Answer to Question #201386 in Calculus for Simphiwe Dlamini

Question

Consider the R² −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).

(b) Sketch the contour curve C together with the line L in R².

(c) Find an equation for the tangent plane V to the graph of f at (x,y) = (1,4).

Answer 1

(a) Here, slope of tangent at x=1 is:

$$C:xy=4\\\Rightarrow y=\dfrac{4}{x}\\\ \\\Rightarrow \dfrac{dy}{dx}=\dfrac{-4}{x^2}\\\ \\\Rightarrow \dfrac{dy}{dx}|_{x=1}=-4$$

$\therefore$ Equation of tangent line L to C at (1,4) is

$y-4=\dfrac{dy}{dx}|_{x=1}\times (x-1)\\\ \\\Rightarrow y-4=(-4)(x-1)\\\Rightarrow \boxed{y=-4x+8}$

(b)

(c) Here,

$f(x,y)=xy\\\Rightarrow f_x(x,y)=y\\\Rightarrow f_x(1,4)=4\ and\\f_y(1,4)=1$

Equation of tangent plane at (1,4) is

$z=f_x|_{(1,4)}(x-1)+f_y|_{(1,4)}(y-1)\\\Rightarrow z=4(x-1)+1(y-1)\\\Rightarrow z= 4x+y-5$

Hence, $\boxed{z-4x-y=-5}$ is equation of tangent plane V at (1,4)