Math 311

These problems are provided to help you study. The presence of a
problem on this handout does not imply that there *will* be a
similar problem on the test. And the absence of a topic does not
imply that it *won't* appear on the test.

1. Find the domain of the function .

2. Find the domain and range of .

3. Compute .

4. Show that is undefined.

5. Compute by converting to polar coordinates.

6. Show that is defined and find its value.

7. Define by

Determine whether f is continuous at .

8. Compute the following partial derivatives:

(a) and .

(b) and .

(c) , if

(d) , if

9. Let

Construct the Taylor series for f at the point , writing terms through the order.

10. For a differentiable function ,

Use a -degree Taylor approximation at to approximate .

11. Find the tangent plane and the normal line to the surface

12. Find the tangent plane to the surface

13. Use a linear approximation to at the point to approximate .

14. Let .

(a) Find a unit vector at which points in the direction of most rapid increase.

(b) Find the rate of most rapid increase at .

15. Find the gradient of and show that it always points toward the origin.

16. Let . Find the directional derivative of f at the point in the direction of the vector .

17. Find the rate of change of at the point in the direction toward the origin. Is f increasing or decreasing in this direction?

18. The rate of change of at is 2 in the direction
*toward* and is in the
direction of the vector . Find .

19. Calvin Butterball sits in his go-cart on the surface

If his go-cart is pointed in the direction of the vector , at what rate will it roll downhill?

20. Find the tangent plane to at the point .

21. Suppose that and are given by

Find and .

22. Let r and be the standard polar coordinates variables. Use the Chain Rule to find and , for .

23. Suppose and , , . Use the Chain Rule to write down an expression for .

24. Suppose that , , and . Use the Chain Rule to find an expression for .

25. Locate and classify the critical points of

26. Locate and classify the critical points of

27. Find the critical points of

You do not need to classify them.

28. Find the points on the sphere which are closest to and farthest from the point .

29. A rectangular box (with a bottom and a top) is to have a total surface area of , where . Show that the box of largest volume satisfying this condition is a cube with sides of length c.

30. (a) Find the critical points of

(b) Express w as a function of x and y by eliminating z, then consider the behavior of w for . Explain why the critical points in (a) can't give absolute maxes or mins.

31. Find the largest and smallest values of subject to the constraint .

1. Find the domain of the function .

Since the denominator of the fraction can't be 0, the domain is

It consists of all points except those lying on the lines or .

2. Find the domain and range of .

Since the expression inside the square root must be positive, the function is defined for . Therefore, the domain is the set of points such that --- that is, the interior of the cylinder of radius 1 whose axis is the z-axis. (There are no restrictions on z.)

To find the range, note that . Also,

Hence,

This shows that every output of f is greater than or equal to 1.

On the other hand, suppose . Then

This shows that every number greater than or equal to 1 is an output of f.

Hence, the range of f is the set of numbers w such that .

3. Compute .

4. Show that is undefined.

If you approach along the x-axis ( ), you get

If you approach along the line , you get

Since the function approaches different values as you approach in different ways, the limit is undefined.

5. Compute by converting to polar coordinates.

Set . As , I have . So

6. Show that is defined and find its value.

Therefore,

Hence,

7. Define by

Determine whether f is continuous at .

Since ,

Therefore, f is not continuous at .

8. Compute the following partial derivatives:

(a) and .

(b) and .

(c) , if

(d) , if

(a)

(b)

(c)

(d)

9. Let

Construct the Taylor series for f at the point , writing terms through the order.

At ,

The series is

10. For a differentiable function ,

Use a -degree Taylor approximation at to approximate .

The -degree Taylor approximation is

Hence,

11. Find the tangent plane and the normal line to the surface

When ,

The point of tangency is .

The normal vector is

The normal line is

The tangent plane is

12. Find the tangent plane to the surface

and give the point of tangency: .

Next,

Thus,

The normal vector is given by

The tangent plane is

13. Use a linear approximation to at the point to approximate .

, so the point of tangency is . A normal vector for a function is given by

Hence, the tangent plane is

Substitute and :

14. Let .

(a) Find a unit vector at which points in the direction of most rapid increase.

(b) Find the rate of most rapid increase at .

(a) Find a unit vector at which points in the direction of most rapid increase is .

(b) Find the rate of most rapid increase at is .

15. Find the gradient of and show that it always points toward the origin.

is the * radial vector* from the
origin to the point . Since is a negative
multiple of this vector always points *inward*
toward the origin.

16. Let . Find the directional derivative of f at the point in the direction of the vector .

Hence,

17. Find the rate of change of at the point in the direction toward the origin. Is f increasing or decreasing in this direction?

First, compute the gradient at the point:

Next, determine the direction vector. The point is , so the direction toward the origin is

Make this into a unit vector by dividing by its length:

Finally, take the dot product of the unit vector with the gradient:

f is increasing in this direction, since the directional derivative is positive.

18. The rate of change of at is 2 in the
direction *toward* and is in the
direction of the vector . Find .

The direction from *toward* the point
is given by the vector . This vector has length 4, so

The vector has length 5, so

Thus, .

I have two equations involving and . Solving simultaneously, I obtain and . Hence, .

19. Calvin Butterball sits in his go-cart on the surface

If his go-cart is pointed in the direction of the vector , at what rate will it roll downhill?

The rate at which he rolls is given by the directional derivative. The gradient is

Since ,

20. Find the tangent plane to at the point .

Write . (Take the original surface and drag everything to one side of the equation.) The original surface is , so it's a level surface of w. Since the gradient is perpendicular to the level surfaces of w, it follows that must be perpendicular to the original surface.

The gradient is

The vector is perpendicular to the tangent plane. Hence, the plane is

21. Suppose that and are given by

Find and .

22. Let r and be the standard polar coordinates variables. Use the Chain Rule to find and , for .

23. Suppose and , , . Use the Chain Rule to write down an expression for .

This diagram shows the dependence of the variables.

There are 3 paths from u to t, which give rise to the 3 terms in the following sum:

24. Suppose that , , and . Use the Chain Rule to find an expression for .

By the Chain Rule,

Next, differentiate with respect to t, applying the Product Rule to the terms on the right:

Since and are functions of x and y, I must apply the Chain Rule in computing their derivatives with respect to t. I get

25. Locate and classify the critical points of

Set the first partials equal to 0:

Solve simultaneously:

Test the critical points:

26. Locate and classify the critical points of

Set the first partials equal to 0:

Solve simultaneously:

\overfullrule=0 pt

Test the critical points:

27. Find the critical points of

You do not need to classify them.

Set the first partials equal to 0:

Solve simultaneously:

28. Find the points on the sphere which are closest to and farthest from the point .

The (square of the) distance from to is

The constraint is .

The equations to be solved are

Note that if in the first equation, the equation becomes , which is impossible. Therefore, , and I may divide by x.

Solve simultaneously:

Test the points:

is closest to and is farthest from .

29. A rectangular box (with a bottom and a top) is to have a total surface area of , where . Show that the box of largest volume satisfying this condition is a cube with sides of length c.

Suppose the dimensions of the box are x, y, and z. Then the volume is

The surface area is

The constraint is

Set up the multiplier equation:

This gives the equations

Note that satisfies the constraint and gives a volume of . Thus, the solution to the problem certainly has . If any of x, y, or z is 0, the volume is 0, which is not a max. So I may assume .

Note that this also implies that , so I may divide by .

Now solve the equations:

The critical point is , which is a cube with sides of length c.

30. (a) Find the critical points of

(b) Express w as a function of x and y by eliminating z, then consider the behavior of w for . Explain why the critical points in (a) can't give absolute maxes or mins.

The constraint is

Set up the multiplier equation:

This gives the equations

Solve the equations:

Test the points:

(b) Solving the constraint for z gives . Then

Consider the behavior of w along the line :

The factor is positive. As , the term becomes large and negative, so . As , the term becomes large and positive, so .

This means that you can find values of x, y, and z satisfying the constraint for which w is arbitrarily big or small. Hence, the critical points found in (a) can't be absolute maxes or mins.

31. Find the largest and smallest values of subject to the constraint .

The constraint is .

Set up the multiplier equation:

This gives two equations:

Solve those equations simultaneously with the constraint:

Test the points:

*To be conscious that you are ignorant is a great step to
knowledge.* - *Benjamin Disraeli*

Copyright 2020 by Bruce Ikenaga