Math 2414 Activity 4 (Due by July 24)
1.a)Show why L’Hopital’s Rule fails for .
b)Find using the inequality which is valid for .
2.a) Show why L’Hopital’s Rule fails for .
b)Find using the inequality which is valid for .
3. Find the value of c so that .
{Hint: Find , and figure out the value of c.}
4. Find the value of c so that .
{Hint: Find , and figure out the value of c.}
5. If you try L’Hopital’s Rule on the following limits, you don’t get anywhere. Find the limits by trying something else.
a) b) c)
6. Find two functions f and g with so that
a) b) c)
{Hint:Keep the functions very simple.}
7. For what values of a and b is the following equation true?
{Hint: , so start applying L’Hopital’s Rule.}
8. Find the following limits:
a) {Hint: In order to apply L’Hopital’s Rule, you need to show that . It suffices to show that .}
b)
9. Using the graphs of the functions f and g, evaluate the following limits:
a) b) c)
d) e) f)
g) h) i)
{Hint: L’Hopital’s Rule}
j) {Hint: L’Hopital’s Rule}
10. Use L’Hopital’s Rule to evaluate . {Hint: Use the Fundamental Theorem.}
11. Find . L’Hopital’s Rule won’t work, so try something else.
{Hint:.}
12.a)Suppose that f has a continuous second derivative. Use L’Hopital’s Rule twice to find.
b) Suppose that fhas a second derivative(not necessarily continuous). Use L’Hopital’s Rule once to find .
{Hint: , and
.}
c) Find a function where the limit exists at, but the function doesn’t have a second derivative at .
{Hint: Try .}
13. Find , for .
a)Case I::
, so differentiate top and bottom:
Now you figure out , and get the result.
b)Take care ofCase II:
14. Find by observing the following:
The last expression is a Riemann sum of or .
15. has the form , but L’Hopital’s rule can’t be applied. Convert the limit into a definite integral to find its value.
16. Suppose that , and . Evaluate .
17. Find {Hint: Integrate first.}
18. Find the value of a so that the following equation is true without using L’hopital’s Rule.
{Hint: Use the fact that and .}
19. If , where b is a nonzero real number, then what must be the value of a?
{Hint: .}
20. If f has a continuous derivative with and , then use L’Hopital’s Rule to find .
21. Suppose that f and g are differentiable functions with , and . Find the value of .
22. Suppose that f and g are differentiable functions with .
a) If , then find .
b) If , then find .
23. Suppose that f and g are differentiable functions with .
a) If , then find .
b) If , then find .
24.a) Show that for , , but doesn’t exist.
b) Show that if and both exist as numbers, then find .
{Hint: , so , so apply L’Hopital’s Rule.}
c) If , where A is a number, show that and .
{Hint: , so , so apply L’Hopital’s Rule.}
25. If and is continuous at , then find .
{Hint: . , so .}
26. Find .
{Hint: and .
, but , so let’s focus on . .}
27. If you try to find the points of intersection of the graphs of the functions and , you might get a graph that looks like the following:
From which you might conclude that there are two intersection points, but you would be mistaken.
a) Use L’Hopital’s Rule to find .
b) Why does the result of part a) imply that there are at least three points of intersection of the two graphs?
28.a) Find {Hint: Simplify.}
b) Find {Hint: Simplify.}
29. Find . {Hint: .}
30. Find a) and b) {Hint: .}
31. Evaluate the following limits for n a positive integer:
a) {Hint: .}
b) {Hint: .}
32. Find.
33. Findvalues of a and b so that .
34. Find , where c is any real number.
35. Find , where c is any real number.
36. a) If , then what can be determined about ?
b) Find an example of functions f and g with , but doesn’t exist.
{Hint: Let and .}
c) If , then what can be determined about ?
d) Find an example of functions f and g with , but doesn’t exist.
{Hint: Let and .}
e) If , then what can be determined about ?
Evaluate the following integrals using integration by parts:
37. 38.
39. 40.
{Hint: Start with .}
41. Suppose that and that is continuous on the interval . Use integration by parts to show that for some constant C.
42. Suppose that for a certain function it is known that and . Use integration by parts to find the value of .
43. Consider the identity . If you assume that has a continuous second derivative, let and and perform integration by parts on the right side with and , you get
. If you simplify, you get .
a) Assume that f has a continuous third derivative. Use integration by parts with and in the integral on the right to extend the formula further.
b) Extend it again.
c) If f has ncontinuous derivatives, generalize the formula.
44. Suppose that and . Use integration by parts to find .
45. Suppose that and that is continuous.
a) Use integration by parts to find the maximum possible value of .
{Hint: Let.}
b) What are the only functions,, for which attains its maximum value?
46. The function is continuous on . Here’s why: For t and h in ,
, and since , we get that . Show that the equation has at least one solution on the interval .
{Hint: Apply the Intermediate Value Theorem on the interval .}
47. Let’s use integration by parts to find a formula for , for .
For , and, you get
.
So you get that , or more simply . Now .
a) Use the formula to find the values of .
b) Try to find a direct formula for .
48. Let’s try something similar for , for .
For , , you get
.
So you get that , or more simply .
.
a) Use the formula to find the values of .
b) Try to find a direct formula for .
49. Let , for .
For ,, you get .
So you get . .
a) Use the formula to find the values of .
b) Try to find a direct formula for .
c) For , , so we know that . From this, determine .
d) Using the result in c), let in your formula in part b) to find a limit equal to the famous number e.
50.a) Let , for . Find a formula that relates and using integration by parts on with and .
{Hint: Use.}
b) Evaluate and .
51. Evaluate using integration by parts twice.
52. Evaluate . Start with the substitution .
53. Evaluate . .
54. Find .
{Hint: , and
.
Putting things together, we get that . But for , so we also get that .}
55. Evaluate .
{Hint: Let .}
56. Let . Evaluate .
{Hint: .}
57. Suppose that f is a decreasing continuously differentiable function on . Show that .
{Hint: .}
58. Perform integration by parts on in order to evaluate .
59.a)Integration by parts is the reversal of the Product Rule. See if you can reverse the Quotient Rule to find a formula for .
{Hint: .}
b) Obtain the same formula using integration by parts on .
60. Find . Find . Find .
{Hint: .}
61. Suppose that f is a function defined on with a continuous second derivative. Also suppose that for . Show that is positive.
{Hint: Integrate by parts twice, and use the fact that .}
62. Let .
a) Find the value of .
b)For , use integration by parts twice to get an equation involving and .
c) Use the equation from part b) and the answer to part a) to find the exact value of .
63. For , find the exact value of .
64. Find the values of a, b, c, d, and e so that .
65. Find .
66. Find .