SPRING BREAK...FINALLY!!!! Just a few more weeks to go!
This week mrs. robinson answered questions that we needed help with. Monday and tuesday helped me a lot! Wednesday I went to NIcholls Orientation, and Thursday I went to the St. Charles Drunk Driving Skit. Today's Good Friday and it offically starts our Spring Break!!!
So i'll go over implicit derivatives because I used to understand them, then after learning some other things i got them confused and intertwined with one another. So, here it goes...
steps to IMPLICIT DERIVATIVES:
1. Take the derivative like normal.
2. For each y-term, you put y' or dy/dx behind it.
3. Solve for dy/dy or y'.
For Example:
36x^2 + 2y = 9
36x + 2(dy/dx) = 0
2(dy/dx) = -36x
-36x/2
dy/dx = -18x
the Second Implicit Derivative...basically the same way!
steps to SECOND IMPLICIT DERIVATIVES
1. Take the derivative of the first derivative
2. Put d^2y/dx^2 for dy/dx
3. Simplify
4. Plug in values
so an example using the same equation is:
we have to take the first implicit derivative FIRST...[durr]
36x^2 + 2y = 9
36x + 2(dy/dx) = 0
2(dy/dx) = -36x
-36x/2
dy/dx = -18x <-- that's our first implicit derivative
now in order to take the second implicit derivative we just take the second derivative of dy/dx = -18x..
dy/dx = -18x
dy/dx = -18
so since we know that dy/dx is -18x we can plug that in
-18x = -18
when solving for x we get 1!
NOW, i understand and grasp that...but for what i really don't get is still Angle of Elevation..and Optimization..and I know there's other things, but most of them are either formula's to memorize [which i'm working on] or things that i understand and just make careless mistakes!
~ElliE~
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Optimization steps:
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2. Write an equation for what you are maximizing or minimizing (this is the primary equation) and find a secondary equation.
3. Solve secondary equation in term of one variable, then plug that in for the primary equation.
4. Take derivative of plugged in primary equation, set equal to zero, and solve for the variable remaining.
5. Plug back into the secondary to find other variable.
Example: We need to enclose a field with a fence. We have 500 feet of fencing material and a building in on one side of the field and so won't need any fencing. Determine the dimensions of the field that will enclose the largest area.
Your primary formula will be the area of a rectangle formula so
A = xy
You are given 500 ft as the fencing material or perimeter so your secondary formula will be 500 = x+2y
(It is not 2x because the building is on one side and does not require fencing)
Now solve your secondary for one variable: x=500-2y
Plug into primary: A= (500-2y) (y)
Distribute y in: 500y - 2y^2
Derivative: 500 - 4y
Set equal to zero and solve for y: 500-4y = 0
y = 125
Plug y back into secondary to find x: 500 = x + 2(125)
x= 250
The dimensions of the largest area are 250 X 125.
For angle of elevation, you sort of use the steps for related rates. The best thing to do is to draw the picture and label what is given. Most likely you will have to use the Pythagorean theorem to find a leg or the rate a leg is changing. Once that is found you take the derivative of the whole Pythagorean theorem and solve for the unknown. Hope this helps!
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