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# Week 10 Problem Set A.
## Reading.
Please read Thomas 4.2 to 4.4, and 4.7 for this week. The topics are
- Rolle's theorem (4.2)
- Applying Rolle's theorem (along with intermediate value theorem) to determine the number of roots to a function
- Mean Value Theorem (4.2)
- Consequences of the Mean Value Theorem
- Corollary 1. Zero derivative on an interval implies constant function on that interval. (4.2)
- Corollary 2. Uniqueness of antiderivatives up to an additive constant. (4.2)
- Corollary 3. Unchanged sign of derivatives on an interval implies monotonicity on that interval. (4.3)
- Antiderivatives (4.7)
- Concavity and curve sketching (4.4)
## Exam 2 corrections.
Do the exam 2 corrections, same as before. Upload them on Canvas when done and when you done all correctly, I will give you half the points back.
## Something to do.
Start making a sheet of theorems and tests/methods/rules that we have encountered so far. This will be a good review for you later.
## Problems.
### Rolle's theorem and the mean value theorem.
1. Write down exactly the statement of Rolle's theorem.
2. Follow either the notes in class or in the text: Prove Rolle's theorem.
3. Write down exactly the statement of the mean value theorem (MVT).
4. Follow either the notes in class or in the text: Prove the mean value theorem by using Rolle's theorem.
5. For each of the following, find the value or values $c$ that satisfies the equation $$
\frac{f(b)-f(a)}{b-a}=f'(c)
$$ in the conclusion of the mean value theorem for the function and intervals given:
1. $f(x)=x^{2}+2x-1$ where $[a,b]=[0,1]$
2. $f(x) = x + \frac{1}{x}$ where $[a,b]=[\frac{1}{2},2]$
3. $f(x)=x^{3}-x^{2}$ where $[a,b]=[-1,2]$
### Determining number of roots to a function on an interval.
These are applications of the intermediate value theorem (to show existence of roots) and Rolle's theorem (to show nonexistence of roots).
1. For each of the following, show the given function $f(x)$ has **exactly one root** on the indicated interval. (Note: Exactly one root means having at least one root, and there cannot be two roots.)
1. $f(x) = x^{4} + 3x + 1$ on $[-2,-1]$
2. $f(x) = \sqrt{x} + \sqrt{1+x} -4$ on $(0,\infty)$
3. $\displaystyle f(x)= x + \sin^{2}(\frac{x}{3})-8$ on $(-\infty,\infty)$
4. $f(x) = \sec(x) - \frac{1}{x^{3}} + 5$ on $(0 , \pi / 2)$
2. In class we showed that quadratic polynomial (degree 2 polynomial) can have at most two real roots by repeated application of Rolle's theorem. Show that a cubic polynomial (degree 3 polynomial) can have at most three real roots.
### Corollaries of the MVT.
There are three main corollaries of the MVT that are quite useful:
> **Corollary 1. (Zero derivative implies constant)**
> If $f$ is continuous on $[a,b]$ and differentiable on $(a,b)$, and that $f'(x)= 0$ for all $x \in (a,b)$, then $f(x)$ is a constant function on $(a,b)$.
> **Corollary 2. (Uniqueness of antiderivatives up to an additive constant)**
> If $f$ and $g$ are continuous on $[a,b]$ and differentiable on $(a,b)$, and that $f'(x)=g'(x)$ for all $x\in (a,b)$, then there is a constant $C$ such that $f(x)=g(x) + C$ for all $x\in(a,b)$.
> **Corollary 3. (Fixed signed derivative implies strict monotonicity)**
> Let $f$ be continuous on $[a,b]$ and differentiable on $(a,b)$.
> If $f'(x) > 0$ for all $x \in(a,b)$, then $f(x)$ is strictly increasing on $[a,b]$
> If $f'(x) < 0$ for all $x\in(a,b)$, then $f(x)$ is strictly decreasing on $[a,b]$
Either follow the work we did in class or from textbook:
1. Prove corollary 1 using the mean value theorem.
2. Prove corollary 2 using corollary 1.
3. Prove corollary 3 using the mean value theorem.
### Applications of MVT.
1. **Temperature change** It took 14 sec for a mercury thermometer to rise from $-19^{\circ}\text{C}$ to $100^{\circ}\text{C}$ when it was taken from a freezer and placed in boiling water. Show that somewhere along the way the mercury was rising at the rate of $8.5^{\circ}\text{C}/\text{sec}$.
2. A trucker handed in a ticket at a toll booth showing that in 2 hours she had covered 159 mi on a toll road with speed limit 65 mph. The trucker was cited for speeding. Why?
3. Show that for any numbers $a$ and $b$, we have the inequality $$
|\sin(b)-\sin(a)|\le|b-a|
$$ (Hint: Consider applying MVT to $f(x)=\sin(x)$ on $[a,b]$)
4. Show that for any number $x$, we have the inequality $$
|\cos(x) - 1| \le|x|
$$ (Hint: Consider applying MVT to $f(x) = \cos(x)$ on $[0,x]$)
### Antiderivatives.
If we have two functions $F(x)$ and $f(x)$ such that $$
F'(x) = f(x)
$$on some interval $I$, then we say
- $f(x)$ is the derivative of $F(x)$
- $F(x)$ is an antiderivative of $f(x)$
on that interval $I$.
By corollary 2 above, it tells us that all antiderivatives of a function $f(x)$ are just constant shifts of each other. So knowing one antiderivative of $f(x)$ tells us what are **all** the antiderivatives of $f(x)$.
1. In each of the following, find all possible functions $y$ with the given derivative $y'$:
1. $y' = x$
2. $y' = x^2$
3. $y'=x^{3}$
4. $y' = 2x-1$
5. $y' = 3x^{2} + 2x-1$
6. $y' = -\frac{1}{x^{2}}$
7. $\displaystyle y' = \frac{1}{x^{3}}+\frac{1}{x^{2}}$
8. $\displaystyle y' = \frac{1}{\sqrt{x}}$
9. $y' = \sin(2x)$
10. $\displaystyle y' = \cos(\frac{x}{2}) + \sin(3x)$
11. $y' = \sqrt{x}$
12. $y' = \sec^{2}(x)$
2. For each of the following, find the function $f(x)$ with the given information about its derivative and some point:
1. $f'(x)=2x-1$ with $f(0)=0$, what is $f(x)$?
2. $\displaystyle f'(x) = \frac{1}{x^{2}}+2x$ with $f(-1) = 1$, what is $f(x)$?
3. $f'(x)=\sec(x)\tan(x)-1$ with $f(0)=0$, what is $f(x)$?
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