Tangent Planes, Approximations, and Differentiability Homework

Directions: Write legibly and in pencil. Complete the homework on time and by yourself. For each problem, write the instructions, label the solution, show all steps, and write the final answer in a sentence. Do not turn in your scratch work. Staple your pages together, in the correct order, and use this page as a cover sheet.

(1) Determine the standard-form equation for the tangent plane to the surface given by the function at the point   

(2) Determine the standard-form equation for the tangent plane to the surface given by the function at the point   

(3) Determine the standard-form equation for the tangent plane to the surface given by the function at the point   

(4) Determine the total differential of the function

(5) Determine the total differential of the function

(6) Determine the total differential of the function

(7) Determine the total differential of the function

(8) Show that the function is differentiable on

(9) Show that the function is differentiable on

(10) Find an equation for each horizontal tangent plane to the surface

(11) Find the tangent plane to the given surfaces at the given point.

    (i) Determine the standard-form equation for the tangent plane to the surface at .

    (ii) Determine the standard-form equation for the tangent plane to the surface at .

(12) Determine the total differential of the functions

.

(13) Use linear approximation (differentials) to find approximate values for

and      

(14) Show that the functions and are differentiable.

(15) Show that if and are sufficiently close to zero and is differentiable at then  

;

and use this approximation for the expressions and around

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