Linear Approximations of Functions

1. Find the linear approximation of the function near the point (1,1).
2. Find the tangent plane to the surface at the point (1,1,4).
3. What is the relationship in general between the linear approximation to a function f(x,y) near a point (a,b) and the tangent plane to the surface z=f(x,y) at the point (a,b,f(a,b).
4. Find the normal line to the surface at (1,1,4).
5. Display the tangent plane, the surface, and the normal line, near the point (1,1,4). Adjust various parameters to get an informative plot.
6. Plot the function giving the percentage error in the linear approximation over the rectangular region 0<x<2, 0<y<2.
   ((1,1) over which the maximum percentage error is at most 1%.
   • Find an even smaller rectangle, centered at (1,1) over which the maximum percentage error is at most 0.01%.
   • Compute the value of f(x,y) and the linear approximation function at the point (1.01,0.99). What is the percentage error in the approximation?

 
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