Question

9. (Sections 2.12, 3.2, 7.5, 7.8) Consider the R 2 − R function f defined by f (x, y) = xy and let C be the contour curve of f at level 4. (a) Find a Cartesian equation for the tangent L to C at (x, y) = (1, 4). (4) (b) Sketch the contour curve C together with the line L in R 2 . (3) (c) Find an equation for the tangent plane V to the graph of f at (x, y) = (1, 4). (4) Hints: • Study Definition 3.2.5 and Remarks 3.2.6 again and note that the contour curves of f lie in the XY-plane. Also study the more general Definition 7.8.4 and read Remarks 7.8.5. • Use Theorem 7.8.1 to find a vector n which is perpendicular to L and then use Definition 2.12.1 to find an equation for L, OR use Definition 7.8.6 directly. Note that in the case of n = 2, the formula in Definition 7.8.6 gives a Cartesian equation for the tangent to a contour curve. • Study Definition 7.5.4 and use the formula (7.2) to find an equation for V OR define a function g for which the graph of f is a contour surface

Answer 1

$xy=4$

a)

$y-y_0=f'(x_0)(x-x_0)$

$f'(x)=-4/x^2$

$f'(x_0)=f'(1)=-4$

Equation of the tangent:

$y-4=-4(x-1)$

b)

c)

Equation of the tangent plane:

$z-f(x_0,y_0)=f_x(x_0,y_0)(x-x_0)+f_y(x_0,y_0)(y-y_0)$

$f(1,4)=4$

$f_x=y,\ f_y=x$

$z-4=4(x-1)+y-4$

$z=4(x-1)+y$