Functions of a Real Variable
We will cover following topics
Introduction
Calculus is the mathematical study of continuous change of variables and their implications on functions which are dependent on these variables. Calculus has two major branches:
- Differential Calculus
- Integral Calculus
Differential Calculus is concerned with instantaneous rates of change and slopes of curves, whereas Integral Calculus is concerned with accumulation of quantities and the areas under & between the curves.
In this course, we will study about single and multi-variable calculus. We assume, for all our calculations and results, that \(f\) is a real variable function.
Limits
Let \(f(x)\) be a function defined on an open interval around \(x_{0}\). We say that the limit of \(f(x)\) as \(x\) approaches \(a\) is \(L\), if for every \(\varepsilon>0\), \(\exists\) \(\delta>0\) such that \(\forall \text{ }x\),
\[0 < \vert x-x_{0} \vert < \delta \Longrightarrow \vert f(x)-L \vert < \varepsilon\]We write \(\lim _{x \rightarrow x_{0}} f(x)=L\) to denote that limit of \(f(x)\) as \(x\) approaches \(a\) is \(L\).
Indeterminate Forms
Indeterminate forms are expressions involving two functions and whose limits cannot be derived by simply using the limits of the individual functions. The following indeterminate forms are encountered while calculating limits of functions:
- Quotient Indeterminate Forms: \(\dfrac{0}{0}\) and \(\dfrac{\infty}{\infty}\); can be solved by L’Hopital’s rule.
- Product Indeterminate Forms: \(0 \cdot \infty\); can be solved by converting to quotient indeterminate form and applying L’Hopital’s rule.
- Subtraction Indeterminate Forms: \(\infty-\infty\); can be solved by converting to quotient indeterminate forms and applying L’Hopital’s rule.
- Exponential Indeterminate Forms: \(0^0\), \(\infty^0\) and \(1^\infty\); can be solved by finding the logarithm of the limit asked.
Some of the forms which are not indeterminate are:
- Quotient: \(\dfrac{0}{\infty}\) and \(\dfrac{1}{\infty}\) have limit as 0.
- Product: \(\infty \cdot \infty\) has limit as \(\infty\).
- Exponential: \(0^{\infty}\) and \(\infty^{\infty}\) have the limits as \(0\) and \(\infty\) respectively.
Continuity
A function \(f(x)\) is said to be continuous at \(x=a\) if:
\[\lim _{x \rightarrow a} f(x)=f(a)\]Ex 1: \(f(x)=1 / x\) is continuous on \((-\infty, 0)\) and on \((0, \infty)\),
Ex 2: \(f(x)=\sin x\) is continuous on \((-\infty,\infty)\), and
Ex 3: \(f(x)=\csc x\) is continuous on \((0, \pi)\), \((\pi, 2\pi)\), \((2\pi, 3\pi)\), … etc.
Intermediate Value Theorem: Let \(f(x)\) be continuous on \([a,b]\) and let \(k\) be any number between \(f(a)\) and \(f(b)\), where \(f(a) \neq f(b)\). Then, there always exists a number \(c\) in the open interval \((a,b)\) such that \(f(c)=k\).
Differentiability
The derivative of \(f(x)\) is defined by \(f'(x)=\lim _{h \rightarrow 0} \dfrac{f(x+h)-f(x)}{h}\). The following notations are used to denote the derivative of a function \(y=f(x)\): \(y'\), \(f'(x)\), \(\dfrac{df}{dx}\), \(\dfrac{dy}{dx}\) and \(Df(x)\).
- \(f'(a)\) is the instantaneous rate of change of \(f(x)\) at \(x=a\).
- If \(y=f(x)\), then \(m=f'(a)\) represents the slope of the tangent to the line \(y=f(x)\) at \(x=a\) and the equation of the tangent at \(x=a\) is given as \(y=f(a)+f^{\prime}(a)(x-a)\).
- If \(f(t)\) is the position of an object at time \(t\), then \(f'(a)\) gives the velocity of the object at \(t=a\).
- A list of common derivatives is given below:
| \(\dfrac{d}{d x}(x)=1\) | \(\dfrac{d}{d x}(\csc x)=-\csc x \cot x\) | \(\dfrac{d}{d x}\left(a^{x}\right)=a^{x} \ln (a)\) |
| \(\dfrac{d}{d x}(\sin x)=\cos x\) | \(\dfrac{d}{d x}(\cot x)=-\csc ^{2} x\) | \(\dfrac{d}{d x}\left(\mathbf{e}^{x}\right)=\mathbf{e}^{x}\) |
| \(\dfrac{d}{d x}(\cos x)=-\sin x\) | \(\dfrac{d}{d x}\left(\sin ^{-1} x\right)=\dfrac{1}{\sqrt{1-x^{2}}}\) | \(\dfrac{d}{d x}(\ln (x))=\dfrac{1}{x}, x>0\) |
| \(\dfrac{d}{d x}(\tan x)=\sec ^{2} x\) | \(\dfrac{d}{d x}\left(\cos ^{-1} x\right)=-\dfrac{1}{\sqrt{1-x^{2}}}\) | \(\dfrac{d}{d x}(\ln \vert x \vert)=\dfrac{1}{x}, x \neq 0\) |
| \(\dfrac{d}{d x}(\sec x)=\sec x \tan x\) | \(\dfrac{d}{d x}\left(\tan ^{-1} x\right)=\dfrac{1}{1+x^{2}}\) | \(\dfrac{d}{d x}\left(\log _{a}(x)\right)=\dfrac{1}{x \ln a}, x>0\) |
Mean Value Theorem
Let \(f(x)\) be a function satisfying two conditions:
i. \(f(x)\) is continuous on \([a,b]\), and
ii. \(f(x)\) is differentiable on \((a,b)\). Then, according to Mean Value Theorem \(\exists\) a number \(c\) such that:
\[f^{\prime}(c)=\dfrac{f(b)-f(a)}{b-a}\]where \(a< c< b\).
Taylor’s Theorem with Remainders
According to Taylor’s Theorem, if a function \(f\) can be differentiated \((n+1)\) times in an interval \(I\) containing \(c\), then \(\forall\) \(x \in I\), \(\exists\) \(k\) between \(x\) and \(c\) such that:
\[\quad f(x)=T_n(x)+R_n(x)\]where \(R_{n}(x)=\dfrac{f^{(n+1)}(k)}{(n+1) !}(x-c)^{n+1}\) is the remainder or error between the actual function value and the estimate using the Taylor Polynomial \((T_n(x)\).
We have the following relation for upper bound on \(R_n(x)\):
\[\vert R_{n}(x) \vert \leq \dfrac{ \vert x-c \vert ^{n+1}}{(n+1) !} \max \vert f^{(n+1)}(k)\vert\]Maxima and Minima
The maxima and minima of a function \(f\) can be obtained by equating its derivative to 0 and then check for second-order derivative at the roots.
(i) For a maxima, \(\dfrac{d y}{d t}=0\) and \(\dfrac{d^{2} y}{d t^{2}}< 0\)
(ii) For a minima, \(\dfrac{d y}{d t}=0\) and \(\dfrac{d^{2} y}{d t^{2}}> 0\)
(iii) For a saddle point, \(\dfrac{d y}{d t}=0\) and \(\dfrac{d^{2} y}{d t^{2}}= 0\)
PYQs
Limits
1) Let \(f: \left[ 0,\dfrac{\pi}{2} \to R \right]\) be a continuous function such that
\[f(x)=\dfrac{cos^2 x}{4x^2-\pi^2}, 0\leq x\leq\dfrac{\pi}{2}\]Find the value of \(f\left( \dfrac{\pi}{2} \right)\).
[2019, 10M]
2) Determine if \(\lim_{z \to 1} (1-z) \tan\dfrac{\pi z}{2}\) exists or not. If the limit exists, then find its value.
[2018, 10M]
3) Find the limit \(\lim_{n \to \infty} \dfrac{1}{n^2}\sum_{r=0}^{n-1} \sqrt{n^2-r^2}\).
[2018, 10M]
4) Evaluate the following limit
\[\lim _{x \rightarrow a}\left(2-\dfrac{x}{a}\right)^{\tan \left(\dfrac{\pi x}{2 a}\right)}\][2015, 10M]
5) Define a sequence \(s_{n}\) of real numbers by \(s_{n}=\sum_{i=1}^{n} \dfrac{(\log (n+i)-\log n)^{2}}{n+i}\). Does \(\lim _{n \rightarrow \infty} s_{n}\) exist? If so, compute the value of this limit and justify your answer.
[2012, 20M]
6) Evaluate \(\lim _{x \rightarrow 2} f(x)\) where
\[f(x)=\left\{\begin{array}{cc}{\dfrac{x^{2}-4}{x-2}} & {, x \neq 2} \\ {\pi} & {, x=2}\end{array}\right.\][2011, 8M]
7) Find the value of \(lim_{x \rightarrow 1} \ln (1-x) \cot \dfrac{\pi x}{2}\)
[2008, 12M]
Continuity
1) Let \(f\) be a real function defined as following:
\(f(x)=x, -1\leq x< 1\) and
\(f(x+2)=x, \forall x \in R\). Show that \(f\) is discontinuous at every odd integer.
[2003, 12M]
2) Let \(f(x)=\left\{\begin{array}{ll}{x^{p} \sin \dfrac{1}{x},} & {x \neq 0} \\ {0} & {, x=0}\end{array}\right.\)
Obtain condition on \(p\) such that \(f\) is continuous at \(x=0\).
[2002, 7M]
3) Let \(f\) be defined by setting \(f(x)=x\) if \(x\) is rational and \(f(x)=1-x\) if is irrational. Show that \(f\) is continuous at \(x=\dfrac{1}{2}\) but is discontinuous at every other point.
[2001, 12M]
Differentiability
1) Is \(f(x)=\vert \cos x \vert + \vert \sin x \vert\) is differentiable at \(x=\dfrac{\pi}{2}\)? If yes, then find its derivative at \(x=\dfrac{\pi}{2}\). If no, then give a proof of it.
[2019, 15M]
2) Let \(f\) be a function defined on \(\mathbb{R}\) such that \(f(0)=-3\) and \(f^{\prime}(x) \leq 5\) for all values of \(x\) in \(\mathbb{R}\). How large can \(f(2)\) possibly be?
[2011, 10M]
3) Let \(f(x)\) be defined by \(f(x)=\sin \vert x\vert\), \((x \in(-\pi, \pi))\). Is \(f\) continuous on \((-\pi, \pi)\)? If it is continuous, then is it differentiable on \((-\pi, \pi)\)?
[2007, 12M]
4) Find \(a\) and \(b\) so that \(f^{\prime}(2)\) exists where \(f(x)=\left\{\begin{array}{l}{\dfrac{1}{\vert x \vert}, \text { if } \vert x \vert >2} \\ {a+b x^{2}, \text { if } \vert x \vert \leq 2}\end{array}\right.\)
[2006, 12M]
5) Show that \(x-\dfrac{x^{2}}{2} < \log (1+x)< x-\dfrac{x^{2}}{2(1+x)}\), \(x>0\).
[2004, 12M]
6) For all real numbers \(x\), \(f(x)\) is given as \(f(x)=\left\{\begin{array}{ll}{e^{x}+a \sin x,} & {x<0} \\ {b(x-1)^{2}+x-2,} & {x \geq 0}\end{array}\right.\)
Find the values of \(a\) and \(b\) at which \(f(x)\) is differentiable at \(x=0\).
[2003, 12M]
7) Show that \(\dfrac{b-a}{\sqrt{1-a^{2}}} \leq \sin ^{-1} b-\sin ^{-1} a \leq \dfrac{b-a}{\sqrt{1-b^{2}}}\) for \(0 < a < b < 1\).
[2002, 12M]
8) Let \(f(x)=\left\{\begin{array}{ll}{x^{p} \sin \dfrac{1}{x},} & {x \neq 0} \\ {0} & {, x=0}\end{array}\right.\)
Obtain condition on \(p\) such that \(f\) is differentiable at \(x=0\).
[2002, 8M]
Mean Value Theorem
1) Prove that between two real roots of \(e^{x} \cos x+1=0\), a real root of \(e^{x} \sin x+1=0\) lies.
[2014, 10M]
2) A twice differentiable function \(f(x)\) is such that \(f(a)=0=f(b)\) and \(f(c)>0\) for \(a<c<b\). Prove that there be is at least one point \(\xi\), \(a<\xi< b\) for which \(f^{\prime \prime}(\xi)< 0\).
[2010, 12M]
3) Suppose that \(f^{\prime}\) is continuous on \([1,2]\) and that \(f\) has three zeroes in the interval \((1,2)\), show that \(f^{\prime \prime}\) has at least one zero in the interval \((1,2)\).
[2009, 12M]
4) If \(f\) is the derivative of some function defined on \([a, b]\), prove that there exists a number \(\eta \in [a, b]\) such that \(\int_{a}^{b} f(t) d t=f(\eta)(b-a)\).
[2009, 12M]
5) Prove that an equation of the form \(x^n = \alpha\), where \(n \in N\) and \(\alpha>0\) is a real number, has a positive root.
[2004, 15M]
Taylor’s Theorem with Remainders
1) Find the values of \(a\) and \(b\) such that \(\lim_{x \rightarrow 0} \dfrac{a \sin ^{2} x \times b \log \cos x}{x^{4}}=\dfrac{1}{2}\).
[2006, 12M]
Maxima and Minima
1) Find the maximum and the minimum value of the function \(f(x)=2x^3-9x^2+12x+6\) on the interval \((2,3)\).
[2019, 15M]
2) Find the shortest distance from the point \((1,0)\) to the parabola \(y^2=4x\).
[2018, 13M]
3) Let \(p\) and \(q\) be positive real numbers such that \(\dfrac{1}{p}\) + \(\dfrac{1}{q}\) =1. Show that for real numbers \(a\), \(b \geq0\),
\[ab \leq \dfrac{a^p}{p} + \dfrac{b^q}{q}\][2012, 12M]
4) Consider the set of triangle having a given base and a given vertex angle. Show that the triangle having the maximum area will be isosceles.
[2002, 15M]