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# Week 7 Problem Set A.
## Task to do.
Do exam 1 corrections, and upload them onto canvas. Catch up all material, so you won't get lost in the new concepts!
## Reading.
Thomas Chapters 3.1, 3.2, 3.3, 3.4, 3.5
## Problems.
### Differentiability and its limit definition.
1. First, complete this mantra: "If a function $f$ is differentiable at point $p$, then it is..........."
2. Answer true or false for the following. If it is false, give a counterexample:
1. If a function $f$ is continuous at point $p$, then it is differentiable at point $p$.
2. If a function $f$ is differentiable at point $p$, then it is continuous at the point $p$.
3. Using the **limit definition of derivative** (the limit of a difference quotient), compute the derivative $f'(x)$ for each of the following function:
1. $\displaystyle f(x)=\frac{1}{x}$
2. $f(x)= \sqrt{x}$
4. For the function $\displaystyle f(x)=\frac{x}{x-2}$, do the following:
1. find the **slope** of the function at the point $(3,3)$, using the limit definition of the derivative.
2. Find the equation of the tangent line to the graph of $y=f(x)$ at the point $(3,3)$.
5. Give scenarios where a function $f(x)$ is **not** differentiable at a point $p$.
6. Determine if each of the following function is differentiable at $x=0$:
1. $f(x)=\begin{cases} 2x-1 & x\ge 0\\ x^{2}+2x-7 & x<0\end{cases}$
2. $g(x)=\begin{cases} x^{2/3} & x\ge 0 \\ x^{1/3} & x<0\end{cases}$
7. Let $A$ and $B$ be some real number, and consider the function $$
f(x)=\begin{cases}
x-Ax^{2} , & x\ge3 \\
1-Bx, & x < 3
\end{cases}
$$Find values $A,B$ such that $f$ is differentiable at $x=3$. (Hint: If it is differentiable at $x=3$, it is also what at $x=3$? So don't forget these conditions that you can set up to solve for $A$ and $B$.)
### Continuity and differentiability of some wild functions.
1. Consider the function $f(x)=\begin{cases}\displaystyle \sin(\frac{1}{x}), & x\neq0 \\0, & x=0\end{cases}$
1. Is $f(x)$ continuous at $x=0$? Why or why not?
2. Is $f(x)$ differentiable at $x=0$? Why or why not?
2. Consider the function $f(x)=\begin{cases}\displaystyle x\sin(\frac{1}{x}), & x\neq0 \\0, & x=0\end{cases}$
1. Is $f(x)$ continuous at $x=0$? Why or why not?
2. Is $f(x)$ differentiable at $x=0$? Why or why not?
3. Consider the function $f(x)=\begin{cases}\displaystyle x^{3}\sin(\frac{1}{x}), & x\neq0 \\0, & x=0\end{cases}$
1. Is $f(x)$ continuous at $x=0$? Why or why not?
2. Is $f(x)$ differentiable at $x=0$? Why or why not?
4. Consider the function $f(x)=\begin{cases}\displaystyle x^{n}\sin(\frac{1}{x^{m}}), & x\neq0 \\0, & x=0\end{cases}$, try to give conditions for $n$ and $m$ so that the function $f(x)$ would be differentiable at $x=0$.
### A real wild function: Weierstrass's nowhere differentiable continuous function.
As mathematicians slowly solidifies the definition of continuity and differentiability, they thought: Is it possible for a function to be continuous everywhere, but differentiable nowhere? Weierstrass came up with the following: $$
f(x) = \sum_{n=0}^{\infty} \left( \frac{2}{3} \right)^{n}\cos(9^{n}\pi x)
$$This is a function that has infinitely many cosines added together. As it turns out, this converges everywhere. Unfortunately for us, we cannot use a computer to add up infinitely many times, but rather only finitely many times. We will try nonetheless.
1. Use DESMOS (or graphing technology of your choice), graph the sum of above function for the first 10 terms (n=0 to n=9), and sketch (or take a screenshot) what you get for the function on the interval $[-5,5]$. Describe what you see.
2. Try zooming in at various places of the graph, does it appear to be a "line" anywhere?
3. Can you explain why, when actually adding up all infinitely many such cosine functions together, the resulting function is "nowhere differentiable"?
Look up Weierstrass's function on wikipedia if you are interested.
### Differentiability on a closed interval.
If a function is defined on a closed interval, we can choose to define the function is differentiable at the end points if the corresponding one-sided derivative exist. (e.g., if the function is defined on $[a,b]$, then it is differentiable at $x=a$ if the right-derivative $f'(a^{+})$ exists, while it is differentiable at $x=b$ if the left-derivative $f'(b^{-})$ exists)
Similarly, we can choose to define the function to be continuous at $x=a$ if it is right-continuous at $x=a$, and continuous at $b$ if it is left-continuous at $x=b$.
Now, each of the following shows a graph of a function over some domain $D$. At what domain points does the function appear to be: (1) differentiable? (2) continuous but not differentiable? (3) neither continuous nor differentiable?
1. ![[smc-fall-2023-math-7/week-7/---files/Pasted image 20231012010130.png]]
2. ![[smc-fall-2023-math-7/week-7/---files/Pasted image 20231012010139.png]]
3. ![[smc-fall-2023-math-7/week-7/---files/Pasted image 20231012010147.png]]
4. ![[smc-fall-2023-math-7/week-7/---files/Pasted image 20231012010155.png]]
5. ![[smc-fall-2023-math-7/week-7/---files/Pasted image 20231012010205.png]]
6. ![[smc-fall-2023-math-7/week-7/---files/Pasted image 20231012010229.png]]
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