Chapter 25

Without evaluating the following definite integrals give bounds within which they lie, that is, give two numbers, one less than and one more than the value of the integral. (a) \(\int_{0}^{\pi / 2} \sin ^{5} x d x . \quad\) (b) \(\int_{0}^{\pi} \sin ^{5} x d x\). (c) \(\int_{0}^{1}\left(x^{10}+9 x^{9}\right) d x . \quad\) (d) \(\int_{0}^{1} e^{x^{2}} d x\). (e) \(\int_{0}^{1} e^{-x^{2}} d x\).

$$ \text { Evaluate } \int_{0}^{2} f(x) d x \text { where } f(x)=\left\\{\begin{array}{r} x^{2} \text { for } 0 \leq x<1 \\ 3 x^{2} \text { for } 1 \leq x \leq 2 . \end{array}\right. $$

What does the statement that \(\int_{0}^{2} 3 x^{2} d x\) exists mean?

Determine the limits of the following sequences by using the theorems on limits of sequences cited in the text: (a) \(s_{n}=\frac{n}{n^{2}+2} \quad\) (b) \(s_{n}=\frac{n+1}{n+2}\) (c) \(s_{n}=\frac{n^{2}-3 n+1}{2 n^{2}-4 n+1} \quad\) (d) \(s_{n}=\frac{2 n+1}{3 n+1}\)

What are the first five terms in the sequence whose \(n\)th term is the following: (a) \(n^{2}\). (b) \(\frac{2 n+1}{n+2}\) (c) \((-1)^{n} \frac{1}{n}\) (d) \(\frac{n^{2}-1}{2 n^{2}+3 n+1}\)

Find the limits called for in the following problems by applying the theorems on limits: (a) \(\lim _{x \rightarrow 2} \frac{x+4}{x}\). Ans. 3 . (b) \(\lim _{x \rightarrow 2} 5 x\) (c) \(\lim _{x \rightarrow 3} x^{2}\) (c) \(\lim _{x \rightarrow 3} x^{2} \quad\) Ans. 9 . (d) \(\lim _{x \rightarrow 2} \frac{x^{2}-5 x}{x^{2}+4 x+6}\). (e) \(\lim _{\Delta x \rightarrow 0} 3 x_{0}^{2}+3 x_{0} \Delta x+(\Delta x)^{2} . \quad\)

State in your own words what one means by the statement that \(y\) is a function of \(x\).

Given that \(F(u)=\int_{0}^{u} x d x\), calculate \(F(3)\).

Evaluate the following: (a) \(\int_{0}^{a} \frac{d x}{\sqrt{a^{2}-x^{2}}} \quad\) (b) \(\int_{0}^{1} \frac{d x}{x^{2}}\). (c) \(\int_{0}^{a} \frac{x d x}{\sqrt{a^{2}-x^{2}}}, \quad\) (d) \(\int_{0}^{2} \frac{d x}{x}\). (e) \(\int_{0}^{1} \frac{d x}{x^{a}}\) for \(0<\mathrm{a}<1 . \quad\) (f) \(\int_{0}^{1} \frac{d x}{x^{a}}\) for \(a>1\). (g) \(\int_{0}^{1} \frac{d x}{(x-1)^{2 / 3}}\). (h) \(\int_{1}^{2} \frac{d x}{(x-1)^{2 / 3}}\). (i) \(\int_{0}^{1} \frac{d x}{(1-x)^{2}}\). Ans. No value. (j) \(\int_{0}^{1} \cot x d x\). (k) \(\int_{0}^{\pi / 2} \sec x d x\) Ans. No value. (1) \(\int_{-1}^{0} \frac{d x}{x^{2}-x}\).

Show by application of the definition of the definite integral and the theorems on sequences that the following hold: (a) \(\int_{a}^{b}[f(x)+g(x)] d x=\int_{a}^{b} f(x) d x+\int_{a}^{b} g(x) d x\). (b) \(\int_{a}^{b} c f(x) d x=c \int_{a}^{b} f(x) d x\). We assume that all the integrals involved exist.

Using the proof of the theorem on the limit of the sum of two functions as a guide, see if you can prove that the limit of the sum of two sequences is the sum of the limits of the separate sequences.

Prove by applying the \(\epsilon-\delta\) definition of a limit of a function that the following hold: (a) \(\lim _{x \rightarrow 2} 4 x=8\). (b) \(\lim _{x \rightarrow 3} 3 x=9\). (c) \(\lim _{x \rightarrow 2} 4 x+7=15\). (d) \(\lim _{x \rightarrow 2} 4=4\). (e) \(\lim _{x \rightarrow 2} x=2\).

What is the domain of a function? The range of a function?

Given that $$ F(u)=\int_{1}^{u}\left(x \sin x+e^{x} \sin x\right)^{2} d x, $$ show by any sound argument, geometrical or analytical, that \(F(u)\) is an increasing function of \(u\). Suggestion: The integrand is positive.

If \(s_{1}, s_{2}, s_{3}, \ldots\) is a sequence of positive numbers whose limit is 0 and if \(t_{1}, t_{2}, t_{3}, \ldots\) is another sequence of positive numbers such that \(t_{n} \leq s_{n}\) for each \(n\), prove that the limit of the second sequence is 0 .

Consider the quadratic equation \(a x^{2}+b x+c=0\). The roots are given by the formula $$ x=\frac{-b \pm \sqrt{b^{2}-4 a c}}{2 a} $$ If \(a\) is 0 , the equation becomes \(b x+c=0\) and there is just the one root, \(x=-c / b\). Show that as \(a\) approaches 0 , one root of the quadratic equation approaches the root of the linear equation and the other root of the quadratic equation becomes infinite. Suggestion: Multiply the numerator and denominator of the expressions for \(x\) by the numerator.

Given that $$ F(u)=\int_{1}^{u} x^{3 / 2} \sin x d x, $$ find \(F^{\prime}(x)\).

If the sequence \(s_{1}, s_{2}, s_{3}, \ldots\) has the limit \(s\) and if \(c\) is a constant, prove that \(c s_{1}, c s_{2}, c s_{3}, \ldots\) has the limit \(c s\).

Show that the values of the first 100 terms of any sequence are immaterial in determining the limit of a sequence.

Since \(\lim _{x \rightarrow 0} \frac{\sin m x}{x}=m\) and \(\lim _{x \rightarrow 0} \frac{x}{\sin n x}=\frac{1}{n}\), what is \(\lim _{x \rightarrow 0} \frac{\sin m x}{\sin n x}\) ? Ans. \(m / n\).

Given a function \(y=f(x)\), suppose we know that for a given \(\epsilon\) there exists a \(\delta\) such that when \(0<|x-a|<\delta\) then \(|y-b|<\epsilon\). Would \(|y-b|\) be less than \(\epsilon\) if we replaced \(\delta\) by any positive number less than \(\delta ?\)

Graph the following functions: (a) \(y=\left\\{\begin{aligned} x^{2} & \text { for } 0 \leq x \leq 1 . \\ x & \text { for } 1 \leq x \leq 5 . \end{aligned}\right.\) (b) \(y=\sqrt{x}\) (c) \(y=\left\\{\begin{array}{ccc}x+2 & \text { for } & 0 \leq x \leq 1 \\ 3 x & \text { for } & 1 \leq x \leq 6\end{array}\right.\)

Evaluate the following expressions: (a) \(\lim _{h \rightarrow 0} \int_{p}^{\pi+h} \sin x d x . \quad\) Ans. \(0 .\) (b) \(\lim _{h \rightarrow 0} \int_{\pi / 2}^{(\pi / 2)+h} \sqrt{\sin x} d x\). (c) \(\frac{d}{d t} \int_{0}^{t^{2}} \sin x d x\). Suggestion: Let \(t^{2}=u\) and use the chain rule. (d) \(\frac{d}{d t} \int_{0}^{t^{2}} \sqrt{\sin x} d x\). (e) \(\lim _{h \rightarrow 0} \frac{1}{h} \int_{a}^{a+h} f(t) d t\). (f) \(\lim _{x \rightarrow 0} \frac{1}{x} \int_{0}^{x} \sqrt{t^{3}+4} d t\). (g) \(\lim _{x \rightarrow 1} \frac{1}{x-1} \int_{1}^{x} \tan ^{-1} t d t\).

If an object is projected along smooth ground with an initial velocity of \(100 \mathrm{ft} / \mathrm{sec}\), but is subject to air resistance that is proportional to the velocity, the velocity of the object at any time \(t\) is given by \(v=100 e^{-k t}\) where \(k\) is a constant which depends on the amount of air resistance. The distance covered by the object in infinite time is given by \(\int_{0}^{\infty} 100 e^{-k t} d t\). Find this distance.

If the sequence \(s_{1}, s_{2}, s_{3}, \ldots\) has the limit \(s\) and \(a\) is a constant, prove that the sequence \(a+s_{1}, a+s_{2}, a+s_{3}, \ldots\) has the limit \(a+s\).

Use the precise definition of the limit of a sequence to prove that the following sequences have the limit indicated: (a) \(0,1, \frac{4}{3}, \frac{6}{4}, \frac{8}{5}, \cdots,(2 n-2) / n \cdots\) has the limit 2 . (b) \(1 \frac{1}{2}, 1 \frac{3}{4}, 1 \frac{7}{8}, \cdots,\left(2^{n+1}-1\right) / 2^{n}, \cdots\) has the limit 2 . (c) \(\frac{1}{2}, \frac{1}{4}, \frac{1}{8}, \cdots, 1 / 2^{n}, \cdots\) has the limit 0 .

If \(f(x)=\left\\{\begin{array}{cl}x^{2}+1 & \text { for } x>0 \\\ -\left(x^{2}+1\right) & \text { for } x<0^{\prime} \text { does } \lim _{x \rightarrow 0} f(x) \text { exist? }\end{array}\right.\)

Prove that if \(f(x)\) has the limit \(b\) as \(x\) approaches \(a\), then \(b\) is unique; that is, there cannot be another limit, say \(c\), in addition to \(b\).

The function \(F(x)=\int_{0}^{x} \frac{d t}{\sqrt{t^{3}+1}}\) cannot be evaluated in terms of elementary functions. Nevertheless, answer the following questions: (a) What is \(F(0)\) ? Ans. 0 . (b) Is \(F\left(-\frac{1}{2}\right)>0\) or \(<0\) ? (c) Does \(F(-3)\) exist? Ans. No. (d) What is \(F^{\prime}(x)\) ? (e) What is \(F^{\prime \prime}(x)\) ? Ans. \(-\frac{3}{2} \frac{x^{2}}{\sqrt{\left(x^{3}+1\right)^{3}}}\). (f) In view of (a), (d), and (e), give an approximate expression for \(F(x)\) around \(x=0\) by using Taylor's theorem. Ans. \(x+F^{\prime \prime \prime}(\mu) \frac{x^{3}}{3 !}, 0<\mu-1 ?\)

If an object is projected along smooth ground with an initial velocity of \(100 \mathrm{ft} / \mathrm{sec}\), but is subject to air resistance that is proportional to the square of the velocity, the velocity of the object at any time \(t\) is given by \(v=100 /(1+100 k t)\) where \(k\) is a constant which depends on the amount of air resistance. The distance covered by the object in infinite time is given by \(\int_{0}^{\infty} 100 d t /(1+100 k t)\). Find this distance.

Show that if a sequence has a limit, the sequence consisting of just the even- numbered terms has the same limit.

Show by application of the precise definition that the sequence \((30)\) of the text cannot have \(1 \frac{1}{2}\) as a limit.

Given that \(g(x)=\left\\{\begin{array}{ll}1 & \text { for } x=0,1,2, \cdots \\\ \frac{1}{x} & \text { for all other values of } x^{\prime}\end{array}\right.\), what are the following? (a) \(\lim _{x \rightarrow 1} g(x)\). Ans. 1 . (b) \(\lim _{x \rightarrow 2} g(x)\). (c) \(\lim _{x \rightarrow 1} \frac{g(x)}{x}\). (d) \(\lim _{x \rightarrow 2} \frac{g(x)}{x}\).

Determine whether the following limits exist and if they do state what they are. No rigorous proof need be given. (a) \(\lim _{x \rightarrow 0} \frac{\sqrt{1-x}}{x}\). (b) \(\lim _{x \rightarrow 1} \frac{\sqrt{2-x}}{x}\). (c) \(\lim _{x \rightarrow 0} \frac{\sqrt{1+x}}{x}\). (d) \(\lim _{x \rightarrow 0} \frac{x^{2}-x}{x}\). (e) \(\lim _{x \rightarrow-1} \frac{x^{3}+1}{x+1}\). (f) \(\lim _{x \rightarrow 2} \frac{x-2}{x^{2}-x-2}\). (g) \(\lim _{x \rightarrow 0} \frac{\sqrt{1+x}-\sqrt{1-x}}{x}\).

Suppose that \(F(x)=\int_{0}^{x}(t+1) \tan ^{-1} \frac{1}{1+t^{2}} d t\) With out evaluating the integral find the following: (a) \(F^{\prime}(x)\). Ans. \((x+1) \tan ^{-1} \frac{1}{1+x^{2}}\) (b) \(F^{\prime}(0)\).

If an object is dropped in a gas for which the resistance is proportional to the velocity and if gravity acts, the velocity at any time \(t\) is given by \(v=(32 / k)\left(1-e^{-k t}\right)\) where \(k\) is a constant that depends on the amount of resistance. The distance fallen by the object in infinite time is given by \(\int_{0}^{\infty} \frac{32}{k}\left(1-e^{-k t}\right) d t\). Find the distance.

Suppose that \(\lim _{x \rightarrow a} f(x)=b\). Show that \(\lim _{n \rightarrow a} f\left(x_{n}\right)=b\) when \(\left\\{x_{n}\right\\}\) is a sequence of numbers whose limit is \(a\).

Prove by application of the precise definition that the limit of the sequence for which \(s_{2 m}=1 / m\) and \(s_{2 m-1}=1 / 2 m\) is 0 .

If \(F(x)=\int_{0}^{x} e^{-r^{2}} d t\), what is \(F^{\prime}(1) ?\)

(a) Calculate the gravitational force that an infinitely extended, thin plane of mass \(M\) per unit volume exerts on a mass \(m\) situated \(h\) units above it. Suggestion: Use the result in formula (36) of Chapter 16 to formulate the improper integral. Ans. \(2 \pi G M m t\). (b) The result in part (a) is independent of \(h\). Can you explain in physical terms why this is reasonable?

Prove that the limit of a sequence is unique, that is, if there is a limit, there is only one limit.

Prove rigorously that 1 is not the limit of the sequence for which \(s_{n}=\) \(1 / n\).

Graph the following functions: (a) \(y=x+\sqrt{(1-x)(2-x)}\) (b) \(y=x+\sqrt{(x-2)(x+1)}\)

A rocket is at rest. Then a force applied instantaneously at \(t=0\) gives the rocket an upward velocity of \(100 \mathrm{ft} / \mathrm{sec}\). Graph the velocity as a function of time from \(t=-\infty\) to \(t=\infty .\) Assume that the acceleration of gravity is \(32 \mathrm{ft} / \mathrm{sec}^{2}\).

The attraction of a spherical shell (idealized as a spherical surface of radius \(R\) ) of mass \(M\) per unit area on a unit particle inside is 0 . The attraction of the shell on a unit particle outside is the same as if the mass were concentrated at the center. If the unit particle outside is \(r\) units from the center, the force of attraction is \(4 \pi R^{2} M / r^{2}\). If the unit particle is just on the shell, the attractive force of the shell on the particle can be shown to be \(2 \pi M\). Graph the force as a function of the distance \(r\) of the unit particle from the center of the shell for \(r \geq 0\).

Sketch the graph of \(y=f(x)\) where \(f(x)=x\) for \(-1 \leq x \leq 1\) and \(f(x)\) is periodic and of period \(2 .\)