Last Minute Thoughts

$e^{2x}+e^{x}\neq e^{3x}$
$e^{2x}\cdot e^{x}=e^{2x+x}=e^{3x}$
In general, $a^b\cdot a^c=a^{b+c}$

It is important you remember
$\int \frac{1}{1+u^2}du=\tan^{-1}(u)$

Know the derivative of all the trigonometric functions: sin(x), cos(x), tan(x), sec(x), csc(x), cot(x).

Recall all the types of limits. Indeterminate types and how to deal with them, see one of the older posts or your textbook! You can use L'Hospital as many times as applicable. Remember the squeeze theorem. And remember $\lim _{x\to 0} \frac{\sin (x)}{x}=1$. Similar forms you can use are $\lim_{x\to 0}\frac{\sin (ax)}{ax}=1$ for any real number $a\neq 0$. $\lim _{x\to \infty} \frac{\sin (1/x)}{1/x}=1$. The last one is same as $\lim _{x\to \infty} x\cdot \sin (1/x)=1$ Note you don't have to remember these, the form is all the same $\lim_{x\to a}\frac{\sin(f(x))}{f(x)}$ where $\lim_{x\to a}f(x)=0$ In general, if you don't have what you want, multiply and divide by what you need.

$a^b=e^{b\cdot \ln(a)}$

Know how to state and use the fundamental theorem of calculus part I and part II.

Know proper use of chain rule.

Know when to put in the integration constant.

Use variables consistently.

Apparently from the one problem, know how to complete the square. A brief summary:
$ax^2+bx$
$=a(x^2+\frac{b}{a}x)$
$=a(x^2+\frac{b}{a}x+(\frac{b}{2a})^2-(\frac{b}{2a})^2)$
$=a(x^2+(\frac{b}{2a})^2)-\frac{b^2}{4a}$
Note this is easier to do when you have numbers.

Example:
$2x^2+8x$
$=2(x^2+4x)$
$=2(x^2+4x+4-4)$
$=2(x+2)^2-8$

Another Example:
$\sin ^2 (x)+6\sin (x)$
$=\sin^2(x)+6\sin(x)+3^2-3^2$
$=(\sin(x)+3)^2-9$

Remember I said in class:
$|f(x)|=\begin{cases}
f(x) & \mbox{if }f(x)\geq0\\
-f(x) & \mbox{if }f(x)<0\end{cases}$

If you know a function is odd, and the interval of integration is on [-a,a] then the integral is 0.

If you know a function is even, and the interval of integration is on [-a,a] then the integral is equal to twice the integral over the interval [0,a].

To show a function has a root use the intermediate value theorem.
To show it has at most a certain number of roots, see how many roots the derivative has. While most problems you will encounter, you should be able to see how many roots the derivative has. Otherwise, take another derivative.