- C Let F X Y Y2 2x22x2 3xy Xy0 If X Y 0 0 If X Y 0 0 I Find Fx And Fy When X Y 0 0 Ii Find Fx 1 (38.24 KiB) Viewed 37 times
c. Let f(x,y)=⎩⎨⎧y2+2x22x2+3xy+xy0 if (x,y)=(0,0) if (x,y)=(0,0) i. Find fx and fy when (x,y)=(0,0) ii. Find fx(0,0) and fy(0,0) iii. Show that fyx(0,0)=fxy(0,0) 2. a. Find ∇f for f(x,y,z)=sin(xyz2)+xy at P=(1,−2,4) b. Suppose that the velocity of the point (x,y,z) in space is given by V(x,y,z)= 1+2x2++y2+3z270 where V is measured in m/s and x,y,z in meters. i. In which direction does the velocity increase test of the point (1,−1,3) ? ii. What is the maximum rate of increase? c. Use the Mean Value Theorem to show that ∥lnp−lnq∥≤3∥p−q∥ for 1/8≤ z<w≤8 d. Determine the first(linear) and second(quadratic) order Taylor Polynomial approximations for f(x,y)=2xey+xy2 near the point (1,0). e. Show that following functions are functionally dependent by using Jacobian matrix. f(x,y)=x2+2y3g(x,y)=x−y 3. a. i. Find a symmetric matrix A such that Q(x)=xAxτ for each x∈R2. Let Q(x)=x12+5x1x2+x22. ii. Determine whether the above matrix is PD, PSD, ND or NSD (Definiteness).