Tangent Planes and Linear Approximations: Examples

1. Find the Tangent plane of the equation below at the point (1,2,2). Also demonstrate that the tangent plane is a good linear approximation of the function using the point (1.1,2.1).  Equation: 4*x^2*y-3*y

Anwser:

First, find f_x and f_y

>	diff(4*x^2*y-3*y,x);

>	diff(4*x^2*y-3*y,y);

Now plug everything into the equation and solve for z to find the tangent plane:

>	solve(z-2=(8*1*2)*(x-1)+(4*1^2-3)*(y-2),z);

Now, put the point (1.1,2.1) into both the original equation and the tangent plane:

>	eval(-16+16*x+y,[x=1.1,y=2.1]);

>	eval((4*x^2*y)-3*y,[x=1.1,y=2.1]);

2. Use differentials to find the change in the following function from (1,1) to (1.01, 0.97):  sqrt(4-x^2-y^2)

Anwser

First, find delta x and delta y                     Delta x = .01    Delta y = -.03

Now, use the definition of the differenital to find the change after first finding f_x and f_y then use eval to put in all of the values:

>	diff(sqrt(4-x^2-y^2),x);

>	diff(sqrt(4-x^2-y^2),y);

>	eval((-x/(4-x^2-y^2)^(1/2))*(.01)+(-y/(4-x^2-y^2)^(1/2))*(-.03),[x=1,y=1]);