Okay, so this week we just reviewed and took AP practice exams and what not... but I was not at school friday so I do not know what we did that day. And since we have not been going over any new material, I am pretty much posting stuff that we learned a long time ago, so I sure hope I am not repeating myself here... but here is my blog anyway.
Related Rates:
1: identify all variables in equations
2: identify what you are looking for
3: sketch and label
4: write an equation involving your variables. (you can only have one unknown so a secondary equation may be given)
5: take the derivative with respect to time.
6: substitute derivative and solve.
Example: the variables x and y are functions of t and related by the equation y=2x^3-x+4 when x=2, dy/dt=-1. Find dy/dt when x=2
alright, so you put down the equation, y=2x^3-x+4.
Then you take the derivative of that, so you get dy/dt=6x^2(dx/dt)-(dx/dt)
then you plug in to find that dy/dt=6(2)^2(-1)-(-1)
and that is further simplified to, dy/dt=-23.
Linearization:
f(x)=f(c)+f'(c)(x-c)
example: Approximate the tangent line to y=x^2 at x=1
you find all the different values: dy/dx=2x dy/dx=2 y=(1)^2=1
then you plug into the formula to get: f(x)=1+2(x-1)
example 2: use differentials to approximate: sq root(16.5)
steps:
1: identify an equation--- f(x)=sq root(x)
2:f(x)+f;(x)dx--- sqrt(x)+ (1/(2sqrt(x)))(dx)
3:determine dx-- .5
4:determine x--- 16
5:plug in--- sqrt(16)+(1/2sqrt(16))(.5)= 4.0625
error= .0005
so yeah, I was not at school friday, did we take another practice ap test or what? Since we took the no calculator portion thursday I guess everyone took the calculator allowed part friday, but I was not there so I do not know. Someone fill me in!! and umm, I suck at integrating fractions!!! could someone tell me how to do that? thanks
Subscribe to:
Post Comments (Atom)
You have to watch for fractions, sometimes it may be ln integration if the top is the derivative of the bottom, or sometimes it may be regular substitution. If it is ln substitution, then your u is what is under the line and your du is the derivaitve of u. Then you revise your equation to satisfy du, and take the ln absolute value of your u and then integrate. If it is substitution, then you must use your substitution rules depending what is given to you in the equation. I hope this helps you some!
ReplyDeleteIntegration is basically the reverse way of a derivative. Also after ever integration you have to put + a constant.
ReplyDeleteExample:
If your integrating x^5 dx you would end up with x^6/6 + C
usually you can use substitution or trig inverse or ln|x| rules for fractions. if not and it is only one term in the denominator, then you make that term's exponent negative and move the term to the numerator
ReplyDeleteusually it is natural log integration. only if the top of the fraction is the derivative of the bottom. sometimes it's just substitution. sometimes you need to use trig inverse rules. u - is the top. du - is the derivative of it. & don't forget if you need to substitute a number in front of the integral
ReplyDelete