MTH101 - Single Variable Calculus

Single Variable Calculus

Main reference book: Calculus by Spivak.
Information on office hours

Assignment 1 Assignment 2 Assignment 3 Assignment 4 Assignment 5
Assignment 6 Assignment 7 Assignment 8 Assignment 9 Assignment 10
Assignment 11 Assignment 12 Assignment 13

Lecture 24 - Integration by substitution

In this lecture we

proved integration by substitution, i.e. $$ \int_{\phi(a)}^{\phi(b)} f(x) dx = \int_a^b f(\phi(t))\phi'(t)dt $$ where $f$ is a continuous function and $\phi$ is a differentiable function whose derivative is continuous.
used this technique of substitution to calculate the following examples: $\int xcos(x^2)dx $ and $\int_0^1 \sqrt{1-x^2}dx$

Lecture 23 - Second Fundamental Thm, Integration by parts

In this lecture we

discussed the concept of an indefinite integral $\int f$ in comparison with the concept of a definite integral $\int_a^b f$. When $f$ is continuous the indefinite integral is nothing but anti-derivative, i.e. a function whose derivative is $f$. Such a function is only determined up to a constant.
discussed the statement of the second fundamental theorem of calculus. This also highlights the usefulness of finding anti-derivatives.
saw how product rule for differentiation combined with the second fundamental theorem leads to the technique of integration by parts $$ \int u'v = uv - \int uv' $$
Using integration by parts to find $$ \int \frac{log \ x}{x^2}dx, \ \ \ \int xcos(x)dx, \ \ \ \int x^2e^x dx. $$

Lecture 22 - First Fundamental Theorem of Calculus

In this lecture we

defined the notion of an integrable function $f:[a,b]\to {\mathbb R}$, i.e. a bounded function such that $$ sup\{L(f,P)\}=inf\{U(f,P)\} $$
proved that the function which is $0$ on rationals and $1$ on irrationals is not integrable on any closed interval.
stated (without proof) that every continuous function on a closed interval is integrable.
for an integrable function $f:[a,b]\to {\mathbb R}$ we defined its indefinite integral as the function $$ F(x) = \int_a^x f. $$
stated the First fundamental theorem of calculus : i.e. if $f$ is continuous, then $F'=f$. We also saw a heuristic proof of this theorem.
Saw an example $ f(x)=x^3-x$ in which the first fundamental theorem of calculus was used to calculate $\int_0^1 f$.

Suggested reading : Chapter 14 of Spivak's Calculus.

Lecture 21 - lower sums and upper sums

In this lecture we

discussed how to define area of a circle as supremum of sum of areas of squares contained in it, or infimum of sum of squares covering it.
for a function $f:[a,b]\to {\mathbb R}$ and a partition $P$ of $[a,b]$ given by $$ a=t_0 < t_1<... < t_n = b $$ defined lower sum $L(f,P$ and upper sum $U(f,P)$.
discussed how $sup\{L(f,P)\} $ and $inf\{U(f,P)\}$ both are candidates for defining the area of region between the graph of $f$ and the X-axis.

Suggested reading : Chapter 13 (Integrals) of Spivak's Calculus.

Lecture 20 - supremum and infimum

In this lecture we

defined upper bound, lower bound, maximum and minimum of a non-empty set of real numbers.
saw example of the set $\{ \frac{n}{n+1}\ | \ n\in {\mathbb N}\}$ which does not have a maximum
defined the notion of a supremum and infimum (this was part of assignment)
stated the following theorem Theorem: a non-empty set which is bounded above always has a supremum.
saw an idea of how to prove the above statement from the theorem that every cauchy sequence is $ {\mathbb R}$ converges.
using the above theorem we proved that every increasing sequence which is bounded above, converges to its supremum.

Suggested reading : Chapter 8 of Spivak's Calculus.

Lecture 19 - l'Hopital's rule

In this lecture we

analyzed the behaviour of a function near a local maximum and proved $f''<0$ at a critical point implies that the point is a local maximum point and a similar statement for local minima. We also saw that the converse of this is not true.
Stated and proved Cauchy's Mean Value theorem and used it to prove (only sketch) l'Hopital's rule.
Used l'Hopital's rule to show \[ \underset{x\to a}{\rm lim} \frac{x}{tan \ x}=1\]

Suggested reading : Chapter 11 (Significance of the derivative) of Spivak's Calculus.

Lecture 18 - Applications of Mean Value Theorem

In this lecture we

proved the following consequences of Mean Value theorem
1. if $f'=0$ then $f$ is a constant
2. if $f'>0$ then $f$ is strictly increasing
3. if $f'<0$ then $f$ is strictly decreasing.

Suggested reading : Chapter 11 (Significance of the derivative) of Spivak's Calculus.

Lecture 17 - Rolle's theorem, Mean value theorem

In this lecture we

stated without proof (see chapter 7) that a continuous function $f:[a,b]:\to {\mathbb R}$ always attains a maximum and minimum. we also saw examples showing this fact fails for open intervals.
discussed a strategy to locate maximum and minimum points of functions defined on an open interval which involves looking at critical points, end-points and points where the function is not differentiable.
saw in detail how to use the above strategy in the case of $f(x)=x^5+x+1$ and $f(x)=x^3-x$ on intervals $[-2,2]$
proved Rolle's theorem and Mean value-theorem (applications to be discussed in the next lecture).

Suggested reading : Chapter 11 (Significance of the derivative) of Spivak's Calculus.

Lecture 16 - local maxima and minima

In this lecture we

defined the concepts of maximum point, minimum point, maximum value, minimum value, local maximum point/value, local minimum point/value, critical points of a function.
studied the above mentioned concepts for the function $f:{\mathbb R}\to {\mathbb R} $ given by $$ f(x)=x^3-x$$
proved that for a function $f$ if $x$ is a local maximum or minimum point, and $f$ is differentiable at $x$, then $f'(x)=0$.

Suggested reading : Chapter 11 (Significance of the derivative) of Spivak's Calculus.

Lecture 15 - Calculating derivatives

In this lecture we discussed the following tools for calculating the derivative of functions

if $f$ and $g$ are differentiable at $a$ then so is $fg$ and $$ (fg)'(a)= f'(a)g(a) + f(a)g'(a) $$ (this was proved in the previous lecture).
if $g$ is differentiable and non-zero at $a$, then $\frac{1}{g}$ is differentiable at $a$ and $$ \big(\frac{1}{g}\big)'(a) = -\frac{g'(a)}{g(a)^2} $$
if $f$ and $g$ are differentiable at $a$ and $g(a)\neq 0$ then $\frac{f}{g}$ is differentiable at $a$ and $$ \big(\frac{f}{g}\big)'(a) = \frac{f'(a)g(a)-f(a)g'(a)}{g(a)^2}$$
if $g$ is differentiable at $a$ and $f$ is differentiable at $g(a)$ then $f\circ g$ is differentiable at $a$ and $$ (f\circ g)'(a) = f'(g(a))g'(a)$$ (proof of this to be continued in the next lecture)

We also studied some examples in which the above results were used to calculate derivatives.
Suggested reading : Chapter 10 (Differentiation) of Spivak's Calculus.

Lecture 14 - Derivatives

In this lecture we

showed that if $f$ is a constant function then its derivative is zero at all points.
proved the product formula for derivatives: if $f$ and $g$ are differentiable at $a$ then so is $fg$ and $$ (fg)'(a)= f'(a)g(a) + f(a)g'(a) $$
introduced the notation $\frac{df}{dx}$.
stated without proof that $\frac{dsin(x)}{dx}=cos \ x, \ \ \frac{dcos(x)}{dx} = -sin(x), \ \ \frac{de^x}{dx}=e^x$.

Lecture 13 - Revision

The goal of this and the next lecture is to solve problems and revise concepts covered so far. In this lecture we

(re)proved that a convergent sequence is bounded. Used it to show that the sequence $n^2$ is not convergent.
proved that the function $f:(0,\infty) \to {\mathbb R}$ defined by $f(x)=\sqrt{x}$ is is differentiable and $f'(a)=\frac{1}{2\sqrt{a}}$.
proved that if $f$ and $g$ are functions such that $$ f(x)< g(x) \ \forall \ x$$ than (assuming the following limits exists) we have $$\underset{x\to a}{\rm lim} \ f(x) \leq \underset{x\to a}{\rm lim} \ g(x).$$ (see assignment 7 for an analogous statement for limits of sequences).
We also discussed an example where $f$ is the constant function $0$ and $$ g(x) = \begin{cases} |x| & \ \ \text{if} \ \ x \neq 0 \\ 1 & \ \ \text{if}\ \ x = 0 \end{cases} $$ which shows that one cannot have a strict inequality in the above result.

Lecture 12 - Derivatives (definition)

In this lecture we

rediscussed the proof of the theorem: $g$ is continuous at $a$ and $f$ is continuous at $g(a)$, then $f\circ g$ is continuous at $a$
saw (see lecture 10) that the function $$ f:(0,1) \longrightarrow {\mathbb R} $$ defined by $$ f(x) = \begin{cases} 0 & \ \ \text{if} \ \ x \ \ \text{is} \ \ \text{irrational} \\ \frac{1}{q} & x = \frac{p}{q} \ \ p,q \ \ \text{are coprime and positive}\end{cases}$$ is continuous at all irrational $a\in (0,1)$ but is not continuous at any rational number.
defined: $f$ is differentiable at a point $a$ in its domain if $$ \underset{h\to 0}{\rm lim} \ \frac{f(a+h)-f(a)}{h} $$ exists. In this case the above limit is denoted by $f'(a)$ and is called the derivative of $f$ at $a$.
showed that the function $f(x)=x^2$ is differentiable at any $a\in {\mathbb R}$ and $f'(a)=2a$.
discussed how $f'(a)$ (if it exists) is equal to the slope of the tangent line to the graph of $f$ at the point $(a,f(a))$.
proved that the function $f(x)=|x|$ is not differentiable at $0$.

Suggested reading : Page 146 to Page 156 of chapter 9 of Spivak's Calculus.

Lecture 11 - Continuous functions

In this lecture we

defined the concept of the right limit $ \Big(\underset{x\downarrow a}{\rm lim} \ f(x)\Big) $ and the left limit $ \Big( \underset{x\uparrow a}{\rm lim} \ f(x) \Big)$.
studied the example of $$ f(x) = \begin{cases} 1 & \ x>0 \\ -1 & x<0 \end{cases}$$ in which $\underset{x\downarrow 0}{\rm lim} \ f(x)$ and $\underset{x\uparrow 0}{\rm lim} \ f(x)$ exists, but $\underset{x\to a}{\rm lim} \ f(x)$ does not exist.
proved that $\underset{x\to a}{\rm lim} \ f(x)$ exists if and only if both the right limit and left limit exist at $a$ and are equal.
defined: $ f $ is continuous at a point $a$ in its domain if $\underset{x\to a}{\rm lim} \ f(x)=f(a)$.
proved that the sum, product and quotient (when denominator is non-zero) of continuous functions is continuous
proved using the above theorem, that all polynomial functions are continuous.
proved that if $g$ is continuous at $a$ and $f$ is continuous at $g(a)$, then $f\circ g$ is continuous at $a$.

Suggested reading : Chapter 6 of Spivak's Calculus.

Lecture 10 - Limits- IV

In this lecture we

We reproved the following statement in detail (without assuming the lemma on page 101, Spivak but arriving at the same estimates): $$ \underset{x\to a}{\rm lim} \ (f\cdot g)(x) = \Big(\underset{x\to a}{\rm lim} \ f(x)\Big) \cdot \Big(\underset{x\to a}{\rm lim} \ g(x)\Big)$$
proved $\underset{x\to a}{\rm lim}\ f(x) = 0 $ for all $a \in (0,1)$ where $$ f:(0,1) \longrightarrow {\mathbb R} $$ is the function defined by $$ f(x) = \begin{cases} 0 & \ \ \text{if} \ \ x \ \ \text{is} \ \ \text{irrational} \\ \frac{1}{q} & x = \frac{p}{q} \ \ p,q \ \ \text{are coprime and positive}\end{cases}$$ The main idea of the proof was to realise that for any positive integer $n$, there are only finitely many values of $x\in (0,1)$ such that $$ f(x) > \frac{1}{n} $$

Suggested reading : Chapter 5 of Spivak's Calculus.

Lecture 9 - Limits- III

In this lecture we

proved the following theorem (page 102 Spivak): Let $ \underset{x\to a}{\rm lim} \ f(x) = \ell\ \ \ , \underset{x\to a}{\rm lim} \ g(x) = m \ \ $ then $$ \underset{x\to a}{\rm lim}\ (f+g)(x) = \ell + m $$ $$ \underset{x\to a}{\rm lim}\ (f\cdot g)(x) = \ell\cdot m$$ $$ \underset{x\to a}{\rm lim} \ \frac{f}{g} = \frac{\ell}{m} \ \ \text{if} \ \ m\neq 0$$ We proved the above theorem assuming the proof of a lemma discussed on page 101 of Spivak. We also recalled several examples discussed in previous classes where we had seen proofs of statements similar to the lemma.

Suggested reading : Chapter 5 of Spivak's Calculus.

Lecture 8 - Limits- II

In this lecture we

we discussed the intuitive meaning of $\underset{x\to a}{\rm lim} \ f(x) = \ell $ in terms of the behaviour of the graph of $f$ over the interval $ (a-\delta, a+\delta) $. Using this, we argued (but did not write down a rigorous proof) that if $$\underset{x\to a}{\rm lim} \ f(x) = \ell \ \ \text{and} \ \ \underset{x\to a}{\rm lim} \ f(x) = m$$ then $\ell = m$. In other words, the limit, if it exists is unqiue.
We discussed and wrote down the proof of the following statements in great detail
1. $\underset{x\to a}{\rm lim} \ x^3 = a^3 $
2. $\underset{x\to a}{\rm lim} \ \frac{1}{x} = \frac{1}{a}, \ \ \text{if} \ \ a\neq 0 $

Suggested reading : Chapter 5 of Spivak's Calculus.

Lecture 7 - Limits- I

In this lecture we

defined/recalled:
A function $f$ approaches limit $\ell $ near $a$ if
for every $\epsilon>0, \ \exists \ \delta>0$
such that, for all $x$, if $ 0<|x-a|< \delta $, then $|f(x)-\ell|<\epsilon $.
introduced the symbol $\underset{x\to a}{\rm lim}\ f(x)$.
discussed graphs of functions in typical situations where $\underset{x\to a}{\rm lim}\ f(x)$ may or may not exist
(see page 90 of the textbook for more on this).
Proved the following
- $\underset{x\to a}{\rm lim}\ x = a$
- $\underset{x\to 0}{\rm lim}\ xsin(\frac{1}{x}) = a$
- $\underset{x\to 0}{\rm lim}\ x^2sin(\frac{1}{x}) = a$
- $\underset{x\to a}{\rm lim}\ x^2 = a^2$
- $\underset{x\to a}{\rm lim}\ x^3 = a^3$ (this proof was incomplete, will be finished in the next class)

Suggested reading : Chapter 5 of Spivak's Calculus.

Lecture 6 - Graphs of functions

In this lecture we

defined the concept of domain of a function.
studied the graphs of the following functions : $ 2x+3, \ \ x-1, \ \ (x-1)^2,\ \ |x|,\ \ sin(\frac{1}{x}), \ \ xsin(\frac{1}{x}), \ \ log(x), \ \ e^x $
defined product, sum and quotient of two functions. we observed that in general to define the quotient of two functions $ \frac{f}{g}$ we have to make our domain smaller by looking at only those $x$ where $g(x)\neq 0$
defined composition of two functions $f\circ g$. We saw examples to see that this is different from the product $f\cdot g$. We also saw that in general $f\circ g \neq g\circ f$.
Let $g(x):= x-c $. Then the graph of $f\circ g $ is obtained by horizontally shifting the graph of $f$ by $c$ to the right. And the graph of $g\circ f$ is obtained by vertically shifting the graph of $f$ by $c$ in the downward direction.
we saw the following definition:
A function $f$ approaches limit $\ell $ near $a$ if
for every $\epsilon>0, \ \exists \ \delta>0$
such that, for all $x$, if $ 0<|x-a|< \delta $, then $|f(x)-\ell|<\epsilon $.

Suggested reading : Examples of functions and their graphs in Chapter 4 of Spivak's Calculus.

Lecture 5 - Functions

In this lecture we

defined the concept of a function $f:{\mathbb R}\to {\mathbb R}$.
Saw various examples of functions.
studied how to draw the graph of a function
defined the concept of a function from a set (not necessarily real numbers) to ${\mathbb R}$, e.g. a function $$ f:{\mathbb N}\to {\mathbb R}$$
Saw that giving a function from $ {\mathbb N} \to {\mathbb R}$ is equivalent to giving an infinite sequence of real numbers.
Defined various notions of intervals in ${\mathbb R}$, e.g, $ (a,b), [a,b], \left(a,b\right], (a,\infty), \left(-\infty,a\right], $ etc.

Suggested reading : Chapter 3 pages 39 to 45, of Spivak's book.

Lecture 4 -Triangle inequality, bounded sequences, Cauchy sequence

In this lecture we

discussed triangle inequality, i.e. $|x + y| \leq |x|+|y| $, where equality holds if and only if $x $ and $y$ has same sign.

Used triangle inequality to show for convergent sequences $$ \underset{n\to \infty}{\rm lim}(x_n+y_n) = (\underset{n\to \infty}{\rm lim}x_n) + (\underset{n\to \infty}{\rm lim}y_n) $$

defined the concept of a bounded sequence. A bounded sequence is a sequence $x_n$ such that there exists a positive real number $C$ satisfying $$ |x_n | \leq C \ \ \forall \ \ n\in {\mathbb N} $$ or equivalently $$ -C \leq x_n \leq C $$
We stated that any convergent sequence is always bounded. (Assignment problem). Note that the converse may not be true.
Defined the notion of a Cauchy sequence.
discussed the Theorem (without proof) that a sequence is Cauchy if and only if it is convergent.

Challenging exercise (will not be asked in the exam): Show there exists a real number whose square is $2$.
(Hint: construct a Cauchy sequence $x_n$ such that the sequence $x_n^2$ converges to $2$ ).

Lecture 3 - Review of convergent sequences

In this lecture we

reviewed the definition of a convergent sequence.

introduced the notation $\underset{n\to \infty}{\rm lim} x_n = x$.

saw how to write the proof of $ \underset{n\to \infty}{\rm lim} \frac{n^2+4}{3n^3} =0 $

Lecture 2- Convergent sequences and limits

Lecture 1- Real Numbers

Office hours/TA information

Name Tut. Batch, Tut. Room, Roll Nos.	Office (in HR4)	Office hours
Aman Jhinga B1, LHC101, 1001-1024	C400	Thu 6pm - 7pm
Girish Kulkarni B5, LHC103, 1025-1048	B210	Tue 2pm - 3pm
Jatin Majithia B2, LHC105, 1049-1072	B205	Thu 5:30pm to 6:30pm
Jyotirmoy Ganguli B6, LHC106, 1073-1096	B210	Fri 5pm - 6pm
Pralhad Mohan Sinde B3, LHC107, 1098-1121	B207	Fri 2pm - 3pm
Rahul Kitture B7, LHC108, 1122-1145	C400	Fri 12:30pm to 1:30pm
Sushil Bhunia B4, LHC201, 1146-1167	B205	Mon 4pm - 5pm
Uday Jagdale B8, LHC301, 1168-1189	C400	Sat 11am to 12pm

Amit Hogadi
Office : B113, HR4
Office hours: Tuesday 1pm to 2pm