Chapter 5

(i) Let \(f:(a, b) \rightarrow E\) be finite, continuous, with a right derivative on \((a, b) .\) Prove that \(q=\lim _{x \rightarrow a^{+}} f_{+}^{\prime}(x)\) exists (finite) iff $$ q=\lim _{x, y \rightarrow a^{+}} \frac{f(x)-f(y)}{x-y} $$ i.e., iff $$ (\forall \varepsilon>0)(\exists c>a)(\forall x, y \in(a, c) \mid x \neq y) \quad\left|\frac{f(x)-f(y)}{x-y}-q\right|<\varepsilon $$ [Hints: If so, let \(y \rightarrow x^{+}\) (keeping \(x\) fixed) to obtain $$ (\forall x \in(a, c)) \quad\left|f_{+}^{\prime}(x)-q\right| \leq \varepsilon . \quad \text { (Why?) } $$ Conversely, if \(\lim _{x \rightarrow a^{+}} f_{+}^{\prime}(x)=q,\) then $$ (\forall \varepsilon>0)(\exists c>a)(\forall t \in(a, c)) \quad\left|f_{+}^{\prime}(t)-q\right|<\varepsilon $$ Put $$ M=\sup _{a

Let \(x=f(t), y=g(t),\) where \(t\) varies over an open interval \(I \subseteq E^{1},\) define a curve in \(E^{2}\) parametrically. Prove that if \(f\) and \(g\) have derivatives on \(I\) and \(f^{\prime} \neq 0,\) then the function \(h=f^{-1}\) has a derivative on \(f[I]\), and the slope of the tangent to the curve at \(t_{0}\) equals \(g^{\prime}\left(t_{0}\right) / f^{\prime}\left(t_{0}\right)\). [Hint: The word "curve" implies that \(f\) and \(g\) are continuous on \(I\) (Chapter \(4, \S 10)\), so Theorems 1 and 3 apply, and \(h=f^{-1}\) is a function. Also, \(y=g(h(x)) .\) Use Theorem 3 of \(\S 1 .]\)

Let $$ f(x)=\sin x \text { and } g(x)=\cos x $$ Show that \(f\) and \(g\) are differentiable on \(E^{1},\) with $$ f^{\prime}(p)=\cos p \text { and } g^{\prime}(p)=-\sin p \text { for each } p \in E^{1} . $$ Hence prove for \(n=0,1,2, \ldots\) that $$ f^{(n)}(p)=\sin \left(p+\frac{n \pi}{2}\right) \text { and } g^{(n)}(p)=\cos \left(p+\frac{n \pi}{2}\right) . $$

Let \(I=[0,2 \pi]\) and define \(f, g, h: E^{1} \rightarrow E^{2}(C)\) by $$ \begin{aligned} f(x) &=(\sin x, \cos x) \\ g(x) &=(\sin 3 x, \cos 3 x) \\ h(x) &=\left(\sin \frac{1}{x}, \cos \frac{1}{x}\right) \text { with } h(0)=(0,1) . \end{aligned} $$ Show that \(f[I]=g[I]=h[I]\) (the unit circle; call it \(\mathrm{A}),\) yet \(\ell_{f} A=2 \pi\) \(\ell_{g} A=6 \pi,\) while \(V_{h}[I]=+\infty\). (Thus the result of Problem 7 fails for closed curves and nonsimple arcs.)

Prove that $$ \ln (1+x)=\sum_{k=1}^{n}(-1)^{k+1} \frac{x^{k}}{k}+R_{n}(x), $$ where \(\lim _{n \rightarrow \infty} R_{n}(x)=0\) if \(-1

Prove that if \(f\) is differentiable at \(p\) then \(\lim _{x \rightarrow p^{+}} \frac{f(x)-f(y)}{x-y}\) exists, is finite, and equals \(f^{\prime}(p)\) $$ \begin{array}{l} \text { i.e., }(\forall \varepsilon>0)(\exists \delta>0)(\forall x \in(p, p+\delta))(\forall y \in(p-\delta, p)) \\ \left|\frac{f(x)-f(y)}{x-y}-f^{\prime}(p)\right|<\varepsilon \end{array} $$ Show, by redefining \(f\) at \(p,\) that even if the limit exists, \(f\) may not be differentiable (note that the above limit does not involve \(f(p))\).

Prove that if \(f: E^{1} \rightarrow E^{*}\) is of class \(\mathrm{CD}^{1}\) on \([a, b]\) and if \(-\infty\frac{f(b)-f(a)}{b-a}\left(x_{0}-a\right)+f(a) ; $$ i.e., the curve \(y=f(x)\) lies above the secant through \((a, f(a))\) and \((b, f(b)) .\)

Prove that if \(f\) has a derivative at \(p\), then \(f(p)\) is finite, provided \(f\) is not constantly infinite on any interval \((p, q)\) or \((q, p), p \neq q\). [Hint: If \(f(p)=\pm \infty\), each \(G_{p}\) has points at which \(\frac{\Delta f}{\Delta x}=+\infty\), as well as those \(x\) with \(\frac{\Delta f}{\Delta x}=-\infty .\) ]

(i) Prove that if \(f\) is constant \((f=c \neq \pm \infty)\) on \(I-Q,\) then $$ \int_{a}^{b} f=(b-a) c \quad \text { for } a, b \in I $$ (ii) Hence prove that if \(f=c_{k} \neq \pm \infty\) on $$ I_{k}=\left[a_{k}, a_{k+1}\right), \quad a=a_{0}

Prove that if \(\int f\) exists on each \(I_{n}=\left[a_{n}, b_{n}\right],\) where $$ a_{n+1} \leq a_{n} \leq b_{n} \leq b_{n+1}, \quad n=1,2, \ldots $$ then \(\int f\) exists on $$ I=\bigcup_{n=1}^{\infty}\left[a_{n}, b_{n}\right] $$ itself an interval with endpoints \(a=\inf a_{n}\) and \(b=\sup b_{n}, a, b \in E^{*}\). [Hint: Fix some \(c \in I_{1}\). Define $$ H_{n}(t)=\int_{c}^{t} f \text { on } I_{n}, n=1,2, \ldots $$ Prove that $$ (\forall n \leq m) \quad H_{n}=H_{m} \text { on } I_{n}\left(\text { since }\left\\{I_{n}\right\\} \uparrow\right) . $$ Thus \(H_{n}(t)\) is the same for all \(n\) such that \(t \in I_{n},\) so we may simply write \(H\) for \(H_{n}\) on \(I=\bigcup_{n=1}^{\infty} I_{n} .\) Show that \(H=\int f\) on all of \(I ;\) verify that \(I\) is, indeed, an interval.]