These problems are supposed to cover the material taught in the first two lectures, you can view the lecture schedule here. Another problem set would be provided for set theory and logic.

Keep in mind that the following problems are designed to keep you creatively engaged. It is okay if you are not able to solve them all, don’t be discouraged. The goal is to get you to think in a different, more creative manner. If you have any doubts, feel free to contact me. There is no right or wrong approach, neither is there any time constraint in which you are expected to solve this. All the best! Have fun!

Some Reading Relevant For This Week

This week we will be going over some fundamental concepts in mathematics which will be useful to study calculus. While the reading itself is not directly relevant to the problems, I do expect you to finish this reading before the class. You do not need to know the material in these readings to solve the problems below.

  1. Tom Apostol, Calculus Volume 1 : Part 1 and 2 of the Introduction
  2. Topology, Munkres — Sections 1 through 7.
  3. Book of Proof, Hammock — Chapter 1

Note that you do not need to solve the questions in these textbooks, if you want to though, it is your choice. The second book is quite difficult, I will admit. But still try to read through. If you can’t follow all the proofs that is quite alright, as long as you understand the idea behind the concepts.

The first lecture would be conducted on Saturday, 18th. I expect you to finish the reading before then. All the best!

Warm Up Problems

  1. Our first task here is to try and understand how various functions behave. If you don’t recall what a function is, think of it as a machine which gives a single output for each input you give it. This machine here I am denoting by $f$, so $f(x)$ means the output received when the input is $x$. Can you give a couple examples each of
    1. functions which are strictly increasing? This means that if $x<y$ then $f(x)<f(y)$ for all $x$ and $y$.
    2. functions which strictly decrease, that is if $x>y$ then $f(x)<f(y)$.
    3. functions which are neither increasing or decreasing.
  2. Based on your answers to the previous question, think about the following. A function is said to have a maximum in certain ‘neighborhood’ if all the values of the function in the points in that particular neighborhood, is $\leq$ the ‘maximum’. This neighborhood can be as small as you want, or as large as you wish.
    1. Can a strictly increasing function have a maximum?
    2. Can a strictly decreasing function have a maximum?
    3. What about functions that are neither increasing nor decreasing? They go up and down and up and down.
  3. Consider functions which are neither decreasing nor increasing. Can we restrict the inputs, so only certain inputs are allowed (the set of all allowed inputs is what we call the domain) such that, over this specified range of inputs the function is increasing or decreasing? Give some examples.
  4. Based on your understanding of increasing and decreasing functions, try to find out the nature of the following curves. If a curve is neither increasing or decreasing over all real numbers, then find the range of values within which it is increasing or decreasing.
    1. the semi circle, $x^2 + y^2 = 1$ where $y>0$.
    2. $x^2$
    3. $x^3$
    4. $cx^3$ where $c$ is some constant real number
    5. $\sin x$
  5. For the functions mentioned in problem 4, try to find the maximum. This may be a regional maximum or it could be a maximum over the entire real numbers.
  6. Try to sketch the following curve yourself using your knowledge of increasing and decreasing curves. Don’t cheat! Calculators and graphing software are prohibited. Try to do this by hand. We are only looking for a qualitative expression.
    1. $(x-5)^2$
    2. $\sin(x-\pi/2)$
    3. $\frac{1}{1+x^2}$
    4. $2^{-x}$
    5. $2^{-x^2}$
    6. $\frac{1}{2^{-x^2}}$