Answer to Question #201386 in Calculus for Simphiwe Dlamini

Question #201386

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).


1
Expert's answer
2021-06-01T13:00:46-0400

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


C:xy=4y=4x dydx=4x2 dydxx=1=4C: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

y4=dydxx=1×(x1) y4=(4)(x1)y=4x+8y-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)=xyfx(x,y)=yfx(1,4)=4 andfy(1,4)=1f(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=fx(1,4)(x1)+fy(1,4)(y1)z=4(x1)+1(y1)z=4x+y5z=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, z4xy=5\boxed{z-4x-y=-5} is equation of tangent plane V at (1,4)




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