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In this part of the project you are going to explore applications of the derivative and the antiderivative using technology. For this part of the project, you will use the two functions that you were assigned for the second part of the project, the polynomial and the other function from a familiar family of functions.
The first task in this part of the project will be to graph again your assigned functions. If you did not have the grids on so that your function looks like it is on graph paper, this time you will need to do that. Place those graphs below the symbolic representation of each of the functions. After you have done that, place an interval of interest (you get to choose it, but it must be continuous) below the symbolic representation of the function. Do this using the equation editor. There must be no points of discontinuity in the interval. Then count the number of squares between the graph of the function and the x-axis in order to estimate the area. You should put your estimate of the area below the typed interval. Make sure that your areas are in appropriate units. Each of the tick marks on the x-axis may represent 10 units and the tick marks on the y-axis may represent 5 units. So each box does not represent an area of 1, but represents an area of 10 X 5 = 50, with appropriate units.
Next up, you will use a different technique to estimate the area between the graph of the function and the x-axis, left endpoint, right endpoint, and midpoint, Reimann Sums. You should use 6 subdivisions to do this for the left, right, and midpoint estimates. [Hint:To make your life easier, you might want to consider making the difference between the upper and lower bound where you will be computing the area to be a multiple of six.] Let’s say your interval is [a,b].
MATH181_project3.png
Then, graph the function and mark the rectangles using the left endpoints of each subinterval.
Repeat using the right endpoints and do again using the midpoints of the subintervals.
Below each of these graphs with rectangles show the summation of values that give you the estimation of the area using the left-endpoints, right endpoints and the midpoint of each subinterval.
The next step involves, using your graphs from above. Discuss in a paragraph, which is a better estimate of the area between the graph of the function and the x-axis. Make sure to mention over estimates, underestimates, etc. Also consider whether the slopes of the tangent lines over your interval (or subintervals) make one of the estimates better than the others.
The last task is to choose 2 points on a continuous portion of the graph of the function. [Hint: Clearly label which function you are working with.] Find the slope of the secant line joining these 2 points.
Then use the equation editor to show the calculation of the slope of the secant line connecting the two points.
Then find a point in the interval that has a tangent line with the same slope as the secant line. [Hint: Use the derivative of the function from above, set that equal to the slope of the secant line, and then solve for a value that is in the continuous interval you chose.] Show these calculations using correct mathematical notation with an equation editor.[1]
Finally copy a graph of the original function below the calculations. Plot the two points on the graph that you chose and graph the secant line, the tangent line and the point whose derivative equals the slope of the secant line.
The last part of this part of the project is to discuss why you chose the technologies that you did choose.
[1] MathType, which interfaces with MicroSoft Word and other word processing packages is available on many of the computers on campus. Also, MicroSoft Word and Google Docs have a built in equation editor that you can use to format your equations / functions. In MicroSoft Word just choose Insert from the top menu bar and then Equation.

In this part of the project you are going to explore applications of the derivative and the antiderivative using technology. For this part of the project, you will use the two functions that you were assigned for the second part of the project, the polynomial and the other function from a familiar family of functions.

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