Let fxy x2 3x xy 32y2 4y 1 a Find the equation of the tan

Solution

Let f(x,y) = x^2 -3x + xy + (3/2)y^2 -4y +1

take implicit derivative

0 = 2x - 3 + x y\' + y + 3y - 4 = 0

at x =2 and y =-1

0 = 4-3 + 2y\' -4-4 = 0

2y\' =8-1 = 7

y\' = slope = 7/2

(a) Find the equation of the tangent plane at x=2, y=-1

y+1 = 7/2(x-2)

2y+2 = 7x-14

(b) Find the equation of the normal line at x = 2, y=-1

y+1 = -2/7(x-2)

7y+7 = -2x+4