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In this section of the project you are going to focus on using derivatives and limits, with some other mathematical ideas you already knew about before calculus. Your professor is going to give you two functions to work with. One of them will be a polynomial and the other from a family of functions that you are familiar with. You should follow this set of instructions for your first function and then go back and repeat the instructions for your second function.
The first task in this part of the project will be to compute the first and second derivatives of each of the functions using technology. You should use an equation editor to write with correct formatting each function with its first and second derivative.[1] Below the symbolic representation of the two original functions you created with the editor, should make a complete graph[2] of each function. You should use the technology to determine the x and y intercepts of the original functions. Make certain to show your calculations using the editor. The intercepts should be labeled on your graph and identified below the symbolic representation of the original functions. Now, label the critical points on the graphs for each of your two functions. You can use technology to help you compute this. Add this information below the space where you identified the x and y intercepts. Below that, show were the functions would have tangent lines that are positive or negative. This should be type set in order to receive any credit. Also, label the points of inflection for each of your two functions. You can use technology to help you compute this. Add this information below the space where you identified the critical points. Below that, show were the functions would be concave up or concave down. This should be type set in order to receive any credit.
Next, for each function identify the x-values where discontinuities occur. These should be identified as removable or non-removable. The intervals of continuity should then be shown, using the equation editor. Following that, identify any asymptotes that exist. Vertical asymptotes should be written as equations using the editor. Horizontal asymptotes should be written using limit notation and the editor. If asymptotes do not exist, that should be noted. After this, identify the end behavior of the function using the equation editor for formatting, if you have not done this with asymptotes already.
Now let’s focus on the first and second derivatives of each function. Create a complete graph[3] of the first derivative and place it below the symbolic representation of the first derivative. Use the technology to determine the x and y intercepts of the first derivative. They should be labeled on the graph and identified below the symbolic representation of the respective first derivative function along with the titles of the axes, units, etc. that are part of the mentioned complete graph.
Then, create a complete graph[4] of the second derivative and place it below the symbolic representation of the second derivative. Use the technology to determine the x and y intercepts of the second derivative. They should be labeled on the graph and identified below the symbolic representation of the respective second derivative function along with the titles of the axes, units, etc. that are part of the mentioned complete graph.
Finally, you should create a complete graph[5] that shows the function and its two derivatives on the same set of axes. Again, do this for each function. (You should have a total of six graphs in your report after this instruction… graphs of the two original functions, graphs of the first derivatives of each function, and graphs of the second derivatives of each function.) Using this graph as a basis, discuss the relationships between the three graphs. Discuss how the intercepts on the graphs of derivatives are related to the graphs of their respective functions. Similarly, discuss how the values of the derivatives are related to the behavior of the original function.

In this section of the project you are going to focus on using derivatives and limits, with some other mathematical ideas you already knew about before calculus. Your professor is going to give you two functions to work with. One of them will be a polynomial and the other from a family of functions that you are familiar with.

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