Single Variable Calculus
Main reference book: Calculus by Spivak.Assignment 1 Assignment 2 Assignment 3 Assignment 4 Assignment 5
Assignment 6 Assignment 7
Lecture 11 and 12 - Maxima/Minima, Rolle's theorem.
Lecture 11 - Derivatives III
Lecture 10 - Derivatives II
Lecture 9 - Derivatives I
Lecture 8 - Limits III
Lecture 7 - Limits II
Lecture 6 - Limits I
Lecture 5 - Functions
Lecture 4 - Convergence of sequences III
Lecture 3 - Convergence of sequences II
Lecture 2 - Convergence of sequences I
Lecture 1 - Real numbers
- Local maxima/minima, global maxima/minima. Examples.
- If \(x\) is a local maximum / minimum point of \(f:(a,b)\to {\mathbb R}\) then \(f'(x)=0.\)
- Find local maximum/minimum points and values of \(f:[-1,2]\to {\mathbb R}\) defined by \(f(x)=x^3-x\). Similar example for \(f(x)=x+1/x\) on \((1/2,2)\).
- Statement and proof of Rolle's theorem.
- Statement of mean value theorem.
Lecture 11 - Derivatives III
- \((1/g)'a = -g'(a)/(g(a)^2\)
- \((f/g)'a = (f'(a)g(a)-f(a)g'(a))/g(a)^2\)
- Statement of Chain Rule : \( (f\circ g)'(a)=f'(g(a))g'(a)\). Example :\(\sin(2x)\)
Lecture 10 - Derivatives II
- \((f+g)'a = f'(a)+g'(a)\)
- \((fg)'a = f'(a)g(a)+f(a)g'(a)\)
- Brief explanation of 'proof by induction' and derivative of \(x^n\) (by induction).
Lecture 9 - Derivatives I
- Example: Equation of tangent to \(y=x^2\) at the point (1,1). Definition of derivative.
- Derivative of constant function, identity function, \(f(x)=x^2\).
- Proof that derivative of \(sin(x)\) is \(cos(x)\).
- Every differentiable function is continuous.
Lecture 8 - Limits III
- \(\underset{x\to 0}{\rm lim}sin(x)/x = 1. \)
- $$ f(x) = \begin{cases} 1 & x\geq 0 \\ 0 & x\leq 0\end{cases}.$$ \(\underset{x\to 0}{\rm lim}f(x) \) does not exist.
- Definition of right limit and left limit.
- Definition of \(\underset{x\to \infty}{\rm lim}f(x) \). Example: \(sin(1/x)/x\).
Lecture 7 - Limits II
- Lemma (..continued from previous lecture): If \(y,y_0\in {\mathbf R}, \ \ y_0\neq 0\) and $$ |y-y_0| < \mathsf{min}\Big\{|y_0|/2, \epsilon \cdot |y_0|^2/2 \Big\}$$ then $$ | \frac{1}{y}-\frac{1}{y_0} | < \epsilon.$$
- Theorem : If \( \underset{x\to a}{\rm lim}\ f(x)=\ell, \underset{x\to a}{\rm lim}\ g(x) = m\) then $$ \underset{x\to a}{\rm lim}\ f(x) + g(x) = \ell + m $$ $$ \underset{x\to a}{\rm lim}\ f(x)\cdot g(x) = \ell \cdot m $$ If \(m\neq 0\), $$ \underset{x\to a}{\rm lim}\ \frac{1}{g(x)} = \frac{1}{m} $$
Lecture 6 - Limits I
- Recall : \(\underset{x\to a}{\rm lim}\ f(x) = \ell\).
- Addition, Multiplication and composition of functions.
- \(\underset{x\to a}{\rm lim}\ x = a\).
- \(\underset{x\to 0}{\rm lim}\ f(x) = 0 \) where $$ f(x) = \begin{cases}x & \ if \ x\neq 0 \\ 1 & \ if \ x=0 \end{cases}$$
- Definition of a continuous function : \(\underset{x\to a}{\rm lim}\ f(x) = f(a) \ \forall \ a \).
- \(\underset{x\to 0}{\rm lim}\ x sin (1/x) =0\).
- Lemma (with proof): $$ |x-x_0|< \epsilon/2 \ \text{ and } |y-y_0|<\epsilon/2 \implies |(x+y)-(x_0+y_0)|< \epsilon.$$ $$ |x-x_0| < min\Big\{ 1, \frac{\epsilon}{2(1+|y_0|)}\Big\} \ \text{ and } |y-y_0| < \frac{\epsilon}{2(1+|x_0|)} \implies |xy-x_0y_0|<\epsilon. $$
- \(\underset{x\to a}{\rm lim}\ (f+g)(x) = \underset{x\to a}{\rm lim}\ f(x) + \underset{x\to a}{\rm lim}\ g(x)\)
Lecture 5 - Functions
- Functions from \({\mathbf N}\to {\mathbf R}\) as sequences and functions from \({\mathbf R}\to {\mathbf R}\)
- Intervals in \({\mathbf R}\) : \( (a,b),[a,b],(a,\infty),(-\infty,b),(a,b],[a,b),(-\infty,\infty)\)
- Graphs of functions : \(x, 2x+3, (x-1)^2, x^3, |x|, x|x|, sin(1/x), xsin(1/x)\) (discussion on graph of \(xsin(1/x)\) to be continued later)
- Definition of \(\underset{x\to a}{\rm lim}\ f(x) = \ell\).
Lecture 4 - Convergence of sequences III
- For convergent sequences \( (x_n),(y_n)\) $$ \underset{n\to \infty}{\rm lim} x_ny_n = \big( \underset{n\to \infty}{\rm lim} x_n \big) \big( \underset{n\to \infty}{\rm lim} y_n \big)$$
- Definition of a bounded sequence.
- Proof that every convergent sequence is bounded (to be continued in next class/tutorial...)
Lecture 3 - Convergence of sequences II
- If \(x\) is a real number such that \(|x|<\epsilon \ \forall \ \epsilon \), then \(x=0\).
- Review of the definition of \((x_n) \) converges to \(x\).
- Proof of \(\underset{n\to \infty}{\rm lim}\ \frac{2n+1}{3n+1} = 2/3\)
- Proof of the fact that \(\underset{n\to \infty}{\rm lim}\ x_n \) does not exist, where \((x_n)\) is the sequence defined by $$ x_n=\begin{cases} 0 & \text{if } n \text{ is odd } \\ 1 & \text{if } n \text{ is even}\end{cases} $$ ( proof to be completed in next class/tutorial)
Lecture 2 - Convergence of sequences I
- Definition of \( x_n \) converges to 0.
- Notation: \( \forall \) and \( \exists\) .
- Definition of \(x_n \) converges to x.
- Notation: \( \underset{n\to \infty}{\rm lim}\ x_n = x\)
Lecture 1 - Real numbers
- The sets \( {\mathbb N}, {\mathbb Q}, {\mathbb R}\)
- Absolute value of real numbers and triangle inequality.
- Sequence of real numbers.
- A brief discussion on what is means for the sequence \( 1/n \) to approach 0.