A. Hints and Answers

**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 .