Many applications of calculus require that you find the maximum or minimum of a function. When these are found over a given interval they are called local extrema. A graphing calculator can find these extrema for you with the push of a couple buttons.
Extreme Roller Coasters
Think about a tall, shiny roller coaster gleaming in the sun on a beautiful summer day. Now, picture waiting at the top of the first hill, your heart is pounding, and you are about to plunge what seems like a thousand feet to the bottom of the hill. In the math world the top and bottom of the hill are called local extrema. This lesson will explain what these extrema are and how you can use a graphing calculator to find them.
What are Local Extrema?
The local extrema of a function are the x values that will give the maximum or minimum y value over a given interval. Local extrema cannot occur at the endpoint of the interval. Graphically a maximum looks like the very top of a hill and a minimum looks like the bottom of a u. Local extrema are different from global extrema in that global extrema are over the entire domain (all possible x values) of the function.
Finding Local Extrema Using Calculus.
Imagine driving over a hill. For a while you are going uphill, you reach the top of the hill, then you start going downhill. There is a moment between going up and going down when you are on flat ground. This moment is your maximum height. The steepness of a hill is called slope. Going uphill is positive slope, going downhill is negative slope. That moment in between, when you are on flat ground, has a slope of zero.
In calculus the derivative of a function is the slope of the tangent line to a point on the curve of the function. To find the maximum or minimum using calculus (the very top of a hill or very bottom of a valley), find the points where the derivative equals zero.
On the TI-83/84/85/89 graphing calculators the buttons that you will need to know to find the maximum and minimum of a function are y=, 2nd, calc, and window.
y=: This button allows you to enter a function into the function bank. This is how you tell the calculator which function you are using. Please clear out any other functions that may already be there.
2nd: This button allows you access to the second function of the buttons. You will need to use calc which is written in a different color. The 2nd button allows access to this function.
Calc: This is found on the top row of the calculator. After selecting this function, the calculator will give you a menu of options. You will want to choose either maximum or minimum.
Window: Many times the local extrema are not visible in the standard window (a 10 x 10 frame). You can adjust the window size to see the part you are interested in.
Steps to Using the Calculator for Extrema
The calculator will present the graph of the function. After you select the interval for the maximum or minimum, it will find the largest or smallest y value contained within that interval.
Enter the function into the y= bank.
Select the 2nd button, followed by the calc button.
Select either maximum or minimum depending on the question.
If the local extrema are not easily visible, adjust the window.
The graph appears on the screen, with the words 'Left Bound?' Move the cursor to the left of the extrema, hit enter.
The screen will now say 'Right Bound?', move the cursor to the right of the extrema, hit enter.
The screen will say 'Guess?', hit enter
The screen will now have the extrema written in terms of x and y on the bottom of the screen.
Find the local extrema of the function f(x) on the interval 1 to 3
If you graph this function using the y= bank in the standard window; you will not see the global extrema on the graph. They are outside of the window.
The question says the extrema are on the interval from 1 to 3. Adjust your window to look at this interval. Go to the 'window' button on the calculator. Choose your X-min to be a little smaller than your interval. Zero will work well for the X-min. Choose your X-max to be a little larger than your interval. In this case four works well for your X-max. You now have a much clearer picture of your local extrema.
Hit 2nd Calc
Choose either maximum or minimum (for this question you will do both)
Follow the screen prompts
The local extrema are a minimum at (2.6,-2.53) and a maximum at (1.44,2.09).
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Jon wants to build a rectangular fence using 73 feet of fencing that he already owns. Three sides of the rectangle will be made of fencing, and his house will serve as the fourth side. The area enclosed in this fence is given by the equation A(x)=73x-2x2, with A(x) being the area in square feet, and x the width in feet. What dimensions would give him the maximum area?
Enter the equation into y= bank
Select a proper window. If you are using the standard window for this graph, you will not see anything useful. What is the question asking you to find? Width and area. Both are measurements that need to be positive. Change your X-min and Y-min to zero (no negatives needed). You also know that he only has 73 ft of fencing, so the width cannot exceed 73 ft. Change the X-max to 73. The y value is the area. Pick an area that would seem reasonable with 73 ft of fencing. You may have to do a little estimating and calculating. I choose 20 by 33 (20+20+33), recalling that the house serves as the fourth side. 20 x 33 =660. So, you can set your Y-max to around 700. If you still can't see the top of the parabola, make the Y-max larger.
2nd Calc, choose maximum
Follow the screen prompts.
The local extrema are (18.25,666.13). These correspond to x and A(x). You will need to calculate the other side (73 - 2*18.25).
The dimensions are 18.25 ft by 36.5 ft, with an area of 666.13 sq ft. (There are two fenced sides that are 18.25 ft and one fenced side that is 36.5 ft.)
When you are finding the local extrema of a function, you are finding the maximum or minimum value of that function over a specific interval. You can use a graphing calculator to find these extrema. The buttons of y=, 2nd, Calc, and Window can reveal the extrema of a function with only a handful of steps. If you enter the equation into the y= equation bank, pick a window that will show the extrema, select 2nd Calc, choose the appropriate minimum or maximum, and follow the on screen prompts; you will have the extrema of a function.
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