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 $\epsilon >0$, $\mathrm{\exists }$ $\delta >0$ such that $\mathrm{\forall }x$,

We write $\underset{x\to x_0}{lim}f(x)=L$ 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:

1. Quotient Indeterminate Forms: ${\displaystyle \frac{0}{0}}$ and ${\displaystyle \frac{\mathrm{\infty }}{\mathrm{\infty }}}$; can be solved by L’Hopital’s rule.
2. Product Indeterminate Forms: $0\cdot \mathrm{\infty }$; can be solved by converting to quotient indeterminate form and applying L’Hopital’s rule.
3. Subtraction Indeterminate Forms: $\mathrm{\infty }-\mathrm{\infty }$; can be solved by converting to quotient indeterminate forms and applying L’Hopital’s rule.
4. Exponential Indeterminate Forms: $0^0$, ${\mathrm{\infty }}^0$ and $1^{\mathrm{\infty }}$; can be solved by finding the logarithm of the limit asked.

Some of the forms which are not indeterminate are:

1. Quotient: ${\displaystyle \frac{0}{\mathrm{\infty }}}$ and ${\displaystyle \frac{1}{\mathrm{\infty }}}$ have limit as 0.
2. Product: $\mathrm{\infty }\cdot \mathrm{\infty }$ has limit as $\mathrm{\infty }$.
3. Exponential: $0^{\mathrm{\infty }}$ and ${\mathrm{\infty }}^{\mathrm{\infty }}$ have the limits as $0$ and $\mathrm{\infty }$ respectively.

Continuity

A function $f(x)$ is said to be continuous at $x=a$ if:

Ex 1: $f(x)=1/x$ is continuous on $(-\mathrm{\infty },0)$ and on $(0,\mathrm{\infty })$,

Ex 2: $f(x)=\sin x$ is continuous on $(-\mathrm{\infty },\mathrm{\infty })$, 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 $f(a)\ne f(b)$. 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 $f^{\prime }(x)=\underset{h\to 0}{lim}{\displaystyle \frac{f(x+h)-f(x)}{h}}$. The following notations are used to denote the derivative of a function $y=f(x)$: $y^{\prime }$, $f^{\prime }(x)$, ${\displaystyle \frac{df}{dx}}$, ${\displaystyle \frac{dy}{dx}}$ and $Df(x)$.

1. $f^{\prime }(a)$ is the instantaneous rate of change of $f(x)$ at $x=a$.
2. If $y=f(x)$, then $m=f^{\prime }(a)$ 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 $y=f(a)+f^{\mathrm{\prime }}(a)(x-a)$.
3. If $f(t)$ is the position of an object at time $t$, then $f^{\prime }(a)$ gives the velocity of the object at $t=a$.
4. A list of common derivatives is given below:

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 $\mathrm{\exists }$ a number $c$ such that:

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 $\mathrm{\forall }$ $x\in I$, $\mathrm{\exists }$ $k$ between $x$ and $c$ such that:

where $R_n(x)={\displaystyle \frac{f^{(n+1)}(k)}{(n+1)!}}(x-c{)}^{n+1}$ 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 $R_n(x)$:

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, ${\displaystyle \frac{dy}{dt}}=0$ and ${\displaystyle \frac{d^{2}y}{d{t}^2}}<0$

(ii) For a minima, ${\displaystyle \frac{dy}{dt}}=0$ and ${\displaystyle \frac{d^{2}y}{d{t}^2}}>0$

(iii) For a saddle point, ${\displaystyle \frac{dy}{dt}}=0$ and ${\displaystyle \frac{d^{2}y}{d{t}^2}}=0$

Limits

Continuity

Differentiability

Mean Value Theorem

Taylor’s Theorem with Remainders

Maxima and Minima