Optimization
The steps are:
1. Determine everything given in the problem and what you are looking to maximize or minimize. This is stated in the problem.
2. Write a primary equation, which is an equation for what you are maximizing or minimizing.
3. Write a secondary equation using the other information given.
4. Solve the secondary equation in terms of one variable (if necessary) then plug that into the primary equation for that variable.
5. Take derivative of the plugged in primary equation, set the equation equal to zero, then solve for the remaining variable.
6. Plug back into the secondary equation to find the other variable.
EXAMPLE:
We need to enclose a field with a fence. We have 500 feet of fencing material and a building on one side of the field so won't need any fencing. Determine the dimensions of the field that will enclose the largest area.
1. P=500 ft, one side does not need fencing, maximizing area (enclose the largest area)
2. Since we are maximizing area, the primary equation will be the area equation for a rectangle. A=xy
3. The secondary equation will be perimeter since that is given in the problem. 500= x+2y
Note: the formula for perimeter is usually 2x+2y, but since one side does not need fencing, it is only x+2y.
4. Solve for one variable: 500=x +2y x= 500 -2y
Now plug into primary equation: A= 500 - 2y (y)
Distribute the y in: 500y - 2y^2
5. Take derivative: 500 - 4y
Set equal to zero and solve for y; 500 -4y = 0 y = 125
6. Plug y into secondary equation to find x: 500 = x + 2y 500 = x+2(125) x= 250
The dimensions of the largest area are 250 X 125.
An easy problem on the AP is the derivative of an integral.
For these problems, just plug in B into the equation and multiply that by the derivative of B.
Example: If f(x) = The integral of (t^2-1)^1/3 dt on the interval [0, x+1], then f'(-4) =
Plug in B: ((x+1)^2 -1 )^1/3 (1)
Next, all you have to do is plug in -4.
((-4+1)^2 -1 ) ^1/3
((-3)^2 -1 ) ^1/3
(9-1)^1/3
8^1/3 = 2
Easy points.
A review on related rates and angle of elevation would be greatly appreciated.
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The steps for related rates are:
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2. Identify the things that you are looking for
3. Sketch a graph and then label that graph
4. Create and write an equation using all of the variables
5. Take the derivative of this equation with respect to time
6. Substitute everything back in
7. Solve the equation
HOPE THIS HELPS!
angle of elevation uses the same steps as related rates but the difference is that you are looking for an angle instead of a rate
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