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
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.]