11th post

okay so this is the eleventh week of calculus and this week was about related rates more than anything. i think im okay a little with related rates i just need to practice.

Related Rates:

1. identify all variables and equations.
2. identify what you are looking for.
3. make a sketch and label.
4. write an equation involving your variables. you can only have an unknown so a secondary equation may be given.
5. take the derivative with respect to time.
6. substitute in derivative and solve.

example: the variable x and y are differentiable functions of t and are related by the equation y=2x^3-x+4 when x=2 dx/dt=1. find dy/dt when x=2.
dy/dt=?
x=2 dx/dt=-1 y=2x^3-x+4
dy/dt=6x^2dx/dt-dx/dt
dy/dt=6(2)^2(-1)+(1)
dy/dt=-23

Implicit Derivatives:

1. involves xs and ys
2.STEPS:
a. take the derivative like normal of both sides
b. everytime you take the derivative of y note it with dy/dx or y1
c. solve for dy/dx
3. if you want the slope you must plug in a x and a y value.

Intermediate Value Theorem:\

1. if f is continuous on [a,b] and k is any number between f(a)and f(b), then there is at least 1 number c when f(c)=k.
* basically you cannot skip any y value

HOW TO FIND THE EQUATION OF A TANGENT LINE:

1. take f1(x)
2. plug x in to find your slope m
3. plug x into f(x)to get y
4. using m and (x,y) plug it into the equation (y-y1)=m(x-x1).

okay im still having trouble with this optimazation stuff so.....

Optimazation:

1. identify primary and secondary equations
*primary- the one your maximizing or minimizing
*secondary- the other one
2. solve secondary for 1 variable and plug into primary .
*if primary only has 1 variable this step is not necessary.
3. take derivative of primary;set=0;solve for x.
4. plug into secondary equation to find other values. check end points if necessary. see examples for which ones you need to check endpoints for.

okay as i said before, im still having trouble figuring out optimazation and implicit derivatives so if someone can please help me out..