Chapter 8

In electromagnetic theory, the magnetic potential \(u\) at a point on the axis of a circular coil is given by $$ u=A r \int_{a}^{\infty} \frac{d x}{\left(r^{2}+x^{2}\right)^{3 / 2}} $$ where \(A, r\), and \(a\) are constants. Evaluate \(u\).

Be sure you have an indeterminate form before applying l'Hôpital's Rule. $$\lim _{x \rightarrow \infty}[\ln (x+1)-\ln (x-1)]$$

$$ \lim _{x \rightarrow 0} \frac{\tan x-x}{\arcsin x-x} $$

There is a subtlety in the definition of \(\int_{-\infty} f(x) d x\) that is illustrated by the following: Show that (a) \(\int_{-\infty}^{\infty} \sin x d x\) diverges and (b) \(\lim _{a \rightarrow \infty} \int_{-a}^{a} \sin x d x=0\).

Be sure you have an indeterminate form before applying l'Hôpital's Rule. $$\lim _{x \rightarrow 0^{+}} \frac{x}{\ln x}$$

$$ \lim _{x \rightarrow 0} \frac{3 x-\sin x}{x} $$

Consider an infinitely long wire coinciding with the positive \(x\)-axis and having mass density \(\delta(x)=\left(1+x^{2}\right)^{-1}\), \(0 \leq x<\infty\). (a) Calculate the total mass of the wire. (b) Show that this wire does not have a center of mass.

Evaluate \(\int_{-1}^{1} \frac{1}{x \sqrt{-\ln |x|}} d x\) or show that it diverges.

Be sure you have an indeterminate form before applying l'Hôpital's Rule. $$\lim _{x \rightarrow 0^{+}}(\ln x \cot x)$$

$$ \lim _{x \rightarrow 0} \frac{\sin x / 2}{x} $$

Be sure you have an indeterminate form before applying l'Hôpital's Rule. $$\lim _{x \rightarrow \infty} \frac{\int_{1}^{x} \sqrt{1+e^{-t}} d t}{x}$$

$$ \lim _{x \rightarrow 0} \frac{x}{e^{2 x}-1} $$

Let \(f\) be a nonnegative continuous function defined on \(0 \leq x<\infty\) with \(\int_{0}^{\infty} f(x) d x<\infty\). Show that (a) if \(\lim _{x \rightarrow \infty} f(x)\) exists it must be 0 ; (b) it is possible that \(\lim _{x \rightarrow \infty} f(x)\) does not exist.

Suppose that \(f\) is continuous on \([0, \infty)\) except at \(x=1\), where \(\lim _{x \rightarrow 1}|f(x)|=\infty\). How would you define $\int_{0}^{\infty} f(x) d x ? \quad

Be sure you have an indeterminate form before applying l'Hôpital's Rule. $$\lim _{x \rightarrow 1^{+}} \frac{\int_{1}^{x} \sin t d t}{x-1}$$

$$ \lim _{x \rightarrow 0} \frac{e^{x}-1}{e^{-x}-1} $$

We can use a computer to approximate \(\int_{1}^{\infty} f(x) d x\) by taking \(b\) very large in \(\int_{1}^{b} f(x) d x\) provided we know that the first integral converges. Calculate \(\int_{1}^{100}\left(1 / x^{p}\right) d x\) for \(p=2,1.1,1.01\), 1, and 0.99. Note that this gives no hint that the integral \(\int_{1}^{\infty}\left(1 / x^{p}\right) d x\) converges for \(p>1\) and diverges for \(p \leq 1\).

Find the area of the region between the curves \(y=(x-8)^{-2 / 3}\) and \(y=0\) for \(0 \leq x<8 .\)

Find each limit. Hint: Transform to problems involving a continuous variable \(x\). Assume that \(a>0\). (a) \(\lim _{n \rightarrow \infty} \sqrt[n]{a}\) (b) \(\lim _{n \rightarrow \infty} \sqrt[n]{n}\) (c) \(\lim _{n \rightarrow \infty} n(\sqrt[n]{a}-1)\) (d) \(\lim _{n \rightarrow \infty} n(\sqrt[n]{n}-1)\)

Find the area of the region between the curves \(y=1 / x\) and \(y=1 /\left(x^{3}+x\right)\) for \(0

Find each limit. (a) \(\lim _{x \rightarrow 0^{+}} x^{x}\) (b) \(\lim _{x \rightarrow 0^{+}}\left(x^{x}\right)^{x}\) (c) \(\lim _{x \rightarrow 0^{+}} x^{\left(x^{x}\right)}\) (d) \(\lim _{x \rightarrow 0^{+}}\left(\left(x^{x}\right)^{x}\right)^{x}\) (e) \(\lim _{x \rightarrow 0^{+}} x^{\left(x^{\left(x^{*} \mid\right.}\right)}\)

Let \(R\) be the region in the first quadrant below the curve \(y=x^{-2 / 3}\) and to the left of \(x=1\). (a) Show that the area of \(R\) is finite by finding its value. (b) Show that the volume of the solid generated by revolving \(R\) about the \(x\)-axis is infinite.

Find \(b\) so that \(\int_{0}^{b} \ln x d x=0\).

Find each limit. (a) \(\lim _{x \rightarrow 0^{+}}\left(1^{x}+2^{x}\right)^{1 / x}\) (b) \(\lim _{x \rightarrow 0^{-}}\left(1^{x}+2^{x}\right)^{1 / x}\) (c) \(\lim _{x \rightarrow \infty}\left(1^{x}+2^{x}\right)^{1 / x}\) (d) \(\lim _{x \rightarrow-\infty}\left(1^{x}+2^{x}\right)^{1 / x}\)

Is \(\int_{0}^{1} \frac{\sin x}{x} d x\) an improper integral? Explain.

For \(k \geq 0\), find $$ \lim _{n \rightarrow \infty} \frac{1^{k}+2^{k}+\cdots+n^{k}}{n^{k+1}} $$ Hint: Though this has the \(\infty / \infty\) form, l'Hôpital's Rule is not helpful. Think of a Riemann sum.

Let \(c_{1}, c_{2}, \ldots, c_{n}\) be positive constants with \(\sum_{i=1}^{n} c_{i}=1\), and let \(x_{1}, x_{2}, \ldots, x_{n}\) be positive numbers. Take natural logarithms and then use l'Hôpital's Rule to show that $$ \lim _{t \rightarrow 0^{+}}\left(\sum_{i=1}^{n} c_{i} x_{i}^{t}\right)^{1 / t}=x_{1}^{c_{1}} x_{2}^{c_{2} \cdots} x_{n}^{c_{n}}=\prod_{i=1}^{n} x_{i}^{c_{i}} $$ Here \(\prod\) means product; that is, \(\prod_{i=1}^{n} a_{i}\) means \(a_{1} \cdot a_{2} \cdots \cdots a_{n}\). In particular, if \(a, b, x\), and \(y\) are positive and \(a+b=1\), then $$ \lim _{t \rightarrow 0^{+}}\left(a x^{t}+b y^{t}\right)^{1 / t}=x^{a} y^{b} $$

Consider \(f(x)=n^{2} x e^{-n x}\). (a) Graph \(f(x)\) for \(n=1,2,3,4,5,6\) on \([0,1]\) in the same graph window. (b) For \(x>0\), find \(\lim _{n \rightarrow \infty} f(x)\). (c) Evaluate \(\int_{0}^{1} f(x) d x\) for \(n=1,2,3,4,5,6\). (d) Guess at \(\lim _{n \rightarrow \infty} \int_{0}^{1} f(x) d x\). Then justify your answer rigorously.

Find the absolute maximum and minimum points (if they exist) for \(f(x)=\left(x^{25}+x^{3}+2^{x}\right) e^{-x}\) on \([0, \infty)\).

(Gamma Function) Let \(\Gamma(n)=\int_{0}^{\infty} x^{n-1} e^{-x} d x, n>0\). This integral converges by Problems 51 and 52 . Show each of the following (note that the gamma function is defined for every positive real number \(n\) ): (a) \(\Gamma(1)=1\) (b) \(\Gamma(n+1)=n \Gamma(n)\) (c) \(\Gamma(n+1)=n\) !, if \(n\) is a positive integer.

The gamma probability density function is $$ f(x)= \begin{cases}C x^{\alpha-1} e^{-\beta x}, & \text { if } x>0 \\ 0, & \text { if } x \leq 0\end{cases} $$ where \(\alpha\) and \(\beta\) are positive constants. (Both the gamma and the Weibull distributions are used to model lifetimes of people, animals, and equipment.) (a) Find the value of \(C\), depending on both \(\alpha\) and \(\beta\), that makes \(f(x)\) a probability density function. (b) For the value of \(C\) found in part (a), find the value of the mean \(\mu\). (c) For the value of \(C\) found in part (a), find the variance \(\sigma^{2}\).

The Laplace transform, named after the French mathematician Pierre-Simon de Laplace (1749-1827), of a function \(f(x)\) is given by \(L\\{f(t)\\}(s)=\int_{0}^{\infty} f(t) e^{-s t} d t\). Laplace transforms are useful for solving differential equations. (a) Show that the Laplace transform of \(t^{\alpha}\) is given by \(\Gamma(\alpha+1) / s^{\alpha+1}\) and is defined for \(s>0\). (b) Show that the Laplace transform of \(e^{\alpha t}\) is given by \(1 /(s-\alpha)\) and is defined for \(s>\alpha\). (c) Show that the Laplace transform of \(\sin (\alpha t)\) is given by \(\alpha /\left(s^{2}+\alpha^{2}\right)\) and is defined for \(s>0\).

Suppose that \(0