Local Linearity 3D Applet Worksheet

This worksheet is designed to build familiarity with the use of the tangent plane as a linear approximation of a function.  This can be done with the applet obtained by following either of the following links:
http://www.slu.edu/classes/maymk/banchoff/LocalLinearity3d.html
Spread windows where you can see them all.  Increase the size of the control panel so that you can see all the readout boxes.

1) The applet opens with the default function f(x,y)=sin(x*y) centered at the point (2, .5) with del set at 1.

• Find  the partials f_x and f_y at (x0,y0)=(2, 0.5) and give the equation of the tangent plane.
• Evaluate f(2.1, 0.4) and compare to the value obtained from the linear approximation.  (Drag the point P1 in the Zoom Graph window until (delx,dely)=(0.,-0.1).)
• With the controls on del, zoom in until the graphs of the function and its tangent plane are indistinguishable to your eye.  What is del?  What is the error term if P1 is moved to the corner of the Zoom Graph plot?
• Zoom out to del = 0.2.  Move P1 around.  Describe the region where the tangent plane is a good approximation.  Move (x0,y0) to (2.5, 2.5) and tell how the size and shape of the good region of approximation has changed.
  

2) Linear approximation is most useful when we have a point P0=(x0,y) and a function f, where f and both of its partials are easy to compute at P0, and we are really interested in f(P1) where P1 is close to P0.

• Estimate (80)^(1/2)+(65)^(1/3).  [You may want to use f(x,y)=x^(1/2)+y^(1/3) which is easy to compute at (81, 64).]  Find the error in your estimate an plot the function and tangent plane to see how reasonable it looks.
• Estimate sin(3)+tan(3.2).  [Trig functions and their derivatives are easy to estimate at the value π.]
  

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Last updated By Mike May, S.J., September 23, 2006.