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. Compute .

2. Compute .

3. Compute .

4. Compute .

5. Compute .

6. Compute .

7. Compute .

8. Compute .

9. Compute .

10. Compute .

11. Compute .

12. Compute .

13. Compute .

14. Compute .

15. Compute .

16. Compute .

17. Suppose

Determine whether is defined. If it is, compute it.

18. Compute .

19. Suppose that

Compute .

20. The picture below shows that graph of a function .

Compute:

(a) .

(b) .

(c) .

(d) .

(e) .

(f) .

(g) .

21. Locate the horizontal asymptotes and the vertical asymptotes of

Justify your answer by computing the relevant limits.

22. Differentiate .

23. Differentiate .

24. Differentiate .

25. Differentiate .

26. Differentiate .

27. Differentiate .

28. Differentiate .

29. Differentiate .

30. Differentiate .

31. Differentiate .

32. Find .

33. Find .

34. Compute .

35. Compute .

36. Find .

37. Find if .

38. Compute .

39. Find .

40. Find if .

41. Compute .

42. Compute .

43. Compute .

44. Compute .

45. Compute .

46. Compute .

47. Compute .

48. Compute the derivative with respect to t of .

49. Compute .

50. Compute for .

51. Compute if .

52. Compute .

53. Compute .

54. f and g are differentiable functions. The values of f, g, , and at and are shown below:

Compute , , and .

55. Find the equation of the tangent line to the graph of at .

56. Find the equation of the tangent line to the graph of at .

57. For what value or values of c does the tangent line to the graph of at pass through the point ?

58. The position of a cheeseburger at time t is given by

Find the value(s) of t for which the velocity is 0. Find the value(s) of t for which the acceleration is 0.

59. For what values of x is the function continuous?

60. For what values of x is the function continuous?

61. Let

Prove or disprove: f is continuous at .

62. Find the value of c which makes the following function continuous at :

63. Prove that has a root between and .

64. Suppose f is continuous, , and . Prove that there is a number x between 2 and 5 such that .

65. Suppose . Use the limit definition of the derivative to compute .

66. Suppose . Use the limit definition of the derivative to compute .

67. Suppose . Use the limit definition of the derivative to compute .

68. The graph of a function is pictured below. For what values of x is f continuous but not differentiable? For what values of f is f not continuous?

69. (a) Find the average rate of change of on the interval .

(b) Find the instantaneous rate of change of at .

70. Give an example of a function which is defined for all x and a number c such that

71. Suppose . Find .

72. Suppose and . Find .

73. Suppose and . Find .

74. Suppose . Find .

75. Compute .

76. Compute .

77. Find the equation of the tangent line to at .

78. * (An alternate approach to logarithmic
differentiation)*

(a) Use the identity to write using and .

(b) Use the Chain Rule to differentiate the expression you obtained in (a). (This gives another way of doing logarithmic differentiation involving powers.)

79. Find if y is defined implicitly by

80. Find the equation of the tangent line to the curve

81. Find the equation of the tangent line to the curve

82. Find at the point on the curve

1. Compute .

Plugging in gives . I factor the top, then cancel common factors of :

2. Compute .

Since plugging in gives , it is
reasonable to suppose that the zeros are being caused by a *common
factor* on the top and the bottom. So factor the top and the
bottom and cancel the 's:

3. Compute .

If I plug in , I get , a *nonzero* number over
0. In this case, the limit does not exist.

(By the way, I hope you didn't try to use L'H\^opital's Rule here. It doesn't apply.)

4. Compute .

This problem is different from the previous one because x is
approaching 3 *from the left*. It would not be incorrect to
say that the limit is undefined, but it is better to say that the
limit is . There is a * vertical
asymptote* at , and the graph goes *downward* (to
) as it approaches the asymptote from the left.

How do you see algebraically that this is what it does? One way is to
plug numbers close to 3, but less than 3, into . For example, if , --- a big negative number. This doesn't
*prove* that it's going to , but it's pretty
good evidence.

Here is how I can analyze it. I know that plugging in gives . Therefore, I should be getting either or --- the reciprocal of something small ( ) should be something big.

Now let x approach 3 from the left. surely goes to 12, a positive number. goes to 0, but since , will be negative. Now , so the answer is .

5. Compute .

The idea is to divide the top and bottom by the highest power of x that occurs in either --- in this case, . Then use the fact that

I have

6. Compute .

There's little question that the limit is undefined; why is it ? One approach is to note that the top and bottom are dominated by the highest powers of x. So this limits looks like

As a check, if you plug into , you get (approximately) .

7. Compute .

As , I have and . So

8. Compute .

As , I have and . So

9. Compute .

10. Compute .

* Method 1.* Multiply the top and bottom by the
conjugate:

* Method 2.* Factor the top:

11. Compute .

12. Compute .

Nothing complicated is going on here --- I just plug in . (By the way --- Do you know how to compute on your calculator? It's approximately -1.12475. You should always give exact answers unless told otherwise, however.)

13. Compute .

14. Compute .

15. Compute .

Remember that

In this problem, I'll let . Now if and only if , so

Here's the solution:

16. Compute .

17. Suppose

Determine whether is defined. If it is, compute it.

The function is made of "two pieces" glued together at ,
I'll compute by computing the limits from the
left and right, and seeing if they're equal. (If they *aren't*
equal, the limit is undefined.)

The left-hand limit is

The right-hand limit is

Since the left- and right-hand limits are equal, is defined, and

18. Compute .

Take the limit as . I have

By the Squeezing Theorem,

19. Suppose that

Compute .

By the Squeezing Theorem, .

20. The picture below shows that graph of a function .

Compute:

(a) .

(b) .

(c) .

(d) .

(e) .

(f) .

(g) .

(a) .

(b) .

(c) is undefined.

(d) .

(e) .

(f) .

(g) .

21. Locate the horizontal asymptotes and the vertical asymptotes of

Justify your answer by computing the relevant limits.

Hence, is a horizontal asymptote at and at .

Factor the bottom:

The function is undefined at , , and at .

Note that

Hence, the graph does *not* have a vertical asymptote at .

Consider the limits at .

I know the one-sided limits will be either or , since plugging in gives , a nonzero number divided by 0.

Take as an example. , , , and . In the last case, since , , i.e. goes to 0 through positive numbers. Since all the factors of are positive as , the quotient must be positive, and the limit is .

Similar reasoning works for the left-hand limit.

Thus,

There is a vertical asymptote at .

Likewise,

There is a vertical asymptote at .

The results are visible in the graph of the function:

22. Differentiate .

23. Differentiate .

24. Differentiate .

25. Differentiate .

26. Differentiate .

27. Differentiate .

28. Differentiate .

29. Differentiate .

30. Differentiate .

31. Differentiate .

32. Find .

Note that

Then

33. Find .

Note that

Then

34. Compute .

35. Compute .

It is *not* a good idea to use the Quotient Rule when either
the top or the bottom of a fraction is a number.

36. Find .

37. Find if .

Rewrite the function as

Then

It is *not* a good idea to use the Quotient Rule when either
the top or the bottom of a fraction is a number.

38. Compute .

For a product of three terms, the Product Rule says

So

39. Find .

Note that

Then

40. Find if .

I applied the Quotient Rule to the original fraction. In taking the derivative of the top, I also needed to apply the Product Rule.

41. Compute .

I applied the Quotient Rule to the original fraction. In taking the derivative of the bottom, I also needed to apply the Product Rule.

42. Compute .

43. Compute .

44. Compute .

Note that

Then

45. Compute .

Note that

Then

46. Compute .

47. Compute .

In going from the second expression to the third, I cancelled a common factor of . If you try to multiply before you cancel, you'll get a big mess, and it will be much harder to simplify.

48. Compute the derivative with respect to t of .

Rewrite the function as

Then

49. Compute .

50. Compute for .

51. Compute if .

I'll compute the first few derivatives until I see the pattern.

Note that I don't get minus signs here; the powers are negative, but the Chain Rule requires the derivative of , which is -1. The two negatives cancel.

Note also that, in order to see the pattern, I did *not*
multiply out the numbers on the top.

Based on the pattern, I see that the power on the bottom is one more than the order of the derivative. On the top, I get the product of the numbers from 2 through the order of the derivative. So

(You can also write the top as (50-factorial).)

52. Compute .

Since is not given, the answer will come out in terms of and .

Note that

Then

53. Compute .

Since is not given, the answer will come out in terms of and .

54. f and g are differentiable functions. The values of f, g, , and at and are shown below:

Compute , , and .

By the Product Rule,

By the Quotient Rule,

By the Chain Rule,

55. Find the equation of the tangent line to the graph of at .

When ,

The point of tangency is .

I have

The tangent line is

56. Find the equation of the tangent line to the graph of at .

When ,

The point of tangency is .

, so

The tangent line is

57. For what value or values of c does the tangent line to the graph of at pass through the point ?

At , I have . Also,

The tangent line to at is

I want this line to pass through the point , so plug the point into the line equation and solve for c:

The values of c are and .

58. The position of a cheeseburger at time t is given by

Find the value(s) of t for which the velocity is 0. Find the value(s) of t for which the acceleration is 0.

The velocity is the derivative of the position:

I have for and .

The acceleration is derivative of the velocity (or the second derivative of the position):

(Note: I differentiated , *not* . Differentiating the first expression is easier than
differentiating the second.)

I have for .

59. For what values of x is the function continuous?

The function will be continuous where it's defined, so I have to find the domain.

I can't divide by 0, and division by 0 occurs when or .

(You can't cancel the 's, because that *assumes* .)

Thus, the domain is , and that is where f is continuous.

60. For what values of x is the function continuous?

The function will be continuous where it's defined, so I have to find the domain.

I can't divide by 0, and division by 0 occurs where . Square both sides: . I get or .

I can't take the square root of a negative number, and this occurs where .

You can solve the inequality by constructing a sign chart.

Alternatively, graph the quadratic. is a parabola opening upward (since it's , which has a positive -term). It has roots at and at . Picture:

From the graph, you can see that for .

I throw out the bad points , , . The domain is or , and that is where f is continuous.

61. Let

Prove or disprove: f is continuous at .

On the other hand, . Since , f is *not* continuous at
.

62. Find the value of c which makes the following function continuous at :

For f to be continuous at , must be defined. This will happen if the left and right-hand limits at are equal. Compute the limits:

Set the left and right-hand limits equal and solve for c:

This value of c will make the left and right-hand limits equal to 17, so in this case, . Since , it follows that , and f is continuous at .

63. Prove that has a root between and .

f is continuous, since it's a polynomial.

Since f is negative when and f is positive when , the Intermediate Value Theorem implies that for some number c between -1 and 0.

64. Suppose f is continuous, , and . Prove that there is a number x between 2 and 5 such that .

and are continuous, so is continuous.

Since is continuous, and since 110 is between 104 and 125, the Intermediate Value Theorem says that there is a number x between 2 and 5 such that .

65. Suppose . Use the limit definition of the derivative to compute .

Now

So

66. Suppose . Use the limit definition of the derivative to compute .

I'll simplify the quotient , then plug it into the limit.

Therefore,

Hence,

67. Suppose . Use the limit definition of the derivative to compute .

68. The graph of a function is pictured below. For what values of x is f continuous but not differentiable? For what values of f is f not continuous?

f is continuous but not differentiable at and at , since at those points the graph is unbroken but has corners.

f is not continuous at and at .

69. (a) Find the average rate of change of on the interval .

(b) Find the instantaneous rate of change of at .

(a)

(b)

70. Give an example of a function which is defined for all x and a number c such that

There are lots of possible answers to this question. For example, let

Then

Of course, the condition means that the
function is * not continuous* at c. The value of
(or even its existence) does not depend on the value
of (or even its existence).

71. Suppose . Find .

Let . Swap x's and y's and solve for y:

Therefore, .

72. Suppose and . Find .

First, notice that means . I'll use the formula

Setting , I get

73. Suppose and . Find .

First, notice that means . I'll use the formula

Setting , I get

74. Suppose . Find .

It looks like I need in this problem, so I'll do that first. By the Chain Rule,

I'll use the formula

Setting , I get

I need to find . Remember that

So I want to find a number A such that , or

If you try to solve this algebraically for A, you might get as far as

But you probably don't know any method or formula for solving this kind of equation.

However, the problem was intended to be doable, so if you can't solve systematically, it must be intended that you should use trial and error. And if it's trial and error, the answer is probably not something like " ". So try a few small numbers for A --- plug them into and see if you can find one that works.

If you do this, you find that solves the equation. Thus, , so . Now going back to (*), I get

75. Compute .

Let . Take the log of both sides, then differentiate:

Note that in the next-to-the-last step, I changed to . If you don't do this, you must have the 's around . Writing " " is not correct.

76. Compute .

Let . Take the log of both sides, then differentiate:

77. Find the equation of the tangent line to at .

When , I have , so the point of tangency is .

Use logarithmic differentiation to find :

Plug in :

The line is

78. * (An alternate approach to logarithmic
differentiation)*

(a) Use the identity to write using and .

(b) Use the Chain Rule to differentiate the expression you obtained in (a). (This gives another way of doing logarithmic differentiation involving powers.)

(a)

(b) I'll differentiate using the Chain Rule. The outer function is the exponential ; to differentiate the inner function , I'll use the Product Rule.

In other words,

79. Find if y is defined implicitly by

Differentiate implicitly and solve for y:

80. Find the equation of the tangent line to the curve

Differentiate implicitly:

* Note:* If you have a point to plug in, plug the
point in * before* you solve for .

Plug in , :

The tangent line is

81. Find the equation of the tangent line to the curve

Differentiate implicitly:

* Note:* If you have a point to plug in, plug the
point in * before* you solve for .

Plug in , :

The tangent line is

82. Find at the point on the curve

Differentiate implicitly:

Plug in and and solve for :

Differentiate (*) implicitly:

Plug in , , and and solve for :

*He who climbs onto the roof should not push away the ladder.*
- *Karol Irzykowski*

Copyright 2020 by Bruce Ikenaga