the derivative for functions of one variable studied back in Calculus I is taken over by the notion of a partial derivative for functions of several variables. Geometrically, partial derivatives are easy to understand. Imagine there is a surface in 3-space with equation z = f ( x, y ), and that ( a, b, c ) is one point on that surface, so that the point on the surface directly above ( a, b ) in the x , y -plane has z coordinate c = f ( a, b ). Now, take a plane parallel to the x , z -plane, passing through b on the y -axis. This plane will intersect the surface in a curve. See figure 1B in section 15.3. On that curve the y coordinate never changes: it stays constant at b . The z coordinate is computed from z = f ( x, b ) for each x . The partial derivative of f at the point ( a, b ) is the slope of the tangent line to that curve at the point ( a, b, c ). To compute that partial derivative, we need only differentiate z = f ( x, b ) with respect to x in the good old-fashioned Calculus I way. For example, if z = x 2 + y 2 , then one point on this paraboloid would be (2 , 3 , 13). The curve of intersection described above would have equation z = x 2 + 9. The derivative of z with respect to x is 2 x . So, putting in the x value of 2 at the point in question, we see the slope of the tangent line is 4. Another way to the say the same thing: to compute the partial derivative of z with respect to x , compute the derivative of f ( x, y ) in the usual way, pretending y is a constant. The particular value y = b can be substituted in after the differentiation has been carried out. That’s a bit more efficient. So, if z = y 2 x + x 3 sin y , then the partial derivative of z with respect to x at any point ( x, y ) would be given by the expression y 2 + 3 x 2 sin y , where we have thought of y as a constant and x as the variable while computing the derivative. At the point on the surface above the point (2 , π ) in the x , y -plane the slope of the curve of intersection described above would be π 2 . The notation for the partial derivative of z with respect to x is ∂z ∂x , so the example above would be written as ∂z ∂x = y 2 + 3 x 2 sin y . Other common symbols for that partial are f x ( x, y ) = y 2 + 3 x 2 sin y and ∂x ( y 2 x + x 3 sin y ) = y 2 + 3 x 2 sin y . If the surface is sliced by a plane parallel to the y , z -plane, so that x has a fixed value along the curve of intersection, then the slopes at various points along that curve are determined by taking the derivative of z = f ( x, y ), this time treating x as a constant and y as the variable. The result is called the partial derivative of z with respect to y . In the example above, we would write f y ( x, y ) = ∂z ∂y = 2 yx + x 3 cos y. Geometrically, the value of the partial derivative of z with respect to x at a point tells us the rate at which the surface is rising or falling at that point in the x direction . Imagine the positive x axis point east and the positive y axis pointing north, in the example above. We were standing on the mountain directly above the point in the x , y
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