Deck 15: Differentiation in Several Variables

Question 1

Let the functions and be defined as and Does the horizontal trace of in the plane intersect the vertical trace of in the plane If so, where do the traces intersect?

Answer 1

The traces intersect at the point

Question 2

Describe and sketch the level curves of the function

Answer 2

the hyperbolas

Question 3

Find the range of the function .

Answer 3

Question 4

Find the level curves of the function , and sketch a contour map.

Question 5

Find the domain of the function , and sketch it in the plane.

Question 6

Evaluate the limit or state that it does not exist.

A)
B)

Question 7

Find the level curves of the function , and sketch a contour map.

Question 8

Let be the function defined by Describe the shape of the level surface

Question 9

Evaluate the limit or state that it does not exist.

Question 10

Evaluate the limit or state that it does not exist.

A)
B)

Question 11

Evaluate the limit or state that it does not exist.

A)
B)

Question 12

Find the level curves of the function ,
and sketch a contour map.

Question 13

Evaluate the limit or state that it does not exist.

Question 14

Find the domain of the function and sketch it in the plane.

Question 15

Find the range of the function restricted to the domain .

Question 16

Find the domain of the function , and sketch it in the plane.

Question 17

Find the domain and range of the function .

Question 18

Find the domain and range of the function .

Question 19

Evaluate the limit or state that it does not exist.

A)
B)

Question 20

Let be the function .

A) Find .
B) Is continuous?

Question 21

If , then is:

A) .
B) .
C) .
D) .
E) none of the above.

Question 22

Let .
Which of the following statements is correct?

A) and are continuous.
B) is continuous, but is not.
C) is continuous, but is not.
D) and are not continuous.
E) is continuous at the origin, hence so are and .

Question 23

Compute the following limit or explain why it does not exist.

Question 24

Determine whether the following function is continuous.

Question 25

Which of the following statements hold for

A) is continuous, and the partial derivatives and exist everywhere.
B) is continuous, but the partial derivatives and do not exist at the origin.
C) is continuous and exists everywhere, but does not exist at the origin.
D) and do not exist at points and .
E) is not continuous at the origin.

Question 26

Let .
Compute the partial derivatives and at all the points where they exist.

Question 27

Which of the following functions satisfies the heat equation

A)
B)
C)
D)
E) None of the above.

Question 28

Find and given that .

Question 29

Let be the following function.

A) Determine whether is continuous at the origin.
B) Compute the partial derivatives and .

Question 30

Find the partial derivatives , and of the function .

Question 31

Let .
Which of the following statements is correct?

A) and are continuous.
B) is continuous but is not continuous at the origin.
C) is continuous but is not continuous at the origin.
D) Both and are not continuous at the origin.
E) Since is continuous, and are also continuous.

Question 32

Compute the partial derivatives and of the function .

Question 33

Determine whether the following function is continuous.

Question 34

Let , and compute and .

Question 35

Find all the functions such that the function is a solution of the differential equation .

Question 36

Define the function by Is continuous at the point

Question 37

The function satisfies which differential equation?

A)
B)
C)
D)
E) None of the above

Question 38

Let .
Define the function by Is continuous on the domain

Question 39

Determine whether the following function is continuous.

Question 40

Which of the following functions satisfies the differential equation

A)
B)
C)
D)
E) and

Question 41

Find the derivative of at the point with respect to the vector , where , , and .

Question 42

Find the linearization of at the point and use it to approximate

Question 43

Find the derivative of at the point in the direction of the vector , where , and .

Question 44

Find an equation of the tangent plane to the graph of the function at the point .

A)
B)
C)
D)
E) None of the above

Question 45

Find the linearization of at the point and use it to approximate .

Question 46

Let .

A) Find and , if they exist.
B) Use the definition of differentiability at a point to determine whether is differentiable at the origin.

Question 47

Find the linearization of at the point and use it to approximate

Question 48

Let .

A) Find the linearization of at the origin.
B) Estimate by and find the error using a calculator.

Question 49

Find and if .

Question 50

Find the linearization of at the point and use it to approximate .

Question 51

Let be the function defined by .

A) Determine whether is continuous.
B) Compute whether and exist.
C) Determine whether is differentiable at .

Question 52

Find the point on the surface where the tangent plane is parallel to the plane .

Question 53

Find the gradient of the function

Question 54

The temperature at the point in a room is given by A fly is standing on a table on the plane at the point .
Find the direction the fly should move in order to feel the maximum rate of increase in temperature:

A) if it flies.
B) if it is just walking on the table.

Question 55

Find the linearization of at the point and use it to approximate

Question 56

Let .

A) Is differentiable at the origin?
B) Does have a tangent plane at the origin? If so, find its equation.

Question 57

Which of the following functions is differentiable at the origin?

A)
B)
C)
D)
E) None of the above

Question 58

If and change by and , respectively, from and , then the approximate change in the angle is which of the following?

A)
B) 0.464
C) 0.003
D) 1.373
E) 0.002

Question 59

Find the tangent plane to the graph of the function at the point .

Question 60

The tangent plane to the surface is parallel to the plane and contains the point .
Find the value of .

Question 61

Given that , find and .

Question 62

The plane best fits the points and if the sum of at these points is minimized by and The plane that best fits the points , , , and is:

A) .
B) .
C) .
D) .
E) .

Question 63

Let Find if

Question 64

Let . Calculate , where , , and .

Question 65

The temperature at on a metal plate is .

A) A heat seeking insect is placed at the point . In what direction should it move in order to feel the greatest increase in heat? What is the rate of change of temperature in this direction?
B) The insect is placed at the point and is very hot. In what direction should it move in order to cool off at the fastest rate?
What is the rate of change of temperature in this direction?

Question 66

Let be differentiable, and let be the function .
Compute the partial derivatives , , and .

Question 67

Given that , compute the partial derivatives , , and .

Question 68

Which of the following statements is correct for the function and a nonzero vector

A)
B)
C)
D)
E) None of the above

Question 69

Find the critical points of for and analyze them using the second derivative test.

Question 70

Let .

A) Do and exist? If so, find them.
B) Use the definition of directional derivative to compute for .
C) Examine whether holds, and conclude the differentiability of at the origin.

Question 71

A metal plate is heated so that its temperature at a point is .
A bug is placed at the point .

A) The bug heads toward the point . What is the rate of change of temperature in this direction?
B) In what direction should the bug head in order to warm up at the fastest rate? Find the rate of change of temperature in this direction.
C) In what direction should the bug head in order to cool off at the fastest rate? Find the rate of change of temperature in this direction.

Question 72

Let Find if

Question 73

The function is defined near the point by the equation The maximum value of the directional derivative of (among all possible directions) is which of the following?

A)
B) 5
C) 3
D)
E) There is no maximum value.

Question 74

Find the tangent plane to the surface at the point on the surface.

Question 75

The plane best fits the points and if the sum of at these points is minimized by and Find the plane that best fits the points , and .

Question 76

Let .
Find the critical points of and analyze them using the second derivatives.

Question 77

Let be a differentiable function, and let be the function Compute at .

Question 78

A radioactive radiation with strength is suddenly discharged. A man standing at the point must run away, in the direction of maximum decrease of radiation.

A) What direction should he choose?
B) The man decided to run along the path . Find the directional derivative of in the direction of the path at .

Question 79

Let .

A) Find the parametrization for the plane using the parameters and .
B) Compute and .

Question 80

Given that find and