Problem 1

Find the equation of the tangent plane to at the point .

Problem 2

Find the implicit equation of the tangent plane (ensure the coefficient of is 1) and the parametric equation of the normal line to the surface at the point .

Problem 3

Consider the function , determine the gradient

Problem 4

Consider the function

(a)

Find

(b)

Find a function which has a zero level set which is equal to the graph of and such that the coefficient of in is 1.

(c)

Find the gradient of .

(d)

Use to find a vector normal to the graph of at the point .

(e)

Find an equation for the tangent plane to at the point .

Problem 5

If the gradient of is and the point lies on the level surface , find an equation for the tangent plane to the surface at the point .

Problem 6

Find level set at of the tangent plane where the coefficient of is 1 and the parametric equation of the normal line to the surface at the point .

Problem 7

Consider the surface

(a)

Find the unit normal vector to the surface at the point with positive first coordinate.

(b)

B. Find the equation of the tangent plane to the surface at the given point. Express your answer in the form , normalized so that .

Problem 8

A differentiable function has the properties that , , and . Find the equation of the tangent plane at the point on the surface where and .

Problem 9

Find the linear approximation to at the point .

Problem 10

An unevenly heated metal plate has temperature in degrees Celsius at a point . If , , and , estimate the temperature at the point .

Problem 11

Use a linear approximation to estimate the value of , and compare that estimate with the exact value and compute the percentage error.

Problem 12

Suppose is a function of three variables

(a)

Find the gradient of

(b)

Evaluate the gradient at the point

(c)

Find the rate of change of at in the direction of the vector .