tangent_approx

tangent_approx - MIT OpenCourseWare http/ocw.mit.edu 18.02...

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MIT OpenCourseWare http://ocw.mit.edu 18.02 Multivariable Calculus Fall 2007 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms .
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TA. The Tangent Approximation 1. Partial derivatives Let w = f (x, y) be a function of two variables. Its graph is a surface in xyz-space, as pictured. w Fix a value y = yo and just let x vary. You get a function of one variable, (1) w = f (x, yo), the partial function for y = yo. Its graph is a curve in the vertical plane y = yo, whose slope at the point P where x = xo is given by the derivative d (2) f dx x Y O , Or afl . x xo (XO,YO) We call (2) the partial derivative of f with respect to x at the point (xo, yo); the right side of (2) is the standard notation for it. The partial derivative is just the ordinary derivative of the partial function - it is calculated by holding one variable fixed and differentiating with respect to the other variable. Other notations for this partial derivative are the first is convenient for including the specific point; the second is common in science and engineering, where you are just dealing with relations between variables and don't mention the function explicitly; the third and fourth indicate the point by just using a single subscript. Analogously, fixing x = xo and letting y vary, we get the partial function w = f (xo, y), whose graph lies in the vertical plane x = xo, and whose slope at P is the partial derivative off with respect to y; the notations are The partial derivatives d f ldx and d f ldy depend on (xo, yo) and are therefore functions of x andy. Written as aw/dx, the partial derivatrive gives the rate of change of w with respect to x alone, at the point (xo, YO): it tells how fast w is increasing as x increases, when y is held constant. For a function of three or more variables, w = f (x, y, z, . . .), we cannot draw graphs any more, but the idea behind partial differentiation remains the same: to define the partial derivative with respect to x, for instance, hold all the other variables constant and take the ordinary derivative with respect to x; the notations are the same as above:
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2 18.02 NOTES Your book has examples illustrating the calculation of partial derivatives for functions of two and three variables. 2. The tangent plane. For a function of one variable, w = f (x), the tangent line to its graph at a point (XO, WO) is the line passing through (xo, wo) and having slope ($1 . For a function of two variables, w = f (x, y), the natural analogue is the tangent plane to the graph, at a point (xo, yo, wo). What's the equation of this tangent plane? Referring to the picture of the graph on the preceding page, we see that the tangent plane (i) must pass through (xo , yo, WO) , where wo = f (xo , YO); (ii) must contain the tangent lines to the graphs of the two partial functions
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This note was uploaded on 05/04/2011 for the course MATH 18.02 taught by Professor Auroux during the Spring '08 term at MIT.

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tangent_approx - MIT OpenCourseWare http/ocw.mit.edu 18.02...

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