Home | | Latest | About | Random
# Week 1 Tuesday Problems
Reading: Chapter 6.1. And preview 6.2$\ast$ and $6.3\ast$
A word on homework. Homework is designed to help you gain further understanding, in addition to just regurgitating information. Write all your work **neatly**(! so you and I can read them) on pieces of paper or digitally. By each exam I will ask you to compile your homework as one portfolio and turn them in. Do your best, think about each problem, do as many as you can, and discuss with your peers. And if you run out of problems to do, there are more in the textbook (and many other textbook) for you to practice! You can do it!
## Problems.
A python code to generate a table of powers of $\gamma$:
```python
gamma = 0.999
for N in range(1500):
print(N,gamma**N)
```
1. Try out Napier's method to estimate the product of $0.612$ with $0.888$, using a base say $\gamma = 0.999$. To get a tabulated table of the powers of $\gamma$, you may use Excel or whatever computer program/calculator. Or, you can copy and paste the above Python code into onecompiler.com: (In onecompiler.com, you need to first change the program from "JAVA" to "PYTHON", then paste in above code, and then run it.) Experiment with different values of $\gamma$.
2. Now, estimate the product of 5132 and 7777 by slightly modifying Napier's method. How good is your approximation?
3. Using a calculator (or computer, or wolfram alpha), compute the quantity $(1 + \frac{1}{N})^N$ for the following various values: $N=1,10,100,1000,10000$. Do these quantities seem to approach Euler's constant $e$?
4. Suppose $f$ is invertible with domain $\mathbb R$ and range $\mathbb R$, and that $f(3)=7$. Calculate $$f^{-1}(7) + f(f^{-1}(20)) - f^{-1}(f(4)) + f(f^{-1}(f(3)))$$ (Note: There is no missing information here!)
5. For each of the functions below, determine if it is one-to-one on its domain or not. Explain and show your work, and give compelling reasons besides "it passes or fails the horizontal line test". (For some of these, you might want to recall how does the fundamental theorem of calculus work!)
1. $f(x)=x^5 + x^3 + x - 5$ on domain $\mathbb R$
2. $\displaystyle f(x) = \frac{x+1}{x^2+1}$ on domain $\mathbb R$
3. $\displaystyle f(x)=\int_3^{x}[2+\cos(t)]dt$ on domain $\mathbb R$
4. $\displaystyle f(x)=\int_3^{x^2}[2+\cos(t)]dt$ on domain $\mathbb R$
5. $\displaystyle f(x)=\int_2^{x^3}[2+\cos(t)]dt$ on domain $\mathbb R$
6. If a function if one-to-one, is it necessarily strictly increasing or strictly decreasing? Come up with TWO different examples of functions, satisfying **all four** of the following conditions **(1)** the domain of $f$ is $\mathbb R$, **(2)** the range of $f$ is also $\mathbb R$, **(3)** $f$ is one-to-one, **(4)** but $f$ neither strictly increasing nor strictly decreasing on its domain. What noticeable (calculus) feature these functions that you gave are lacking? (Experiment and draw some pictures to help you.)
7. Recall in class we partially proved why a continuous one-to-one function $f$ on an interval must be strictly increasing or strictly decreasing. Using it as a guide (and the intermediate value theorem), prove the following fact:
- Let $f(x)$ be continuous on an interval $I$, and that there exists three points $a < b < c$ in $I$ such that $f(a) > f(b) < f(c)$. Then $f$ cannot be one-to-one.
///