A. Hints and Answers

Exercise 0.3:

The Rhind value is $256/81 = 3.1604\ldots$

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:

$1^3+2^3+\cdots n^3 = \frac{n^2(n+1)^2}{4}.$

Exercise 2.18:

area $T(a) =\displaystyle { \frac{2}{3} a^{\frac{3}{2}} }$.

Exercise 2.27:

$.027027027\ldots = \frac{1}{37}$.

Exercise 2.36:

I let $${O_j = B(a^{j\over N}, a^{j-1\over N}; 0, a^{-{2j\over N}})}$$ and $${I_j = B(a^{j\over N},a^{j-1\over N}; 0 ,a^{-{2(j-1) \over N}})}$$.

Exercise 3.20:

Recall $(R\mbox{$\Longrightarrow$}S)\mbox{$\Longleftrightarrow$}(S$ or not $R)$.

Exercise 5.61:

$S_1^{ab}[{1\over t}] = S_1^a[{1\over t}] \cup S_a^{ab}[{1\over t}]$. (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 ${1\over 3}$. 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) $(1+{3\over n})^{2n} = ( (1+{3/n})^n)^2$.

Exercise 6.97:

$(1-{c\over n})^n = {(1-{c^2\over n^2})^n \over (1+ {c\over n})^n}$.

Exercise 7.16:

Take $c = {b\over a}$ in lemma 7.13.

Exercise 7.18:

$A_0^af = A_0^{1\over n} f + A_{1\over n}^af$. Show that $A_0^{1\over n} f$ is small when $n$ is large.

Exercise 8.14:

$e){x + 1 \over x} = 1 + {1\over x}$. Not all of these integrals exist.

Exercise 8.16:

Show that $\sum(f,P,S) \leq \sum(g,P,S)$ for every partition $P$ of $[a,b]$ and every sample $S$ for $P$.

Exercise 8.28:

$g$ is the sum of an integrable function and a spike function.

Exercise 8.32:

$f$ is not piecewise monotonic. It is easy to see that $f$ is integrable on $[1,2]$. If you can show it is integrable on $[0,1]$then you are essentially done.

Exercise 8.34:

b) $(b-a)^3/6$.

Exercise 8.41:

For any partition $P$ of $[0.1]$ you can find a sample $S$ such that $\sum(R,P,S) = 0.$

Exercise 8.46:

In equation 8.44, replace $r$ by ${1\over R}$, , and replace $a$ and $b$ by $RA$ and $RB$.

Exercise 8.48:

$\alpha(E_{ab}) = \pi ab.$

Exercise 8.50:

If $a = 1/4$ then both areas are approximately 3.1416

Exercise 8.55:

area = $4\pi$.

Exercise 8.57:

The areas are $5/12$ and $1$.

Exercise 8.58:

The area is ${37\over 12}$. Some fractions with large numerators may appear along the way.

Exercise 9.20:

The last two formulas are obtained from the second by replacing $t$ by $t/2$.

Exercise 9.29:

I used exercise 9.28 with $x={\pi \over 6}$ to find $\cos({\pi \over 6})$. You can also give a more geometric proof.

Exercise 9.44:

You will need to use (9.24).

Exercise 9.48:

$\alpha(S_0^\pi(\sin)) = 2$.

Exercise 9.49:

area $= \sqrt{2}$.

Exercise 9.69:

$\int_0^{\pi/2}\sin(x)dx=1;$ $\int_0^{\pi/2} \sin^2(x)dx = \pi/4;$ $\int_0^{\pi/2}\sin^4(x)dx = 3\pi/16.$

Exercise 10.25:

$f'(a) = -{1\over a^2}.$

Exercise 10.26:

See example 10.9 and 9.26.

Exercise 10.27:

$f'(a) = {1 \over (a+1)^2}.$

Exercise 10.28:

$y=2x-4;$ $y=-6x-4.$

Exercise 11.6:

I used formula 9.25

Exercise 11.15:

${d\over dt}(\vert-100t\vert) = {100 t \over \vert t\vert}.$

Exercise 11.21:

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

Exercise 11.24:

$f'(x) = \ln(x)$, $g'(x) = {ad-bc \over (cx+d)^2}$, $k'(x) = 2(2x+3)(x^2+3x+11)$

Exercise 11.29:

$(g\circ (g \circ g))(x) = ((g\circ g)\circ g)(x) = x$ for $x \in \mbox{{\bf R}}\setminus{0,1}$. If you said $(f\circ f)(x) = x$, calculate both sides when $x = -1$.

Exercise 11.40:

Use the definition of derivative. $h'(2) = 0$.

Exercise 11.43:

$g'(x) = -\tan(x)$, $h'(x) = \tan(x)$, $k'(x) = \sec(x)$, $l'(x) = -\csc(x)$, $m'(x) = 9x^2\ln(5x)$, $n'(x) = {\sqrt{x^2+1}\over x}$ (It requires a lot of calculation to simplify $n'$), $p'(x) = {x^2 \over x+4}$, $q'(x) = \sin(\ln(\vert 6x\vert)).$

Exercise 12.14:

d) Such a function $k$ 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 $F-G$.

Exercise 12.35:

Yes.

Exercise 12.36:

You can apply the chain rule to the identity $f(-x)=f(x)$.

Exercise 13.14:

The function to minimize is $f(x) =$distance $((0,{9 \over 2}),(x,x^2))$.

Exercise 13.15:

You may get a complicated equation of the form $\sqrt{f(x)} = \sqrt{g(x)}$ to solve. Square both sides and the equation should simplify.

Exercise 14.5:

Apply the intermediate value property to $f -f{p}$.

Exercise 14.9:

One of the zeros is in $[1,2]$.

Exercise 14.10:

I showed that if temp$(A) <$ temp$(B) <$ temp$(D)$, then there is a point $Q$ in $DC \cup CA$ such that temp$(Q) =$ temp$(B)$.

Exercise 14.11:

if temp$(A) <$ temp$(B) <$ temp$(C) <$ temp$(D)$, find two points different from $B$ that have the same temperature as $B$.

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 $A$ and $B$.

Exercise 14.54:

$f'(x) = 2\sqrt{a^2-x^2}$; $h'(x) = \arccos(ax)$; $n'(x) = (a^2+b^2)e^{ax}\sin(bx)$; $p'(x)=a^3x^2e^{ax}$.

Exercise 14.55:

It is not true that $l(x) = x$ for all $x$. Note that the image of $l$ is $[-{\pi \over 2},{\pi\over 2}]$.

Exercise 15.5:

$g^{(k)}(t) = tf^{(k)}(t) + kf^{(k-1)}(t)$.

Exercise 15.8:

Use the antiderivative theorem twice.

Exercise 15.9:

$(fg)^{(3)} = fg^{(3)} + 3f^{(1)} g^{(2)} + 3f^{(2)} g^{(1)} + f^{(3)}g$.

Exercise 15.13:

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

Exercise 15.22:

$h(t) = h_0 + v_0t - {1 \over 2}gt^2.$

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 ${1 \over {2n+1}}$ for all $n\in\mbox{${\mbox{{\bf Z}}}^{+}$}$.

Exercise 16.8:

The result is known if $p<q$. To get the result when $q<p$, apply 16.6 to $f$ on $[q,p]$.

Exercise 16.13:

Not all of these integrals make sense. $K'(x)=1$ for all $x \in \mbox{{\bf R}}$. $L'(x) = 1$ for $x \in \mbox{${\mbox{{\bf R}}}^{+}$}$. $L'(x) = -1$ for $x \in \mbox{${\mbox{{\bf R}}}^{-}$}$. $L(0)$ is not defined.

Exercise 17.16:

b) $-\ln(\vert\csc(e^x) + \cot(e^x)\vert)$; f) ${1\over 2}\ln(\vert\sin(2x)\vert)$. i) Cf example 9.68i.j) You did this in exercise 9.69.

Exercise 17.31:

b) ${1\over 2}e^x(\sin(x) -\cos(x)).$ When you do the second integration by parts, be careful not to undo the first. c) $x\arctan(x) - {1\over 2}\ln(1+x^2)$. Let $g'(x) = 1$. d) and e) can be done easily without using integration by parts. f) If $r=-1$ the answer is ${1\over 2}(\ln(\vert x\vert))^2$.

Exercise 17.42:

c) Let $u = \sqrt{x}$. You will need an integration by parts. d) Let $u = \ln(3x)$. e) Remember the definition of $2^x$.

Exercise 17.49:

a) ${1\over2}a^2 \arcsin({x\over a}) + {1\over 2}x\sqrt{a^2-x^2}.$ If you forget how to find $\int\cos^2(\theta)d\theta$, review example 9.53. Also recall that $\sin(2x) = 2\sin(x)\cos(x)$. b) $\ln({x + \sqrt{a^2+x^2} \over a}).$ c) and d) do not require a trigonometric substitution.

Exercise 17.53:

a) ${5\over 4}.$ b) ${\pi \over 6}.$ c)${4\over 15}.$

Exercise 17.54:

${2\pi \over 3} + \sqrt{3}.$

Exercise 17.64:

(g) $\ln(\sqrt{x^2+2x+2} + x + 1).$ First complete the square, and then reduce the problem to $\int {1\over \sqrt{u^2 + 1}}du$.