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 w pictured 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 2 d fdx x Y O xo Or af l x 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 a t P is the partial derivative o f f with respect t o y the notations are The partial derivatives d f l d x 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 18 02 NOTES 2 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 a t 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 this will hold if the plane has the same slopes in the i and j directions as the surface does Using these two conditions it is easy to find the equation of the tangent plane The general equation of a plane through XO yo wo is Assume the plane is not vertical then C w wo getting 3 w wo a x 0 0 so we can divide through by C and solve for b y yo a AIC b B I C The plane passes through xo yo wo what values of the coefficients a and b will make it also tangent to the graph there We have a slope of plane 3 in the i direction slope of graph in the i direction by putting y yo in 3 by ii above by the definition of partial derivative similarly Therefore the equation of the tangent plane to w f x y at 0 yo is Your book has examples of calculating tangent planes using 4 3 The approximation formula The most important use for the tangent plane is to give an approximation that is the basic formula in the study of functions of several variables almost everything follows in one way or another from it The intuitive idea is that if we stay near 0 yo wo the graph of the tangent plane 4 will be a good approximation to the graph of the function w f x Y Therefore if the point x y is close to 0 yo f x y wo g 2 0 0 height of graph height of tangent plane 2 Y YO 0 TA T H E TANGENT APPROXIMATION 3 The function on the right side of 5 whose graph is the tangent plane is often called the linearization of f x y a t xo yo it is the linear function which gives the best approxima tion to f x y for values of x y close to 50 YO An equivalent form of the approximation 5 is obtained by using A notation if we put then 5 becomes This formula gives the approximate change in w when we make a small change in x and y We will use it often The analogous approximation formula for a function w f x y z of three variables would be 7 AW M AX E AY AZ if Ax A A i Z0 Unfortunately for functions of three or more variables we can t use a geometric argument for the approximation formula 7 for this reason it s best to recast the argument for 6 in a form which doesn t use tangent planes and geometry and therefore can be generalized to several variables This is done at the end of this Chapter TA for now let s just assume the truth of 7 and its higher dimensional analogues Here are two typical examples of the use of the approximation formula Other examples are in the Exercises In the rest of your study of partial differentiation you will see how the approximation formula is used to derive the important theorems and formulas Example 1 Give a reasonable square centered at 1 I over which the value of w x3y4 will not vary by more than f 1 Solution We use 6 We calculate for the two partial derivatives and therefore evaluating the partials at 1 l and using 6 we get Thus if 1 Ax1 5 Ol and Ay 1 Ol we should have which is within the bounds So the answer is the square with center at 1 l given by 18 02 NOTES 4 E x a m p l e 2 The sides a b c of a rectangular box have lengths measured to be respectively 1 2 and 3 To which of these measurements is the volume V most sensitive Solution V abc and therefore by the approximation formula 7 AV bcAa acAb abAc z 6Aa 3Ab 2Ac NN a t 1 2 3 thus it is most sensitive to small changes in side a since Aa occurs with the largest coefficient That is if one at a time the …
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