Question

Problem 8. True or False (You do not need to explain your answer but must indicate clearly “T” or “F.” You will lose points in your T or F is not clearly indicated.) (a)_ Given z=f(x,y), fxy(x, y) is always equal to fxx (x, y). (b)_ Tangent plane to z= f(x, y) at (x,,y.) always approximates the value of z near (xo, y.). (c) __ If fx and f, exists, then z= f(x,y) is differentiable. (d) __If f{(1,2)=0, then fx, (1, 2) = 0. (e) Ixy The tangent plane to f(x,y)={ 2 if (x, y)+(0,0) x² + y2 at the origin is the x - y plane. if (x, y) =(0,0)

Answer 1

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Solutions th) TRUE : home thean One The order of differentiasitily commutes as per се как clairaut's сосем 4 11 4 5* * Continuous. to. Tangent plane to the sunfare a=fleng TRUE Û about (xo,go) is: Z = f(x0,40) + of tro, yo)'(x-70) + of (xsyo) .(y-yo). so. It approximate the value of Z about (20,%) S. TRUE f(x,y) must be continuous and fx and - fy must exist at(ay). Jets, FALLE! Counter- example, Suppose: f(x,y) = x²-ry so that fn (4) - of a 2x=y Af (1,2): f (1,3)+227 -2 2-20. Bul? fayden tommer 27 (237) 2-1 70 So, finy (1,2) = -1 40 e FALSE: flory) is NOT differenbable as limit itself does nof exist. let us make (x,y) pase through the line ya mx so that if appwaches (0,0) alorg Too yame, m -they: lin. f(x 1. hey xxmx (x,y) 100 xxo x²+mber itme which depends upon the value of an and linit is not unique. - linif cloesnof exist. → f(x,y) is not differentialle > fx,fy does not & Tangent plane does not exist. exist. WWWWWWWWWWWWWWW