Next: B. Proofs of Some Up: Math 111 Calculus I Previous: Bibliography   Index

Exercise 0.3:
The Rhind value is

Exercise 1.7:
Look at the boundary.

Exercise 1.10:
If a set has no endpoints, then it contains all of its endpoints and none of its endpoints.

Exercise 2.10:

Exercise 2.18:
area .

Exercise 2.27:
.

Exercise 2.36:
I let and .

Exercise 3.20:
Recall or not .

Exercise 5.61:
. (Draw a picture.)

Exercise 5.80:
Consider a partition with a fairly large number of points.

Exercise 6.33:
The assertion is false.

Exercise 6.59:
(part e) The limit is . It simplifies matters if you factor both the numerator and the denominator. The sequence in part g) is a translate of the sequence in part f).

Exercise 6.69:
All four statements are false.

Exercise 6.94:
a) .

Exercise 6.97:
.

Exercise 7.16:
Take in lemma 7.13.

Exercise 7.18:
. Show that is small when is large.

Exercise 8.14:
. Not all of these integrals exist.

Exercise 8.16:
Show that for every partition of and every sample for .

Exercise 8.28:
is the sum of an integrable function and a spike function.

Exercise 8.32:
is not piecewise monotonic. It is easy to see that is integrable on . If you can show it is integrable on then you are essentially done.

Exercise 8.34:
b) .

Exercise 8.41:
For any partition of you can find a sample such that

Exercise 8.46:
In equation 8.44, replace by , , and replace and by and .

Exercise 8.48:

Exercise 8.50:
If then both areas are approximately 3.1416

Exercise 8.55:
area = .

Exercise 8.57:
The areas are and .

Exercise 8.58:
The area is . Some fractions with large numerators may appear along the way.

Exercise 9.20:
The last two formulas are obtained from the second by replacing by .

Exercise 9.29:
I used exercise 9.28 with to find . You can also give a more geometric proof.

Exercise 9.44:
You will need to use (9.24).

Exercise 9.48:
.

Exercise 9.49:
area .

Exercise 9.69:

Exercise 10.25:

Exercise 10.26:
See example 10.9 and 9.26.

Exercise 10.27:

Exercise 10.28:

Exercise 11.6:
I used formula 9.25

Exercise 11.15:

Exercise 11.21:
You can use the definition of derivative, or you can use the product rule and the reciprocal rule.

Exercise 11.24:
, ,

Exercise 11.29:
for . If you said , calculate both sides when .

Exercise 11.40:
Use the definition of derivative. .

Exercise 11.43:
, , , , , (It requires a lot of calculation to simplify ), ,

Exercise 12.14:
d) Such a function does exist.

Exercise 12.15:
a) Use extreme value property.

Exercise 12.27:
Proof is like given proof of corollary 12.26.

Exercise 12.31:
Apply corollary 12.26 to .

Exercise 12.35:
Yes.

Exercise 12.36:
You can apply the chain rule to the identity .

Exercise 13.14:
The function to minimize is distance .

Exercise 13.15:
You may get a complicated equation of the form to solve. Square both sides and the equation should simplify.

Exercise 14.5:
Apply the intermediate value property to .

Exercise 14.9:
One of the zeros is in .

Exercise 14.10:
I showed that if temp temp temp, then there is a point in such that temp temp.

Exercise 14.11:
if temp temp temp temp, find two points different from that have the same temperature as .

Exercise 14.17:
You may want to define some of these functions using more than one formula.

Exercise 14.41:
Use the extreme value property to get and .

Exercise 14.54:
; ; ; .

Exercise 14.55:
It is not true that for all . Note that the image of is .

Exercise 15.5:
.

Exercise 15.8:
Use the antiderivative theorem twice.

Exercise 15.9:
.

Exercise 15.13:
You will need to use a few trigonometric identities, including the reflection law (9.18).

Exercise 15.22:

Exercise 15.29:
Use theorem 15.27 and corollary 12.26

Exercise 16.2:
You probably will not be able to find a single formula'' for this. My function has a local maximum at for all .

Exercise 16.8:
The result is known if . To get the result when , apply 16.6 to on .

Exercise 16.13:
Not all of these integrals make sense. for all . for . for . is not defined.

Exercise 17.16:
b) ; f) . i) Cf example 9.68i.j) You did this in exercise 9.69.

Exercise 17.31:
b) When you do the second integration by parts, be careful not to undo the first. c) . Let . d) and e) can be done easily without using integration by parts. f) If the answer is .

Exercise 17.42:
c) Let . You will need an integration by parts. d) Let . e) Remember the definition of .

Exercise 17.49:
a) If you forget how to find , review example 9.53. Also recall that . b) c) and d) do not require a trigonometric substitution.

Exercise 17.53:
a) b) c)

Exercise 17.54:

Exercise 17.64:
(g) First complete the square, and then reduce the problem to .

Next: B. Proofs of Some Up: Math 111 Calculus I Previous: Bibliography   Index
Ray Mayer 2007-09-07