Math 161

These problems are intended to help you study for the final. However,
you shouldn't assume that each problem on this handout corresponds to
a problem on the final. Nor should you assume that if a topic
*doesn't* appear here, it *won't* appear on the final.

1. Compute .

2. Compute .

3. Compute .

4. Compute .

5. Compute .

6. Compute .

7. Compute .

8. Compute the following limit: .

9. Compute the following limit: .

10. Compute the following limit: .

11. Compute the following limit: .

12. Compute the following limit: .

13. Compute the following limit: .

14. Compute the following limit: .

15. Compute the following limit: .

16. Compute the following limit: .

17. Compute the following limit: .

18. Compute the following limit: .

19. Compute the following limit: .

20. Compute the following limit: .

21. Compute the derivative: .

22. Compute the derivative: .

23. Compute the derivative: .

24. Compute the derivative: .

25. Compute the derivative: .

26. Compute the derivative: .

27. Compute the derivative: .

28. Compute the derivative: .

29. Compute , where .

30. Compute , where .

31. Compute when and , where .

32. Find the points on the curve where .

33. Compute the following derivatives:

(a) .

(b) .

(c) .

(d) .

(e) .

(f) .

34. Prove that for .

35. Graph .

36. Graph .

The derivatives are

37. Compute .

38. Compute .

39. Compute .

40. Compute .

41. Compute .

42. Compute .

43. Compute .

44. Compute .

45. Compute , if and .

46. Compute .

47. Compute .

48. Compute .

49. Compute .

50. Suppose and . Assuming that f has a differentiable inverse, what is ?

51. Find for .

52. Given that , what is ?

53. Find the largest interval containing on which the function has an inverse .

54. The position of a bowl of potato salad at time t is

(a) Find the velocity and the acceleration .

(b) When is the velocity equal to 0? When is the acceleration equal to 0?

55. A population of flamingo lawn ornaments grows exponentially in Calvin's yard. There are 20 after 1 day and 60 after 4 days. How many are there after 6 days?

56. A bacon, sausage, onion, mushroom, and ham quiche is placed in a oven. The initial temperature of the quiche is ; after 10 minutes, the quiche's temperature is . What is the quiche's temperature 25 minutes after being placed in the oven?

57. A hot pastrami sandwich with a temperature of is placed in a room to cool. After 10 minutes, the temperature of the sandwich is . When will the temperature be ?

58. 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.

59. Find the area of the region between and from to .

60. Find the area of the region bounded by

61. Approximate the area under from to using 20 rectangles of equal width, and using the midpoints of each subinterval to obtain the rectangles' heights.

62. Find the exactly value of .

63. Write the following sum using summation notation, then approximate its value to 5 decimal places:

64. Calvin runs south toward Phoebe's house at 2 feet per second. Bonzo runs east away from Phoebe's house at 5 feet per second. At what rate is the distance between Calvin and Bonzo changing when Calvin is 50 feet from the house and Bonzo is 120 feet from the house?

65. A bird flies at a constant speed of 16 feet per second at a constant height of 48 feet. Its path takes it directly over a camera, which turns to track the bird. At what rate is the acute angle between the ground and the line of sight from the camera to the bird changing 4 second after it has passed above the camera?

66. Find the dimensions of the rectangle with the largest possible perimeter that can be inscribed in a semicircle of radius 1.

67. A window is made in the shape of a rectangle with an isosceles right triangle on top.

(a) Write down an expression for the *total area* of the
window.

(b) Write down an expression for the *perimeter* of the window
(that is, the length of the *outside* edge).

(c) If the perimeter is given to be 4, what value of s makes the total area a maximum?

68. A cylindrical can with a top and a bottom is to be made with square inches of sheet metal with no waste. What values for the radius r and the height h give the can of largest volume?

69. A rectangular box with a square bottom * and no
top* has a volume of 55296 cubic inches. What values of the
length x of a side of the bottom and the height y give the box with
the smallest total surface area (the area of the bottom plus the area
of the sides)?

70. (a) Find the absolute max and the absolute min of on the interval .

(b) Find the absolute max and the absolute min of on the interval .

71. Use a limit of a rectangle sum to find the exact area under from to .

72. (a) Compute .

(b) Compute .

73. Use the definition of the derivative as a limit to prove that .

74. Compute .

Hint: Write the limit as a difference quotient that gives the derivative of a certain function.

75. Let

76. Suppose that and . Use differentials to approximate to 5 places.

77. A differentiable function satisfies and . Use differentials to approximate .

78. Use 3 iterations of Newton's method starting at to approximate a solution to .

79. Suppose that f is a differentiable function, , and

Prove that .

80. Prove that the function has exactly one root.

\centerline{\bigssbold Solutions to the Review Problems for the Final}

1. Compute .

2. Compute .

3. Compute .

The limit represents the derivative of :

Hence, by the Power Rule,

4. Compute .

5. Compute .

means that we're approaching 4 from the right --- that is, through numbers larger than 4.

For x close to 4 but larger than 4 --- e.g. --- is negative. Since the square root of a negative number is undefined, the limit is undefined.

6. Compute .

Plugging in gives . Moreover,

Hence, the limit is undefined.

7. Compute .

8. Compute the following limit: .

9. Compute the following limit: .

10. Compute the following limit: .

11. Compute the following limit: .

L'H\^opital's rule doesn't apply, because plugging in gives 1, which is not an indeterminate form.

12. Compute the following limit: .

Set , so

Then

So

13. Compute the following limit: .

14. Compute the following limit: .

15. Compute the following limit: .

16. Compute the following limit: .

Set . Then

Therefore,

Hence,

17. Compute the following limit: .

This is an indeterminate form of type , so I can apply L'H\^opital's rule:

18. Compute the following limit: .

Let . Then

Hence,

Therefore,

19. Compute the following limit: .

20. Compute the following limit: .

Let , so

Then

Therefore,

21. Compute the derivative: .

22. Compute the derivative: .

23. Compute the derivative: .

24. Compute the derivative: .

25. Compute the derivative: .

26. Compute the derivative: .

27. Compute the derivative: .

28. Compute the derivative: .

29. Compute , where .

Differentiate implicitly, then solve for :

30. Compute , where .

Use logarithmic differentiation:

31. Compute when and , where .

Differentiate implicitly:

Plug in and :

Now differentiate (*) implicitly:

Note that the term produces *three* terms
when the Product Rule is applied.

Now set , , and :

32. Find the points on the curve where .

Differentiate implicitly:

Set :

Substitute into and solve for x:

gives , so . The point is .

gives , or . First, gives . Second, gives . The points are and .

The points are , , and .

33. Compute the following derivatives:

(a) .

(b) .

(c) .

(d) .

(e) .

(f) .

(a)

(b)

(c)

(d)

(e)

(f)

34. Prove that for .

The picture shows that the result seems to be true. However, a picture is not a proof.

Let . f measures the distance between the two curves. The derivative is . at ; since and , the critical point is a local min. Since it is the only critical point, it is an absolute min.

Thus, the minimum vertical distance between the curves occurs at , when it is . Since this is an absolute min, it follows that for --- that is, for .

Hence, for .

35. Graph .

The function is defined for all x (you can take the cube root of any number).

Since , the x-intercepts are and . The y-intercept is .

The derivatives are

(In working with these kinds of expressions, it is better to have a sum of terms when you're differentiating. On the other hand, it is better to have everything together in factored form when you set up the sign charts. Notice how I used these two forms in the derivatives above.)

for and at . is defined for all x.

The function increases for and decreases for . There is a local max at , .

at . is undefined at .

The graph is concave down for and for . The graph is concave up for . and are inflection points.

Since the function is defined for all x, there are no vertical asymptotes.

The graph goes downward on the far left and far right. There are no horizontal asymptotes.

36. Graph .

The derivatives are

The domain is .

The x-intercepts are and .

The y-intercept is .

The derivatives were given:

for and is undefined for . Since y is undefined for , this cannot be a max or a min.

The function increases for . It decreases for and for .

is a local max.

for and is undefined for . Since y is undefined for , this cannot be an inflection point.

The function is concave up for . It is concave down for and for .

is an inflection point.

By considering highest powers of x on the top and bottom,

Hence, is a horizontal asymptote at and .

37. Compute .

38. Compute .

39. Compute .

40. Compute .

41. Compute .

42. Compute .

43. Compute .

44. Compute .

45. Compute , if and .

46. Compute .

47. Compute .

I'll use the formula

In our problem,

48. Compute .

49. Compute .

50. Suppose and . Assuming that f has a differentiable inverse, what is ?

51. Find for .

First, note that .

Also, , so .

Hence,

52. Given that , what is ?

First, . Then

I need to find . Suppose . Then , so

I can't solve this equation algebraically. This is a case where you
need to remember that *this is a problem in a math course*.
The equation must have a solution, and probably an "easy"
one --- you would not expect a solution to be something like
" ".

So one approach you could take is to *assume* that there's an
easy solution, and use trial-and-error: Try , , , and so on. Doing
so, you find that works. Check: .

Another way to find a solution is by drawing the graph (of ):

It looks like is a solution.

Thus, , so . Hence,

53. Find the largest interval containing on which the function has an inverse .

The derivative is

for and . is defined for all x. Here is the sign chart for :

f decreases for , and this is the largest interval containing on which f is always increasing or always decreasing. Therefore, the largest interval containing on which the function has an inverse is .

54. The position of a bowl of potato salad at time t is

(a) Find the velocity and the acceleration .

(b) When is the velocity equal to 0? When is the acceleration equal to 0?

(a)

(b) The velocity is 0 at and at . The acceleration is 0 at . at .

55. A population of flamingo lawn ornaments grows exponentially in Calvin's yard. There are 20 after 1 day and 60 after 4 days. How many are there after 6 days?

Let F be the number of flamingos after t days. Then

When , :

When , :

Divide by and solve for k:

Plug this back into :

Hence,

When ,

56. A bacon, sausage, onion, mushroom, and ham quiche is placed in a oven. The initial temperature of the quiche is ; after 10 minutes, the quiche's temperature is . What is the quiche's temperature 25 minutes after being placed in the oven?

The temperature of the oven is and the initial temperature is . So

When , the temperature is :

Thus,

Remember that " " is another way of writing " ". Also, notice that I wrote the "t" to the left of the value for k, because I don't mean t to multiply the " ".

Set :

57. A hot pastrami sandwich with a temperature of is placed in a room to cool. After 10 minutes, the temperature of the sandwich is . When will the temperature be ?

Let T be the temperature at time t. The initial temperature is , and the room's temperature is 70, so

When , :

Hence,

Set and solve for t:

58. 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.

The curves intersect at the point , . (You can see this by solving and simultaneously.)

Divide the region up into horizontal rectangles. A typical rectangle
has width . The *right* end of a rectangle is
on ; the *left* end of a rectangle is
on (i.e. ). Therefore, the
length of a rectangle is , and the area of a
rectangle is .

The area is

59. Find the area of the region between and from to .

Find the intersection point:

is the intersection point between 0 and 4.

Use vertical rectangles. Between and , the top curve is and the bottom curve is . Between and , the top curve is and the bottom curve is . The area is

60. Find the area of the region bounded by

Find the intersection points:

The curves intersect at and at . The left-hand curve is and the right-hand curve is . Using horizontal rectangles, the area is

61. Approximate the area under from to using 20 rectangles of equal width, and using the midpoints of each subinterval to obtain the rectangles' heights.

The width of each rectangle is .

The midpoints start at 3.05 and go to 4.95 in steps of size 0.1.

The calculator command to compute the sum is:

* sum(seq((x - sin(x)) 2, x, 3.05, 4.95,
0.1)) * 0.1*

The answer is .

62. Find the exactly value of .

Note that is the area of a quarter of a circle of radius 3. So

63. Write the following sum using summation notation, then approximate its value to 5 decimal places:

The calculator command to do the sum is:

* sum(seq((3 + sin(x + 1))/(x 2 + x), x, 1, 40))*

The answer is .

64. Calvin runs south toward Phoebe's house at 2 feet per second. Bonzo runs east away from Phoebe's house at 5 feet per second. At what rate is the distance between Calvin and Bonzo changing when Calvin is 50 feet from the house and Bonzo is 120 feet from the house?

Let x be the distance from Bonzo to the house, let y be the distance from Calvin to the house, and let s be the distance between Calvin and Bonzo.

By Pythagoras,

Differentiate with respect to t:

and (negative,
because his distance from the house is *decreasing*). When
and , . So

65. A bird flies at a constant speed of 16 feet per second at a constant height of 48 feet. Its path takes it directly over a camera, which turns to track the bird. At what rate is the acute angle between the ground and the line of sight from the camera to the bird changing 4 second after it has passed above the camera?

Let be the acute angle between the ground and the line of sight from the camera to the bird, and let x be the distance the bird has flown past the point directly above the camera.

Then

At 16 feet per second, the bird will have flown feet in 4 seconds. By Pythagoras, the hypotenuse of the triangle is . Hence, .

Thus,

66. Find the dimensions of the rectangle with the largest possible perimeter that can be inscribed in a semicircle of radius 1.

The height of the rectangle is y; the width is .

The perimeter of the rectangle is

By Pythagoras, , so . Therefore,

gives a "flat" rectangle lying along the diameter of the semicircle. gives a "thin" rectangle lying along the vertical radius.

, so for . (The negative root does not lie in the interval .)

gives . The dimensions of the rectangle with the largest perimeter are and ; the maximum perimeter is .

67. A window is made in the shape of a rectangle with an isosceles right triangle on top.

(a) Write down an expression for the *total area* of the
window.

(b) Write down an expression for the *perimeter* of the window
(that is, the length of the *outside* edge).

(c) If the perimeter is given to be 4, what value of s makes the total area a maximum?

(a)

(b)

(c) Since ,

Hence,

The extreme cases are and , which gives .

The derivative is

Find the critical point:

Plug the critical point and the endpoints into A:

When , the area is a maximum.

68. A cylindrical can with a top and a bottom is to be made with square inches of sheet metal with no waste. What values for the radius r and the height h give the can of largest volume?

I have

Solving the second equation for h, I get

Plug this into V:

Since is ruled out (as it causes division by 0 in the equation for h), I won't have two endpoints. I will use the Second Derivative Test. I have

Find the critical points by setting and solving:

(I can throw out , since the radius can't be negative.) This gives . In addition.

The critical point is a local max; since it's the only critical point, it's an absolute max.

69. A rectangular box with a square bottom * and no
top* has a volume of 55296 cubic inches. What values of the
length x of a side of the bottom and the height y give the box with
the smallest total surface area (the area of the bottom plus the area
of the sides)?

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

The volume is

Solving for y gives

Plug this into A 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:

gives

Plug into the Second Derivative:

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

70. (a) Find the absolute max and the absolute min of on the interval .

(b) Find the absolute max and the absolute min of on the interval .

(a) The derivative is

for and ; however, only is in the interval . is defined for all x.

The absolute max is at ; the absolute min is at .

(b)

for and is undefined for . Both points are in the interval.

The absolute max is at and the absolute min is at .

71. Use a limit of a rectangle sum to find the exact area under from to .

Divide the interval up into n equal subintervals. Each subinterval has length . I'll evaluate the function at the right-hand endpoints of the subintervals, which are

The function values are

The sum of the rectangle area is

The exact area is

72. (a) Compute .

(b) Compute .

(a)

(b)

73. Use the definition of the derivative as a limit to prove that .

74. Compute .

Hint: Write the limit as a difference quotient that gives the derivative of a certain function.

The limit as h goes to 0, the , and the limits x and remind me of the definition of the derivative. So I make a guess that the limit in the problem is actually the derivative of a function. The problem is to figure out what function is being differentiated ....

Let . Then

Therefore,

75. Let

Is f continuous at ? Why or why not?

The left and right-hand limits agree. Therefore,

However, , so . Hence, f is not continuous at .

76. Suppose that and . Use differentials to approximate to 5 places.

Use

and . Therefore,

77. A differentiable function satisfies and . Use differentials to approximate .

; when , . Therefore,

Hence,

78. Use 3 iterations of Newton's method starting at to approximate a solution to .

Write the equation as . Set , so .

The solution is approximately .

79. Suppose that f is a differentiable function, , and

Prove that .

Apply the Mean Value Theorem to f on the interval :

Then

80. Prove that the function has exactly one root.

First,

By the Intermediate Value Theorem, f has at least one root between and 0.

Suppose f has more than one root. Then f has at least two roots, so let a and b be two roots of f. By Rolle's theorem, f has a critical point between a and b.

However,

Since , . But , so

Thus, f has no critical points. Therefore, it can't have more than one root.

It follows that f has exactly one root.

*The best thing for being sad is to learn something.* -
*Merlyn*, in *T. H. White's* *The Once and Future
King*

Copyright 2020 by Bruce Ikenaga