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Example text
G'(O) = (0,0) . (a, b) O. This verifies, as required, that the chain rule does no t a pply to this function. = ow/ax on the left-hand side means the partial derivative of w(x , y,g(x , y)) with respect to x, holding y constant. T he term ow/ox on the right-hand side means the partial derivative of w(x, y , z ) with respect to x holding yand z constant . These two terms are not equal because different independent variables are held constant. Thus, the reasoni ng is indeed flawed . 24. 6: GRADIENTS A ND D IRE C TIO N A L D ERIVAT IVES GOALS 1.
18sand 8v 18t. 8. (5, 2, - 1 + h, 1). h h -+ O (b) Is it possible to define g(O, 0) so that g(x , y) 1R2? Explain why or why not . = (x 2 + y2) /( x 2 + y) is continuous on all 9. Let u (x, y ) = 2x + y2 + eX. At (1,1) show that u increases faster in the direction parallel the x axis t han in th direction parallel to the 11 axis. 10. A recent survey showed that patient satisfaction, s, at a pharmacy depended mainly up.. three factors - t he waiting time in minutes (t) , the patient's percei v d degree of illness (i) a scale from 0 to 10, and t he dollar cost of the prescription (c) .
T hese are high r-order derivatives, such as second and th ird deriva tives. With several variables, higher-order deri vat ives may be taken with respect to different :::y variables. The notation means (:x ) (~~) , which is also denoted fy x . 2. EquaLity of m ixed partials. If the second partial deri vatives are cont inuous, t hen an iterated partial derivative may be computed in any order. 3. W arning. Note that the theorem on equality of mixed partials requires continuous partial derivatives.