Calculus: The Language of Change

Exercise Set 7.3
Implicit Tangents and Derivatives

by John Robeson

Exercise 1) Implicit Differentiation

Use implicit differentiation to find the equation of the tangent line to
		(((5 x))^2) - (((4 y))^2) = (3^2)
at the point (x,y) = (1,1).  Use the local coordinates (dx,dy) centered at (x,y) = (1,1).

Solution to Exercise 1)

Evaluate the cell below for the solution.

\(Clear[f, x, y, dx, dy]; 
f[x_, y_] := \((5\ x)\)\^2 - \((4\ y)\)\^2; 

    Print["The equation ", f[x, y], "= 9"]; 


    differential\  = \ Dt[f[x, y]] /. {Dt[x] -> dx, Dt[y] -> dy}; 

    Print["has differential  ", differential, " = 0"]; 

x\  = \ 1; 
    \ y\  = \ 1; 

    Print["at (x,y) = (", x, ",", y, "), f[x,y] = c = ", 
      f[x, y]]; 

    Print["and the differential is  ", differential, " = 0"]; 


    curve\  = \ 
      ContourPlot[
        f[x + dx, y + dy] - f[x, y], {dx, \(-3\), 7}, {dy, \(-7\), 3}, 
        ContourShading -> False, \ Contours -> {0}, PlotPoints -> 30, 
        ContourStyle -> {Red}, FrameLabel -> {"dx", "dy"}, 
        DisplayFunction -> Identity]; 


    tanLine\  = \ 
      ContourPlot[differential, {dx, \(-3\), 3}, {dy, \(-3\), 3}, 
        ContourShading -> False, \ Contours -> {0}, ContourStyle -> {Blue}, 
        FrameLabel -> {"dx", "dy"}, DisplayFunction -> Identity]; 


    Show[tanLine, curve, Graphics[{Brown, PointSize[0.05], Point[{0, 0}]}], 
      AspectRatio -> Automatic, DisplayFunction -> $DisplayFunction]; \)