Posts 33 and 34

a few things to rememberrrrrrrr

limit rules:

if the degree on the top is bigger than the degree on the bottom the limit is infinity
if the degree on the top is smaller than the degree on the bottom the limit is zero
if the degree on the top is the same as the degree on the bottom divide the coefficients to find the limit


rules with derivatives:

chain Rule
you start from outside in. let's say you have cos(2x)
first take derivative of cos
then of multiply that by 2x

product Rule
copy the first x derivative of the second + copy the second x derivative of the first

quotient Rule
copy the bottom x derivative of the top - copy the top x derivative of the bottom/bottom ^2


average value:

of f(x)=1/x from x=1 to x=e is
1/e-1[ln (absoulte value of e)- ln (absoulte value of 1)]
1/e-1[1-0]
1/e-1


and implicit derivatives:

1. take the derivative of both sides like you would normally do
2. everytime the derivative of y is taken it needs to be notated with either y^1 or dy/dx
3. solve for dy/dx or y^1 as if you are solving for x


derivatives of integrals:

F(x)= integral from 0 to x^2 sin(t)
sin(x^2)(2x) *the 2x is the derivative of the bounds


and my favorite optimization:

1 Identify primary and secondary equations your primary is the one your or maximizing or minimizing and your secondary is the other equation
2. Solve for your secondary variable and plug into your primary equation if your primary only has on variable this isn’t necessary
3. Plug into secondary equation to find the other value check your end points if necessary


finally fromulas that we all forget with our derivatives:

d/dx c=0 (c is a #)
d/dx cu=cu' (c is #)
d/dx cx=c (c is a #)
d/dx u+v=u'+v'
d/dx uv=uv'+vu'
d/dx u/v=(vu'-uv')/v^2
d/dx sinx=cosx(x')
d/dx cosx=-sinx(x')
d/dx tanx=sec^2x(x')
d/dx secx=secxtanx(x')
d/dx cscx=-cscxtanx(x')
d/dx cotx=-csc^2x(x')
d/dx lnu= 1/u(u')
d/dx e^u=e^u(u')