Mathematical Methods for Physics and Engineering: A Comprehensive Guide

K. F. Riley, M. P. Hobson, S. J. Bence

Chapter 2

Preliminary calculus - all with Video Answers

Educators

Chapter Questions

Problem 1

Obtain the following derivatives from first principles:
(a) the first derivative of $3 x+4$
(b) the first, second and third derivatives of $x^{2}+x ;$
(c) the first derivative of $\sin x$.

Suzanne W.

Suzanne W.

Numerade Educator

Problem 2

Find from first principles the first derivative of $(x+3)^{2}$ and compare your answer with that obtained using the chain rule.

Suzanne W.

Numerade Educator

Problem 3

Find the first derivatives of
(a) $x^{2} \exp x$, (b) $2 \sin x \cos x$, (c) $\sin 2 x,(d) x \sin a x$,
(e) $(\exp a x)(\sin a x) \tan ^{-1} a x$, (f) $\ln \left(x^{2}+x^{-a}\right)$,
(g) $\ln \left(a^{x}+a^{-x}\right)$, (h) $x^{x}$.

Suzanne W.

Numerade Educator

Problem 4

Find the first derivatives of
(a) $x /(a+x)^{2}$, (b) $x /(1-x)^{1 / 2}$, (c) $\tan x$, as $\sin x / \cos x$,
(d) $\left(3 x^{2}+2 x+1\right) /\left(8 x^{2}-4 x+2\right)$

Suzanne W.

Numerade Educator

Problem 5

Use result $(2.12)$ to find the first derivatives of
(a) $(2 x+3)^{-3}$, (b) $\sec ^{2} x,\left(\right.$ c) $\operatorname{cosech}^{3} 3 x$, (d) $1 / \ln x$, (e) $1 /\left[\sin ^{-1}(x / a)\right]$

Suzanne W.

Numerade Educator

Problem 6

Show that the function $y(x)=\exp (-|x|)$ defined by
$$y(x)= \begin{cases}\exp x & \text { for } x<0 \\ 1 & \text { for } x=0 \\ \exp (-x) & \text { for } x>0\end{cases}$$
is not differentiable at $x=0 .$ Consider the limiting process for both $\Delta x>0$ and $\Delta x<0$

Suzanne W.

Numerade Educator

Problem 7

Find $d y / d x$ if $x=(t-2) /(t+2)$ and $y=2 t /(t+1)$ for $-\infty<t<\infty$. Show that it is always non-negative, and make use of this result in sketching the curve of $y$ as a function of $x$

Suzanne W.

Numerade Educator

Problem 8

If $2 y+\sin y+5=x^{4}+4 x^{3}+2 \pi$, show that $d y / d x=16$ when $x=1$.

Suzanne W.

Numerade Educator

Problem 9

Find the second derivative of $y(x)=\cos [(\pi / 2)-a x]$. Now set $a=1$ and verify that the result is the same as that obtained by first setting $a=1$ and simplifying $y(x)$ before differentiating.

Suzanne W.

Numerade Educator

Problem 10

The function $y(x)$ is defined by $y(x)=\left(1+x^{m}\right)^{n}$.
(a) Use the chain rule to show that the first derivative of $y$ is $n m x^{m-1}\left(1+x^{m}\right)^{n-1}$.
(b) The binomial expansion (see section $1.5$ ) of $(1+z)^{n}$ is
$$
(1+z)^{n}=1+n z+\frac{n(n-1)}{2 !} z^{2}+\cdots+\frac{n(n-1) \cdots(n-r+1)}{r !} z^{r}+\cdots
$$
Keeping only the terms of zeroth and first order in $d x$, apply this result twice to derive result (a) from first principles.
(c) Expand $y$ in a series of powers of $x$ before differentiating term by term. Show that the result is the series obtained by expanding the answer given for $d y / d x$ in (a).

P Krishnamurthy

P Krishnamurthy

Numerade Educator

Problem 11

Show by differentiation and substitution that the differential equation
$$
4 x^{2} \frac{d^{2} y}{d x^{2}}-4 x \frac{d y}{d x}+\left(4 x^{2}+3\right) y=0
$$
has a solution of the form $y(x)=x^{n} \sin x$, and find the value of $n$.

Suzanne W.

Numerade Educator

Problem 12

Find the positions and natures of the stationary points of the following functions:
(a) $x^{3}-3 x+3 ;$ (b) $x^{3}-3 x^{2}+3 x ;$ (c) $x^{3}+3 x+3$
(d) $\sin a x$ with $a \neq 0 ;$ (e) $x^{5}+x^{3} ;$ (f) $x^{5}-x^{3}$

Eduard Sanchez

Numerade Educator

Problem 13

Show that the lowest value taken by the function $3 x^{4}+4 x^{3}-12 x^{2}+6$ is $-26$.

Suzanne W.

Numerade Educator

Problem 14

By finding their stationary points and examining their general forms, determine the range of values that each of the following functions $y(x)$ can take. In each case make a sketch-graph incorporating the features you have identified.
(a) $y(x)=(x-1) /\left(x^{2}+2 x+6\right)$
(b) $y(x)=1 /\left(4+3 x-x^{2}\right)$.
(c) $y(x)=(8 \sin x) /\left(15+8 \tan ^{2} x\right)$.

Suzanne W.

Numerade Educator

Problem 15

Show that $y(x)=x a^{2 x} \exp x^{2}$ has no stationary points other than $x=0$, if $\exp (-\sqrt{2})<a<\exp (\sqrt{2})$
$2.16$ The curve $4 y^{3}=a^{2}(x+3 y)$ can be parameterised as $x=a \cos 3 \theta, y=a \cos \theta$.
(a) Obtain expressions for $d y / d x$ (i) by implicit differentiation and (ii) in parameterised form. Verify that they are equivalent.
(b) Show that the only point of inflection occurs at the origin. Is it a stationary point of inflection?
(c) Use the information gained in (a) and (b) to sketch the curve, paying particular attention to its shape near the points $(-a, a / 2)$ and $(a,-a / 2)$ and to its slope at the 'end points' $(a, a)$ and $(-a,-a)$.
2.17 The parametric equations for the motion of a charged particle released from rest in electric and magnetic fields at right angles to each other take the forms
$$
x=a(\theta-\sin \theta), \quad y=a(1-\cos \theta)
$$
Show that the tangent to the curve has slope $\cot (\theta / 2)$. Use this result at a few calculated values of $x$ and $y$ to sketch the form of the particle's trajectory.

Suzanne W.

Numerade Educator

Problem 16

The curve $4 y^{3}=a^{2}(x+3 y)$ can be parameterised as $x=a \cos 3 \theta, y=a \cos \theta$.
(a) Obtain expressions for $d y / d x$ (i) by implicit differentiation and (ii) in parameterised form. Verify that they are equivalent.
(b) Show that the only point of inflection occurs at the origin. Is it a stationary point of inflection?
(c) Use the information gained in (a) and (b) to sketch the curve, paying particular attention to its shape near the points $(-a, a / 2)$ and $(a,-a / 2)$ and to its slope at the 'end points' $(a, a)$ and $(-a,-a)$.

Suzanne W.

Numerade Educator

Problem 17

The parametric equations for the motion of a charged particle released from rest in electric and magnetic fields at right angles to each other take the forms
$$
x=a(\theta-\sin \theta), \quad y=a(1-\cos \theta)
$$
Show that the tangent to the curve has slope $\cot (\theta / 2)$. Use this result at a few calculated values of $x$ and $y$ to sketch the form of the particle's trajectory.

Suzanne W.

Numerade Educator

Problem 18

Show that the maximum curvature on the catenary $y(x)=a \cosh (x / a)$ is $1 / a$. You will need some of the results about hyperbolic functions stated in subsection $3.7 .6$

Suzanne W.

Numerade Educator

Problem 19

The curve whose equation is $x^{2 / 3}+y^{2 / 3}=a^{2 / 3}$ for positive $x$ and $y$ and which is completed by its symmetric reflections in both axes is known as an astroid. Sketch it and show that its radius of curvature in the first quadrant is $3(a x y)^{1 / 3}$.

Suzanne W.

Numerade Educator

Problem 20

A two-dimensional coordinate system useful for orbit problems is the tangentialpolar coordinate system (figure $2.13$ ). In this system a curve is defined by $r$, the distance from a fixed point $O$ to a general point $P$ of the curve, and $p$, the perpendicular distance from $O$ to the tangent to the curve at $P .$ By proceeding as indicated below, show that the radius of curvature at $P$ can be written in the form $\rho=r d r / d p$
Consider two neighbouring points $P$ and $Q$ on the curve. The normals to the curve through those points meet at $C$, with (in the limit $Q \rightarrow P$ ) $C P=C Q=\rho$. Apply the cosine rule to triangles $O P C$ and $O Q C$ to obtain two expressions for $c^{2}$, one in terms of $r$ and $p$ and the other in terms of $r+\Delta r$ and $p+\Delta p .$ By equating them and letting $Q \rightarrow P$ deduce the stated result.

Victor Salazar

Numerade Educator

Problem 21

Use Leibniz' theorem to find
(a) the second derivative of $\cos x \sin 2 x$,
(b) the third derivative of $\sin x \ln x$,
(c) the fourth derivative of $\left(2 x^{3}+3 x^{2}+x+2\right) \exp 2 x$.

Suzanne W.

Numerade Educator

Problem 22

If $y=\exp \left(-x^{2}\right)$, show that $d y / d x=-2 x y$ and hence, by applying Leibniz' theorem, prove that for $n \geq 1$
$$
y^{(n+1)}+2 x y^{(1)}+2 n y^{(n-1)}=0
$$

Suzanne W.

Numerade Educator

Problem 23

(a) By considering its properties near $x=1$, show that $f(x)=5 x^{4}-11 x^{3}+$ $26 x^{2}-44 x+24$ takes negative values for some range of $x .$
(b) Show that $f(x)=\tan x-x$ cannot be negative for $0 \leq x \leq \pi / 2$, and deduce that $g(x)=x^{-1} \sin x$ decreases monotonically in the same range.

Suzanne W.

Numerade Educator

Problem 24

Determine what can be learned from applying Rolle's theorem to the following functions $f(x):$ (a) $e^{x} ;$ (b) $x^{2}+6 x ;$ (c) $2 x^{2}+3 x+1 ;$ (d) $2 x^{2}+3 x+2 ;$ (e) $2 x^{3}-21 x^{2}+60 x+k$. (f) If $k=-45$ in (e), show that $x=3$ is one root of $f(x)=0$, find the other roots, and verify that the conclusions from (e) are satisfied.

Eduard Sanchez

Numerade Educator

Problem 25

By applying Rolle's theorem to $x^{n} \sin n x$, where $n$ is an arbitrary positive integer, show that $\tan n x+x=0$ has a solution $\alpha_{1}$ with $0<\alpha_{1}<\pi / n$. Apply the theorem a second time to obtain the nonsensical result that there is a real $\alpha_{2}$ in $0<\alpha_{2}<\pi / n$, such that $\cos ^{2}\left(n \alpha_{2}\right)=-n^{2} .$ Explain why this incorrect result arises.

Stephen Hobbs

Numerade Educator

Problem 26

Use the mean value theorem to establish bounds
(a) for $-\ln (1-y)$, by considering $\ln x$ in the range $0<1-y<x<1$,
(b) for $e^{y}-1$, by considering $e^{x}-1$ in the range $0<x<y$.

Suzanne W.

Numerade Educator

Problem 27

For the function $y(x)=x^{2} \exp (-x)$ obtain a simple relationship between $y$ and $d y / d x$ and then, by applying Leibniz' theorem, prove that
$$
x y^{(n+1)}+(n+x-2) y^{(n)}+n y^{(n-1)}=0
$$

Suzanne W.

Numerade Educator

Problem 28

Use Rolle's theorem to deduce that if the equation $f(x)=0$ has a repeated root $x_{1}$ then $x_{1}$ is also a root of the equation $f^{\prime}(x)=0$
(a) Apply this result to the 'standard' quadratic equation $a x^{2}+b x+c=0$, to show that the condition for equal roots is $b^{2}=4 a c_{-}$
(b) Find all the roots of $f(x)=x^{3}+4 x^{2}-3 x-18=0$, given that one of them is a repeated root.
(c) The equation $f(x)=x^{4}+4 x^{3}+7 x^{2}+6 x+2=0$ has a repeated integer root. How many real roots does it have altogether?

Eduard Sanchez

Numerade Educator

Problem 29

Show that the curve $x^{3}+y^{3}-12 x-8 y-16=0$ touches the $x$-axis.

Suzanne W.

Numerade Educator

Problem 30

Find the following indefinite integrals:
(a) $\int\left(4+x^{2}\right)^{-1} d x ;$ (b) $\int\left(8+2 x-x^{2}\right)^{-1 / 2} d x$ for $2 \leq x \leq 4$
(c) $\int(1+\sin \theta)^{-1} d \theta ;$ (d) $\int(x \sqrt{1-x})^{-1} d x$ for $0<x \leq 1$

Suzanne W.

Numerade Educator

Problem 31

Find the indefinite integrals $J$ of the following ratios of polynomials:
(a) $(x+3) /\left(x^{2}+x-2\right)$
(b) $\left(x^{3}+5 x^{2}+8 x+12\right) /\left(2 x^{2}+10 x+12\right)$
(c) $\left(3 x^{2}+20 x+28\right) /\left(x^{2}+6 x+9\right)$
(d) $x^{3} /\left(a^{8}+x^{8}\right)$

Victor Salazar

Numerade Educator

Problem 32

Express $x^{2}(a x+b)^{-1}$ as the sum of powers of $x$ and another integrable term, and hence evaluate
$$
\int_{0}^{b / a} \frac{x^{2}}{a x+b} d x
$$

Suzanne W.

Numerade Educator

Problem 33

Find the integral $J$ of $\left(a x^{2}+b x+c\right)^{-1}$, with $a \neq 0$, distinguishing between the cases (i) $b^{2}>4 a c$, (ii) $b^{2}<4 a c$, and (iii) $b^{2}=4 a c$.

Victor Salazar

Numerade Educator

Problem 34

Use logarithmic integration to find the indefinite integrals $J$ of the following:
(a) $\sin 2 x /\left(1+4 \sin ^{2} x\right)$;
(b) $e^{x} /\left(e^{x}-e^{-x}\right)$
(c) $(1+x \ln x) /(x \ln x)$
(d) $\left[x\left(x^{n}+a^{n}\right)\right]^{-1}$

Suzanne W.

Numerade Educator

Problem 35

Find the derivative of $f(x)=(1+\sin x) / \cos x$ and hence determine the indefinite integral $J$ of $\sec x$

Suzanne W.

Numerade Educator

Problem 36

Find the indefinite integrals $J$ of the following functions involving sinusoids:
(a) $\cos ^{5} x-\cos ^{3} x$;
(b) $(1-\cos x) /(1+\cos x)$
(c) $\cos x \sin x /(1+\cos x)$
(d) $\sec ^{2} x /\left(1-\tan ^{2} x\right)$

Victor Salazar

Numerade Educator

Problem 37

By making the substitution $x=a \cos ^{2} \theta+b \sin ^{2} \theta$, evaluate the definite integrals $J$ between limits $a$ and $b(>a)$ of the following functions:
(a) $[(x-a)(b-x)]^{-1 / 2} ;$
(b) $[(x-a)(b-x)]^{1 / 2}$
(c) $[(x-a) /(b-x)]^{1 / 2}$

Suzanne W.

Numerade Educator

Problem 38

Determine whether the following integrals exist and, where they do, evaluate them:
(a) $\int_{0}^{\infty} \exp (-\lambda x) d x$;
(b) $\int_{-\infty}^{\infty} \frac{x}{\left(x^{2}+a^{2}\right)^{2}} d x$
(c) $\int_{1}^{x} \frac{1}{x+1} d x$
(d) $\int_{0}^{1} \frac{1}{x^{2}} d x$
(e) $\int_{0}^{\pi / 2} \cot \theta d \theta$
(f) $\int_{0}^{1} \frac{x}{\left(1-x^{2}\right)^{1 / 2}} d x$.
d x)(d x / d u)$.

Suzanne W.

Numerade Educator

Problem 39

Use integration by parts to evaluate the following:
(a) $\int_{0}^{y} x^{2} \sin x d x$
(b) $\int_{1}^{y} x \ln x d x$
(c) $\int_{0}^{y} \sin ^{-1} x d x$
(d) $\int_{1}^{y} \ln \left(a^{2}+x^{2}\right) / x^{2} d x$.

Victor Salazar

Numerade Educator

Problem 40

Show, by each of the following methods, that the indefinite integral $J$ of $x^{3} /(x+$ 1) $^{1 / 2}$ is
$$
J=\frac{2}{33}\left(5 x^{3}-6 x^{2}+8 x-16\right)(x+1)^{1 / 2}+c
$$
(a) by using repeated integration by parts.
(b) by setting $x+1=u^{2}$ and determining $d J / d u$ as $(d J / d x)(d x / d u)$.

Eduard Sanchez

Numerade Educator

Problem 41

The gamma function $\Gamma(n)$ is defined for all $n>-1$ by
$$
\Gamma(n+1)=\int_{0}^{\infty} x^{n} e^{-x} d x
$$
Find a recurrence relation connecting $\Gamma(n+1)$ and $\Gamma(n)$.
(a) Deduce (i) the value of $\Gamma(n+1)$ when $n$ is a non-negative integer and (ii) the, value of $\Gamma\left(\frac{7}{2}\right)$, given that $\Gamma\left(\frac{1}{2}\right)=\sqrt{\pi} .$
(b) Now, taking factorial $m$ for any $m$ to be defined by $m !=\Gamma(m+1)$, evaluate $\left(-\frac{3}{2}\right) !$

Suzanne W.

Numerade Educator

Problem 42

Define $J(m, n)$, for non-negative integers $m$ and $n$, by the integral
$$
J(m, n)=\int_{0}^{\pi / 2} \cos ^{w} \theta \sin ^{n} \theta d \theta
$$
(a) Evaluate $J(0,0), J(0,1), J(1,0), J(1,1), J(m, 1), J(1, n)$
(b) Using integration by parts prove that, for $m$ and $n$ both $>0$,
$$
J(m, n)=\frac{m-1}{m+n} J(m-2, n) \quad \text { and } \quad J(m, n)=\frac{n-1}{m+n} J(m, n-2)
$$
(c) Evaluate (i) $J(5,3)$, (ii) $J(6,5)$, (iii) $J(4,8)$.

Victor Salazar

Numerade Educator

Problem 43

By integrating by parts twice, prove that $I_{n}$ as defined in the first equality below for positive integers $n$ has the value given in the second equality.
$$
I_{n}=\int_{0}^{\pi / 2} \sin n \theta \cos \theta d \theta=\frac{n-\sin (n \pi / 2)}{n^{2}-1}
$$

Suzanne W.

Numerade Educator

Problem 44

Evaluate the following definite integrals:
(a) $\int_{0}^{\infty} x e^{-x} d x$;
(b) $\int_{0}^{1}\left[\left(x^{3}+1\right) /\left(x^{4}+4 x+1\right)\right] d x$;
(c) $\int_{0}^{\pi / 2}[a+(a-1) \cos \theta]^{-1} d \theta$ with $a>\frac{1}{2}$;
(d) $\int_{-\infty}^{\infty}\left(x^{2}+6 x+18\right)^{-1} d x$

Eduard Sanchez

Numerade Educator

Problem 45

If $J_{r}$ is the integral
$$
\int_{0}^{\infty} x^{r} \exp \left(-x^{2}\right) d x
$$
show that
(a) $J_{2 r+1}=(r !) / 2$,
(b) $J_{2 r}=2^{-r}(2 r-1)(2 r-3) \cdots(5)(3)(1) J_{0}$.

Kyler Gray

Numerade Educator

Problem 46

(a) Find positive constants $a, b$ such that $a x \leq \sin x \leq b x$ for $0 \leq x \leq \pi / 2$. Use this inequality to find (to two significant figures) upper and lower bounds for the integral
$$
I=\int_{0}^{\pi / 2}(1+\sin x)^{1 / 2} d x
$$
(b) Use the substitution $t=\tan (x / 2)$ to evaluate $I$ exactly.

Suzanne W.

Numerade Educator

Problem 47

By noting that for $0 \leq \eta \leq 1, \eta^{1 / 2} \geq \eta^{3 / 4} \geq \eta$, prove that
$$
\frac{2}{3} \leq \frac{1}{a^{5 / 2}} \int_{0}^{a}\left(a^{2}-x^{2}\right)^{3 / 4} d x \leq \frac{\pi}{4}
$$

Suzanne W.

Numerade Educator

Problem 48

Show that the total length of the astroid $x^{2 / 3}+y^{2 / 3}=a^{2 / 3}$, which can be parameterised as $x=a \cos ^{3} \theta, y=a \sin ^{3} \theta$, is $6 a$

Suzanne W.

Numerade Educator

Problem 49

By noting that $\sinh x<\frac{1}{2} e^{x}<\cosh x$, and that $1+z^{2}<(1+z)^{2}$ for $z>0$, show that for $x>0$, the length $L$ of the curve $y=\frac{1}{2} e^{x}$ measured from the origin satisfies the inequalities $\sinh x<L<x+\sinh x$.

Suzanne W.

Numerade Educator

Problem 50

The equation of a cardioid in plane polar coordinates is
$$
\rho=a(1-\sin \phi)
$$
Sketch the curve and find (i) its area, (ii) its total length, (iii) the surface area of the solid formed by rotating the cardioid about its axis of symmetry and (iv) the volume of the same solid.

Geena Pullo

Numerade Educator