Steps:
1. Identify primary and secondary equations
- primary is the one you are maximizing or minimizing
- secondary is the other one
2. Solve secondary equation for one variable and plug into the primary
- If primary equation only has one variable, you can skip this step
3. Take the derivative of primary equation, set equal to zero, then solve for x.
4. Plug into secondary equation to find the other value.
Example:
Find the length and width of a rectangle with a perimeter of 60 meters and a maximum area.
1. Since maximizing the area, the area of a rectangle formula will be the primary. A=lw
The secondary formula will be the perimeter formula because you are given the perimeter.
60 = 2l + 2w
2. Solve secondary equation for one variable:
60 = 2l + 2w
60-2w = 2l
l= 60-2w/2
Then plug back into the primary:
A = 60-2w/2 (w)
3. Multiply the w in to simplify equation:
60w- 2w^2/ 2 = 30w - w^2
Then take the derivative: 30 - 2w
4. Solve for w:
30 = 2w
w = 15
Plug back into secondary to find l:
60= 2l + 2(15)
60 = 2l + 30
30 = 2l
l=15
Implicit Derivatives involve x's and y's
Steps:
1. Take the derivative using same rules of both sides
2. Every time you take the derivative of y, note it with dy/dx or y'.
3. Solve for dy/dx.
Example:
y^3 + y^2 - 5y - x^2 = 4
3y^2 dy/dx + 2y dy/dx - 5 dy/dx -2x = 0
Now solve for dy/dx:
dy/dx = 2x/ 3y^2 + 2y - 5
Example 2:
x^2 + y^2 = 9
2x + 2y dy/dx = 0
dy/dx = -2x/2y
which equals -x/y
I'm still having trouble with using graphs and I forgot how to do linearization so a recap would be splendid.
for linearization know that the key word is approximate:
ReplyDeletef[x] = f[c] + f'[c][x-c]
f[c] - y value
f'[c] - slope
[x-c] - x-number
For example:
approximate the tangent line to y=x^2 at x=1.
you know that c=1
plug in c to the original - y=(1)^2
take the derivative of the equation, giving you 2x. now plug in c to the derivative - y=2(1) ... since you get dy/dx = 2 .. 2 is your slope so now put in point slope form:
f(X)=1+2(x-1)
[well that's what the notes say...hope this helps]
Remember the key word for linearization first: approximate.
ReplyDeleteThe steps are:
1. Identify the equation
2. Use the formula f(x)+f '(x)dx
3. Determine your dx in the problem
4. Then determine your x in the problem
5. Plug in everything you get
6. Solve the equation
sounds easy right?
There's usually a decimal in there for your x and dx.
Ok. So all linearization is is a tangent line. APPROXIMATE is the KEY WORD. Also, it helps to realize that when you see an x value it may be a decimal--another hint.
ReplyDeleteSo when you're given an equation, first take the derivative to find the slope. Once doing this, plug in the whole number x value (which the above steps refer to as c I believe...) to get your actual numerical slope. Then plug that x value into your original function to find the y value.
Now you have a point and a tangent line....Tangent line equation is now called for. And that's it...I think.