Math 161

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 point(s) on closest to the point .

2. Calvin Butterball wants to fence in two equal-size rectangular pens in his yard for his pet fish. (Calvin does not have much luck with pets, for some reason.) As shown in the picture below, one side of each pen will be bounded by an existing stone wall (and will therefore not require any fence).

If Calvin has 300 feet of fence, what should the dimensions of the pens be to maximize the total area?

3. A rectangular box with a bottom and a top consists of two identical partitions which share a common wall. Each partition has a square bottom. If the total volume of the box (i.e the sum of the volumes of the two partitions) is 6272 cubic inches, what dimensions give the box which has the smallest total surface area?

4. A cylindrical can with a top and a bottom is to have a volume of cubic inches. The material for the top and bottom costs 10 cents per square inch, while the material for the sides costs 3 cents per square inch. What dimensions yield a can which costs the least?

5. Calvin Butterball sits in his rowboat 9 miles from a long straight shore. Phoebe Small waits in a car at a point on shore 15 miles from the point on the shore closest to Calvin. Calvin rows to a point on the shore, then runs down the shore to the car.

(Then they drive to the shopping mall, where they purchase two rolls
of duct tape,a tub of margarine, a Led Zeppelin T-shirt, two cans of
*Red Bull*, a copy of *Mother Earth News*, a bowling
ball, a metric hex key set, an ionic air purifier, three
*Cinnabons*, leather pants, a 120 mm case fan, curly fries,
and a bazooka.)

If Calvin can row at 4 miles per hour and run at 5 miles per hour, at what point on shore should he land in order to minimize his total travel time to the car?

6. A rectangular box with a square bottom * and no
top* is made with 972 square inches of cardboard. What values of
the length x of a side of the bottom and the height y give the box
with the largest volume?

7. A rectangular poster has a total area of 288 square inches. The poster consists of a rectangular printed region, surrounded by margins 1 inch wide on the top and bottom and 2 inches wide on the left and right. What dimensions for the printed region maximize the area of the printed region?

8. Compute .

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40. Compute .

41. Use a calculating device to approximate the following sum to at least three decimal places:

42. Write the series in summation form:

(a) .

(b) .

43. (a) Express the following sum in terms of n: .

(b) Find the exact value of .

44. (a) Verify that

(b) Use the result of (a) to evaluate .

45. Approximate using 10 rectangles of equal width and using the left-hand endpoint of each subinterval to obtain the rectangles' heights.

46. Approximate using 10 rectangles of equal width and using the right-hand endpoint of each subinterval to obtain the rectangles' heights.

47. Approximate using 10 rectangles of equal width and using the midpoint of each subinterval to obtain the rectangles' heights.

48. Some values for a function are shown below.

(a) Approximate using 5 rectangles of equal width and using the left-hand endpoints to obtain the rectangle heights.

(b) Approximate using 10 rectangles of equal width and using the right-hand endpoints to obtain the rectangle heights.

49. Compute by writing the integral as the limit of a rectangle sum.

50. (a) Given that , what is ?

(b) Compute .

51. Compute .

52. Compute .

53. Compute .

54. Suppose that , , and . Find .

55. Find functions and such that the antiderivative of
is *not* equal to the antiderivative of
times the antiderivative of .

56. Compute .

57. Compute the exact value of .

58. Compute the exact value of .

Hint: Complete the square in x.

59. Find the total area of the region bounded by and the x-axis.

60. Find the area of the region bounded by

61. Find the area of the region in the first quadrant bounded on the left by , on the right by , and below by the x-axis.

62. Find the area of the region bounded by the graphs of and .

63. Find the area of the region between and , from to .

64. (a) Compute .

(b) Compute .

(c) Compute .

65. (a) Prove that .

(b) Use the Integral Mean Value Theorem to estimate .

1. Find the point(s) on closest to the point .

*Draw a picture. Label the things that are relevant to the
problem.*

*Write down an expression for the thing you're trying to maximize
or minimize.*

The distance from to is

A distance is smallest exactly when its square is smallest. So we can work with the square of the distance instead:

This has the advantage of removing the square root and making it easier to differentiate.

*Look at the extreme cases to determine any endpoints.*

x can be as small as 0, but it can be arbitrarily large. That is, .

Therefore,

for .

The minimum does *not* occur at . To show that is an absolute min, notice that . Therefore, is a local min. It is the
only critical point, so it is an absolute min.

The closest point to on the curve is the point .

2. Calvin Butterball wants to fence in two equal-size rectangular pens in his yard for his pet fish. (Calvin does not have much luck with pets, for some reason.) As shown in the picture below, one side of each pen will be bounded by an existing stone wall (and will therefore not require any fence).

If Calvin has 300 feet of fence, what should the dimensions of the pens be to maximize the total area?

The area is . The amount of fence is , so . Therefore,

The endpoints are and (i.e. ).

The derivative is

for .

gives the absolute max; .

3. A rectangular box with a bottom and a top consists of two identical partitions which share a common wall. Each partition has a square bottom. If the total volume of the box (i.e the sum of the volumes of the two partitions) is 6272 cubic inches, what dimensions give the box which has the smallest total surface area?

Suppose the square base of a partition has sides of length x, and let y be the height of the box.

The total surface area is

The volume is

Plug this into A and simplify:

The only restriction on x is .

The derivatives are

is defined for all . Set and solve:

The corresponding value for y is .

Hence, is a local min. Since it's the only critical point, it's an absolute min.

4. A cylindrical can with a top and a bottom is to have a volume of cubic inches. The material for the top and bottom costs 10 cents per square inch, while the material for the sides costs 3 cents per square inch. What dimensions yield a can which costs the least?

Let r be the radius of the can, and let h be the height.

The total cost is the cost of the sides plus the cost of the top and bottom:

The volume is , so

Plug into C and simplify:

The only restriction on r is that . Since I don't have two endpoints, I'll use the Second Derivative Test. Differentiate:

Find the critical points:

This gives . Now

Hence, is a local min. Since it's the only critical point, it must give an absolute min.

5. Calvin Butterball sits in his rowboat 9 miles from a long straight shore. Phoebe Small waits in a car at a point on shore 15 miles from the point on the shore closest to Calvin. Calvin rows to a point on the shore, then runs down the shore to the car.

(Then they drive to the shopping mall, where they purchase two rolls
of duct tape,a tub of margarine, a Led Zeppelin T-shirt, two cans of
*Red Bull*, a copy of *Mother Earth News*, a bowling
ball, a metric hex key set, an ionic air purifier, three
*Cinnabons*, leather pants, a 120 mm case fan, curly fries,
and a bazooka.)

If Calvin can row at 4 miles per hour and run at 5 miles per hour, at what point on shore should he land in order to minimize his total travel time to the car?

Let x be the distance from the point on shore closest to Calvin to the point where he lands. The distance that he rows is , and the distance that he runs is . Since the time elapsed is equal to the distance divided by the speed, his total travel time is

(The first term is his rowing distance divided by his rowing speed, and the second term is his running distance divided by his running speed.)

The endpoints are (where he rows to the nearest point on the shore) and (where he rows directly to Phoebe).

Differentiate:

Find the critical points by setting :

(I omitted because it isn't in the interval .)

Test the critical point and the end points:

He should row to a point 12 miles from the point closest to shore to minimize his travel time.

6. A rectangular box with a square bottom * and no
top* is made with 972 square inches of cardboard. What values of
the length x of a side of the bottom and the height y give the box
with the largest volume?

The volume is

The area of the 4 sides is , and the area of the bottom is . So

Solving for y gives

Plug this into V and simplify:

Note that , since plugged into gives a contradiction. So the only restriction on x is that .

Since x is not restricted to a closed interval , I'll use the Second Derivative Test.

Compute the derivatives:

Find the critical points:

Since x is a length, it must be positive, so . This gives

Plug into the Second Derivative:

is a local max, but it's the only critical point, so it's an absolute max.

7. A rectangular poster has a total area of 288 square inches. The poster consists of a rectangular printed region, surrounded by margins 1 inch wide on the top and bottom and 2 inches wide on the left and right. What dimensions for the printed region maximize the area of the printed region?

The x be the width of the printed region, and let y be the height. The area of the printed region is

The total area of the poster is 288. The width is and the height is , so

Substituting this in A, I get

The extreme cases are and ; plugging into gives . So the endpoints are and .

Set and solve for x:

gives , but a width can't be negative. gives . Plugging this into gives .

gives an absolute max. The area is maximized when and .

8. Compute .

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29. Compute .

The Product Rule says that

Hence,

30. Compute .

31. Compute .

32. Compute .

33. Compute .

34. Compute .

35. Compute .

36. Compute .

37. Compute .

In the first step, I used the double angle formula

38. Compute .

39. Compute .

I multiply the two terms out, using the rule :

40. Compute .

41. Use a calculating device to approximate the following sum to at least three decimal places:

42. Write the series in summation form:

(a) .

(b) .

(a)

(b)

43. (a) Express the following sum in terms of n: .

(b) Find the exact value of .

(a) Use the formulas

Therefore,

(b)

44. (a) Verify that

(b) Use the result of (a) to evaluate .

(a) Adding the fractions on the right over a common denominator, I have

(b)

45. Approximate using 10 rectangles of equal width and using the left-hand endpoint of each subinterval to obtain the rectangles' heights.

The width of a rectangle is . The left-hand endpoints are

To do the sum on a TI calculator, the calculator command is

* sum(seq((sin(x))/x, x, 1, 4.6, 0.4)) * 0.4*

The answer is .

46. Approximate using 10 rectangles of equal width and using the right-hand endpoint of each subinterval to obtain the rectangles' heights.

The width of a rectangle is . The left-hand endpoints are

To do the sum on a TI calculator, the calculator command is

* sum(seq((sin(x))/x, x, 1.4, 5, 0.4)) * 0.4*

The answer is .

47. Approximate using 10 rectangles of equal width and using the midpoint of each subinterval to obtain the rectangles' heights.

The width of a rectangle is . The midpoints are

To do the sum on a TI calculator, the calculator command is

* sum(seq((sin(x))/x, x, 1.2, 4.8, 0.4)) * 0.4*

The answer is .

48. Some values for a function are shown below.

(a) Approximate using 5 rectangles of equal width and using the left-hand endpoints to obtain the rectangle heights.

(b) Approximate using 10 rectangles of equal width and using the right-hand endpoints to obtain the rectangle heights.

(a) , so the approximation is

(b) , so the approximation is

49. Compute by writing the integral as the limit of a rectangle sum.

The width of a typical rectangle is

I'll use the right-hand endpoints of the subintervals. (You could use the left-hand endpoints or the midpoints; the computation would look different, but the final answer would be the same.)

I'm going from 2 to 4 in steps of size . The diagram shows that right-hand endpoints:

The function is . The rectangle sum is

50. (a) Given that , what is ?

(b) Compute .

(a)

(b)

51. Compute .

52. Compute .

53. Compute .

54. Suppose that , , and . Find .

Since , I have

Thus, .

Since , I have

Therefore, .

55. Find functions and such that the antiderivative of
is *not* equal to the antiderivative of
times the antiderivative of .

There are lots of possibilities. For example, take and . Then

But

Obviously, .

56. Compute .

Here is the graph of .

is the area under the graph and above the x-axis, from to . This is the shaded region in the picture. It is a triangle with height 1 and base 2, so its area is

57. Compute the exact value of .

First, break the integral up into two pieces:

The first integral can be computed directly:

For the second integral, notice that is a semicircle:

The radius is , and it's centered at the origin.

The integral is computing the area of the semicircle, which is half the area of a circle of radius 10:

Therefore,

58. Compute the exact value of .

Hint: Complete the square in x.

You can't compute this integral directly using techniques you know now.

If , then

(I knew to add 9 to both sides, since ,
then . You should have seen this when you took algebra;
it's called *completing the square*.)

The equation represents a circle of radius centered at
. If I go 4 units to the left and right of ,
I get , and , which are the limits on the integral.
Finally, represents the *top*
semicircle of the circle, because always gives the
*positive* square root.

Putting everything together, I see that the integral represents the area of a semicircle of radius 4. Since the area of a circle of radius r is ,

59. Find the total area of the region bounded by and the x-axis.

The curve is positive from 0 to 2 and negative from 2 to 4, so the area is

, so

Using this in the integrals above, I find that the area is

60. Find the area of the region bounded by

is a parabola opening to the right, and is a line.

Find the intersections:

The curves intersect at and .

Use horizontal rectangles. The right end of a horizontal rectangle is on and the left end is on . So the area is

61. Find the area of the region in the first quadrant bounded on the left by , on the right by , and below by the x-axis.

I'll find the area using horizontal rectangles; vertical rectangles would require two integrals.

The left-hand curve is

The right-hand curve is

The region extends from to . The area is

62. Find the area of the region bounded by the graphs of and .

Solve simultaneously:

The top curve is and the bottom curve is . The area is

63. Find the area of the region between and , from to .

Find the intersection points:

On the interval , the curves cross at .

I need two integrals to find the area. From to , the top curve is and the bottom curve is . From to , the top curve is and the bottom curve is . The area is

64. (a) Compute .

(b) Compute .

(c) Compute .

(a) Using the second form of the Fundamental Theorem of Calculus, I get

(b) In this situation, I'm differentiating with respect to x, but the top limit in the integral is --- they don't match. In order to apply the second form of the Fundamental Theorem, I need to use the Chain Rule to "make them match":

(c)

65. (a) Prove that .

(b) Use the Integral Mean Value Theorem to estimate .

(a) Since , it follows that . Hence,

(b) If , then . Note that for , and is defined for all x.

The maximum value of on is e and the minimum value is 1. Thus, .

The Integral Mean Value Theorem says that for some c satisfying ,

The max-min result I derived above shows that . Hence,

*You are all you will ever have for certain.* - *June Havoc*

Copyright 2020 by Bruce Ikenaga