The steps for optimization are as follows:
1. Identify your primary and secondary equations. Primary will be the one the problem is asking you to minimize or maximize. I've also noticed that the secondary will usually be set equal to a number.
2. After finding secondary, solve it for one variable if there are two.
3. Once you have this variable, plug it into your primary equation for the variable you solved the secondary equation for
4. Take the derivative of the equation you just formulated and set it equal to zero (this zero will become one of your answers)
5. Once you come out with your zeros, plug them into your secondary equation and solve for the variable you have left (this will give you your second answer)
The terms for the First Derivative Test:
1. Increasing
2. Decreasing
3. Horizontal Tangent
4. Min/Max
Steps:
1. Take the derivative
2. Set it equal to zero
3. Solve for x to get the possible critical points [[can someone clarify this part???? My notebook says it equals critical points AND mins/maxs/horizontal tangents]]
4. Set up intervals with your x value(s)
5. Plug into your first derivative
6. To find an absolute extrema, plug in the values from step 5 into your original function
Can someone check this for me? I don't know if I'm not in my right mind right now [no comments] or if I'm really failing at this? Or if I just do it and not think about it now?
f(x) = 1/2x - sinx
Find the extrema on the interval (0, 2pi)
f'(x) = 1/2 - cosx
1/2 - cosx = 0
-cosx = -1/2
cosx = 1/2
x=cos^-1(1/2)
x=pi/3 and 5pi/3
Because
| +
-------
| +
cos is positive on the first and forth quadrants
300 degrees in radians is 5pi/3
and
60 degrees in radians is pi/3
well tom we have another ap quiz that im not looking forward to. Some of the stuff that im going to struggle with include the stuff when you have to look at the graphs and determine the first der. or second deriv of the graph. I need help with this
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An example of the graphs would be on AB Model Examination 1 - Section 1 - Part A - Number 22. They give the graph of f'(x) and tell you pick out the graph of f(x).
ReplyDeleteThe f'(x) graph is a horizontal line that meets an increasing line right at the axis. Both lines are above the axis.
Ok, since both lines are above the axis, that means that both lines have to be increasing, so you can eliminate (A), (B), and (E).
You can them eliminate (C) because the derivative of a horizontal line would be a horizontal line at x = 0.
Interpreting a graph is very easy but in order to do it properly one must understand certain properties.
ReplyDelete1. If the original graph is increasing, the slope is positive.
2. If the original graph is decreasing, the slope is negative.
3. An interval with a positive slope on the first derivative means that there is a downward concavity on that interval in the second derivative.
4. An interval with a negative slope on the first derivative means that there is an upward concavity on that interval in the second derivative.
5. Upward concavity (bowl-shaped) is positive.
6. Downward concavity (umbrella-shaped) is negative.
7. There is a horizontal tangent where the slope=0 on the original graph.
8. Wherever there is a horizontal slope there is either a maximum or a minimum value.
9. A x intercept on the first derivative is either a maximum of a minimum on the original graph.