Chapter 7

For a fixed positive number \(\beta\), find $$ \lim _{n \rightarrow \infty}\left[\frac{1^{\beta}+2^{\beta}+\cdots+n^{\beta}}{n^{\beta+1}}\right] $$

Find the unique solution of each of the following differential equations: a. \(\left\\{\begin{array}{ll}F^{\prime}(x)+F(x)=x & \text { for all } x \\\ F(0)=1\end{array}\right.\) b. \(\left\\{\begin{array}{ll}F^{\prime}(x)+4 F(x)=e^{x} & \text { for all } x \\\ F(2)=31\end{array}\right.\) c. \(\left\\{\begin{array}{ll}F^{\prime}(x)+F(x)=x^{2} & \text { for all } x \\\ F(0)=-1\end{array}\right.\)

Evaluate the following integrals: a. \(\int_{1}^{2} x e^{x^{2}} d x\) b. \(\int_{0}^{1}(1-x)^{2} \sqrt{2+x} d x\) c. \(\int_{2}^{3} x^{3} e^{x^{2}} d x\) d. \(\int_{2}^{\pi} x^{2} \cos x d x\)

Find $$ \lim _{n \rightarrow \infty}\left[\frac{1}{n+1}+\frac{1}{n+2}+\cdots+\frac{1}{2 n}\right] $$

For each of the following integrals, verify Corollary 7.20 by direct computation. a. \(\int_{0}^{1}\left(x^{2}+x^{3}\right) d x\) b. \(\int_{2}^{3}(x+1)^{2} d x\)

For numbers \(c\) and \(a\), consider the differential equation $$ \left\\{\begin{array}{ll} F^{\prime}(x) & =c(a-F(x)) \\ F(0) & =0 \end{array} \quad\right. \text { for all } x $$ Prove that the unique solution is given by the formula $$ F(x)=a\left(1-e^{-c x}\right) \quad \text { for all } x $$

Evaluate the following integrals: a. \(\int_{1}^{e}(\ln x)^{2} d x\) b. \(\int_{4}^{5} \frac{1+x}{1-x} d x\) c. \(\int_{4}^{9} \frac{1}{1-x^{2}} d x\) d. \(\int_{3}^{4}\left(\frac{1}{x^{2}-2 x}+\frac{1}{1+\sqrt{x}}\right) d x\)

Find $$ \lim _{n \rightarrow \infty}\left[\sum_{k=1}^{n} \frac{k}{n^{2}+k^{2}}\right] $$

Suppose that the function \(f:[a, b] \rightarrow \mathbb{R}\) is continuous and that its restriction to the open interval \((a, b)\) has a bounded second derivative. Let \(n\) be a natural number Show that $$ \int_{a}^{b} f(x) d x=\left(\frac{b-a}{n}\right)\left[\frac{f(a)}{2}+\sum_{k=1}^{n-1} f\left(a+\frac{k}{n}(b-a)\right)+\frac{f(b)}{2}\right]+E $$ where $$ |E| \leq \frac{(b-a)^{3}}{12 n^{2}} \sup \left\\{\left|f^{\prime \prime}(x)\right| x \text { in }(a, b)\right\\}. $$

Prove that for any two natural numbers \(n\) and \(m\), $$ \int_{0}^{1} x^{m}(1-x)^{n} d x=\int_{0}^{1}(1-x)^{m} x^{n} d x $$

Suppose that the function \(f: \mathbb{R} \rightarrow \mathbb{R}\) has a continuous second derivative. Fix a number \(a\). Prove that $$ \int_{a}^{x} f^{\prime \prime}(t)(x-t) d t=-(x-a) f^{\prime}(a)+f(x)-f(a) $$ for all \(x\)

Suppose that the function \(f: \mathbb{R} \rightarrow \mathbb{R}\) is continuous and that $$ f(x)=\int_{0}^{x} f(t) d t $$ for all \(x\). Prove that \(f(x)=0\) for all \(x\)

Suppose that the function \(f:[0,1] \rightarrow \mathbb{R}\) is integrable. Prove that $$ \lim _{n \rightarrow \infty} \frac{1}{n}\left[f\left(\frac{1}{n}\right)+f\left(\frac{2}{n}\right)+\cdots+f\left(\frac{n-1}{n}\right)+f(1)\right]=\int_{0}^{1} f $$

Suppose that the function \(g: \mathbb{R} \rightarrow \mathbb{R}\) is continuous and that \(g(x)>0\) for all \(x\) Define $$ h(x)=\int_{0}^{x} \frac{1}{g(t)} d t \quad \text { for all } x $$ and let \(J=h(\mathbb{R}) .\) Prove that if \(f: J \rightarrow \mathbb{R}\) is the inverse of \(h: \mathbb{R} \rightarrow \mathbb{R},\) then \(f: J \rightarrow \mathbb{R}\) is a solution of the nonlinear differential equation $$ \left\\{\begin{array}{l} f^{\prime}(x)=g(f(x)) \quad \text { for all } x \text { in } J \\ f(0)=0 \end{array}\right. $$

An approximation rule similar to the Trapezoid Rule is the Midpoint Rule. This rule approximates the integral \(\int_{c}^{d} f(x) d x\) by \((d-c) f([d+c] / 2) .\) Prove that if the function \(f:[c, d] \rightarrow \mathbb{R}\) is continuous and its restriction \(f:(c, d) \rightarrow \mathbb{R}\) has a second derivative, then for some point \(\zeta\) in \((c, d)\) $$ \int_{c}^{d} f(x) d x=(d-c) f\left(\frac{c+d}{2}\right)+\frac{1}{24}(d-c)^{3} f^{\prime \prime}(\zeta) $$ [Hint: Let \(m=(c+d) / 2,\) let \(t_{0}=(d-c) / 2\) and define $$ H(t)=\left[\int_{m-t}^{m+t} f\right]-2 t f(m)-\left(\frac{t}{t_{0}}\right)^{3} \int_{m-t_{0}}^{m+t_{0}} f-2 \frac{t^{3}}{t_{0}^{2}}(f) m \quad \text { for }-t_{0} \leq t \leq t_{0} $$ Apply the Generalized Rolle's Theorem with \(n=1\) to the function \(\left.H:\left[-t_{0}, t_{0}\right] \rightarrow \mathbb{R} .\right]\)

Suppose that the function \(f:[a, b] \rightarrow \mathbb{R}\) is integrable. In order for a sequence \(\left\\{P_{n}\right\\}\) of partitions of the domain \([a, b]\) to be an Archimedean sequence of partitions for \(f\) on \([a, b],\) is it necessary that \(\lim _{n \rightarrow \infty}\) gap \(P_{n}=0 ?\)

For a continuous function \(f:[a, b] \rightarrow \mathbb{R}\) and a partition \(P\) of its domain \([a, b]\) show that both the Trapezoidal Rule and Simpson's Rule approximations of the integral of \(f\) on \([a, b]\) are Riemann sums.

Let \(p\) and \(n\) be natural numbers with \(n \geq 2\). Prove that $$ \sum_{k=1}^{n-1} k^{p} \leq \frac{n^{p+1}}{p+1} \leq \sum_{k=1}^{n} k^{p} $$ (Hint: Use an induction argument on \(n .)\)

(Fermat's Method for Computing \(\left.\int_{1}^{b} x^{\beta} d x .\right)\) Let \(b>1\) and \(\beta \neq-1 .\) Define \(f(x)=x^{\beta}\) for all \(x\) in \([1, b] .\) For each natural number \(n,\) let \(P_{n}=\left\\{x_{0}, \ldots, x_{n}\right\\}\) be the partition of \([1, b]\) defined by \(x_{i}=b^{i / n}\) for \(0 \leq i \leq n\) a. Show that $$ \sum_{i=1}^{n} f\left(x_{i}\right)\left(x_{i}-x_{i-1}\right)=\frac{b^{1 / n}-1}{b^{1 / n}}\left[\frac{1-\left(b^{\beta+1}\right)^{(n+1) / n}}{1-b^{(\beta-1) / n}}-1\right] $$ b. Show that $$ \lim _{n \rightarrow \infty} \frac{1-\left(b^{1 / n}\right)^{\beta+1}}{1-b^{1 / n}}=\beta+1 $$ c. Use (a) and (b) to show that $$ \int_{1}^{b} x^{\beta} d x=\lim _{n \rightarrow \infty} \sum_{i=1}^{n} f\left(x_{i}\right)\left(x_{i}-x_{i-1}\right)=\frac{b^{\beta+1}-1}{\beta+1} $$