Problem 18
The forward difference operators with spacing \(h>0\) are defined by $$ \begin{array}{c} \Delta^{0} f(x)=f(x), \quad \Delta f(x)=f(x+h)-f(x) . \\ \Delta^{n+1} f(x)=\Delta\left[\Delta^{n} f(x)\right], \quad n \geq 1 . \end{array} $$ (a) Prove by induction on \(n:\) If \(k \geq 2, c_{1}, \ldots, c_{k}\) are constants, and \(n \geq 1,\) then $$ \Delta^{n}\left[c_{1} f_{1}(x)+\cdots+c_{k} f_{k}(x)\right]=c_{1} \Delta^{n} f_{1}(x)+\cdots+c_{k} \Delta^{n} f_{k}(x) $$ (b) Prove by induction: If \(n \geq 1\), then $$ \Delta^{n} f(x)=\sum_{m=0}^{n}(-1)^{n-m}\left(\begin{array}{c} n \\ m \end{array}\right) f(x+m h) $$
Problem 18
In Exercises 2.4.2-2.4.40, find the indicated limits. $$ \lim _{x \rightarrow \pi / 2}|\tan x|^{\cos x} $$
Problem 18
Suppose that \(f\) is continuous on \([a, b],
f_{+}^{\prime}(a)<\mu
Problem 19
Let $$ f(x)=\frac{\sin x}{x}, \quad x \neq 0 $$ (a) Define \(f(0)\) so that \(f\) is continuous at \(x=0 .\) HINT: Use Exercise 2.3 .8 . (b) Show that if \(\bar{x}\) is a local extreme point of \(f,\) then $$ |f(\bar{x})|=\left(1+\bar{x}^{2}\right)^{-1 / 2} $$ HINT: Express \(\sin x\) and \(\cos x\) in terms of \(f(x)\) and \(f^{\prime}(x),\) and add their squares to obtain a useful identity. (c) Show that \(|f(x)| \leq 1\) for all \(x .\) For what value of \(x\) is equality attained?
Problem 19
Define (a) \(\lim _{x \rightarrow x_{0}-} f(x)=-\infty\) (b) \(\lim _{x \rightarrow x_{0}+} f(x)=\infty\) (c) \(\lim _{x \rightarrow x_{0}+} f(x)=-\infty\)
Problem 19
(a) Show that if \(f_{1}, f_{2}, \ldots, f_{n}\) are continuous on a set \(S\) then so are \(f_{1}+f_{2}+\) \(\cdots+f_{n}\) and \(f_{1} f_{2} \cdots f_{n}\) (b) Use (a) to show that a rational function is continuous for all values of \(x\) except the zeros of its denominator.
Problem 19
In Evercises \(2.5 .19-2.5 .22, \Delta\) is the forwand difference operator with spacing \(h>0\). Let \(m\) and \(n\) be nonnegative integers, and let \(x_{0}\) be any real number. Prove by induction on \(n\) that $$ \Delta^{n}\left(x-x_{0}\right)^{m}=\left\\{\begin{array}{ll} 0 & \text { if } 0 \leq m \leq n_{+} \\ n ! h^{n} & \text { if } m=n . \end{array}\right. $$ Does this suggest an analogy between "differencing" and differentiation?
Problem 20
In Evercises \(2.5 .19-2.5 .22, \Delta\) is the forwand difference operator with spacing \(h>0\). Find an upper bound for the magnitude of the error in the approximation $$ f^{\prime \prime}\left(x_{0}\right) \approx \frac{\Delta^{2} f\left(x_{0}-h\right)}{h^{2}} $$ (a) assuming that \(f^{\prime \prime \prime \prime}\) is bounded on \(\left(x_{0}-h, x_{0}+h\right) ;\) (b) assuming that \(f^{(4)}\) is bounded on \(\left(x_{0}-h, x_{0}+h\right)\).
Problem 20
Find (a) \(\lim _{x \rightarrow 0+} \frac{1}{x^{3}}\) (b) \(\lim _{x \rightarrow 0-} \frac{1}{x^{3}}\) (c) \(\lim _{x \rightarrow 0+} \frac{1}{x^{6}}\) (d) \(\lim _{x \rightarrow 0-} \frac{1}{x^{6}}\) (e) \(\lim _{x \rightarrow x_{0}+} \frac{1}{\left(x-x_{0}\right)^{2 k}}\) (f) \(\lim _{x \rightarrow x_{0}-} \frac{1}{\left(x-x_{0}\right)^{2 k+1}}\) \((k=\) positive integer)
Problem 20
Let \(n\) be a positive integer and $$ f(x)=\frac{\sin n x}{n \sin x}, \quad x \neq k \pi \quad(k=\text { integer }) $$ (a) Define \(f(k \pi)\) so that \(f\) is continuous at \(k \pi .\) HINT: Use Exercise 2.3 .8 . (b) Show that if \(\bar{x}\) is a local extreme point of \(f,\) then $$ |f(\bar{x})|=\left[1+\left(n^{2}-1\right) \sin ^{2} \bar{x}\right]^{-1 / 2} $$ HINT: Express \(\sin n x\) and \(\cos n x\) in terms of \(f(x)\) and \(f^{\prime}(x),\) and add their squares to obtain a useful identity. (c) Show that \(|f(x)| \leq 1\) for all \(x .\) For what values of \(x\) is equality attained?
