Home | | Latest | About | Random
# Week 3 Wednesday 7/5 brief notes.
## Announcement.
Don't forget, **exam tomorrow in class 7/6 Thursday 6:30PM to 9:15PM**. You can bring a **one page** (you may use both sides) cheat sheet if you wish. No calculators or any other technology.
Also, turn in a portfolio of your homework up to this Monday by tomorrow's exam.
Today we will do a brief review and answer your questions.
Some things to think about:
1. When would a function have an inverse? What does one-to-one mean?
2. Graphically and algebraically, how do $f$ and its inverse $f^{-1}$ relate?
3. If a function is continuous, how can we tell when it is (or is not) one-to-one?
4. If a function is differentiable, how can we tell when it is one-to-one?
5. If $f$ has an inverse $f^{-1}$ and both are differentiable, how do we find the derivative of $f^{-1}$?
6. How do we define the natural logarithm function $\ln(x)$ in our class?
7. What are the main properties of $\ln(x)$, and its derivative?
8. How could we estimate $\ln(n)$ for some positive integer $n$?
9. Is $\ln(x)$ invertible? What do we call its inverse?
10. How do we in our class define the number $e$?
11. How is the function $\exp(x)$ defined? What are its properties and derivative?
12. How do we define the hyperbolic functions $\cosh(x),\sinh(x)$? What are their derivatives and graphs?
13. How do we define and think about the inverses of these trigonometric and hyperbolic functions? What are their derivatives?
14. When is it appropriate to apply L'Hospital rule?
15. How to apply integration by parts? How do we use it to convert an integral to another integral? How to create reductive/inductive formulas?
16. How do we deal with antiderivatives with the following techniques?
1. Basic substitution
2. Integration by parts
3. Integral with rational functions, partial fractions
4. Trigonometric integrals
5. Weierstrass substitution
6. Trigonometric substitutions
17. Miscellaneous things that help:
1. Long division
2. Various trigonometric identities (Pythagorean, double-angle formulas, product-to-sum, etc)
3. Derivatives and antiderivatives of basic trigonometric functions.
18. You can do it ! Do your best!
## Some practice.
**Example.** Consider $h(x)=x+ \sqrt{x}$ on the domain $x > 0$.
(a) Show $h$ is one-to-one.
(b) Does $h$ have an inverse function $h^{-1}$?
(c) Find the value $h^{-1}(6)$ and $h^{-1}(12)$.
(d) Find the derivative: $(h^{-1})'(6)$.
(e) Algebraically, can you solve for $h^{-1}(x)$ as an expression of $x$?
**Example.** Find the following limits $$
\lim_{n\to\infty}(1+n)^{1/\ln(n)} \quad \text{and}\quad\lim_{n\to\infty}(1+n)^{1/\ln(n)}
$$
**Example.** Find the following limit $$
\lim_{x\to\infty} \frac{(\ln(x))^{2}}{x}
$$
**Example.** Find the following limit $$
\lim_{x\to\infty} \frac{\ln(2^{x}+1)}{\ln(3^{x}+1)}
$$
**Example.** Find the following integral $$
\int \frac{x^{3}+4x^{2}+x -1}{x^{3}+x^{2}}dx
$$
**Example.** Find the following integral $$
\int e^{3x}\cosh(2x) dx
$$
**Example.** Find the integral $$
\int \cos^{5}(x)dx
$$
**Example.** Find the integral $$
\int \frac{x^{3}-x+2}{\sqrt{4-5x}}dx
$$
**Example.** Find the following integral $$
\int x\cos^{2}(x)dx
$$
**Example.** Find the following integral $$
\int \frac{1}{1+\sin(x)-\cos(x)}dx
$$
**Example.** Find the following integral $$
\int \frac{dx}{(x^{2}+4x+20)^{3/2}}
$$
**Example.** Show the following reduction formula $$
\int \sinh^{n}(x) dx = \frac{1}{n}\cosh(x)\sinh^{n-1}(x)-\frac{n-1}{n}\int\sinh^{n-2}(x)dx
$$
**Example.** Show the following induction formula $$
\int \frac{1}{\sinh^{n}(x)}dx = -\frac{\cosh(x)}{n \sinh^{n+1}(x)}- \frac{n+1}{n}\int \frac{1}{\sinh^{n+2}(x)}dx
$$