First derivative test:
You have to take the derivative of the function and set it equal to zero. Then you solve for the x values (critical points) and set them up into intervals between negative infinity and infinity. Then you plug in values between those intervals into the first derivative to find max or mins or if the graph is increasing or decreasing.

Second derivative test:
You take the derivative of the function twice and set it equal to zero. You solve for the x values and set them up into intervals between negative infinity and infinity. You plug in numbers between those intervals into the second derivative to see where the graph is concave up, concave down, or where there is a point of inflection.


limit rules:
1. if the highest exponent is the same on the top and bottom then the limit is the top coefficient over the bottom coefficient of the highest exponents.
2. If the highest exponent is on the top then the limit is infinity.
3. But if the highest exponent is on the bottom then the limit is 0.


EXAMPLES:

1. If f(x)=ln((x^2)-1), then f^1(x) *(x^2-1)is really the absolute value of x^2-1

*take derivative
f^1(x)=ln(x^2-1)
(1/x^2-1)(2x)
2x/(x^2)-1


2. d/dx cos^2(x^3)

chain rule!
-2cos(x^3)sin(x^3)(3x^2)
-6x^2cos(x^3)sin(x^3)


3. Let f be a differentiable function such that f(3)=2 and f^1(3)=5. If the tangent line to the graph of f at x=3 is used to find an approximation to a zero of f, that approximation is?

f(3)=2 (3=x, 2=y)
f^1(3)=5 (<--5 slope)

y-2=5(x-3)
y-2=5(3.1-3)
y-2=5(.1)
y=2.5

*approximation=pick the answer choice closest which is 2.6.


I know this is going to sound dumb, buttt i don't remember how to integrate a fraction, i know there is no quotient rule, but i don't know what to do.