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Activity9.1Motivation

In the first activity of this lab you are going to discuss a few questions with your group mates that will hopefully motivate you for one of the topics covered in the lab.

Subsection9.1.1Exercises

1

Discuss how you could use the first derivative formula to help you determine the vertex of the parabola \(y=-2x^2+18x-7\) and then determine the vertex. Remember that the vertex is a point in the \(xy\)-plane and as such is identified using an ordered pair.

2

The curves in Figures 9.1.1 and 9.1.2 were generated by two of the four functions given below. Use the given functions along with their first derivatives to determine which functions generated the curves. Please note that the \(y\)-scales have deliberately been omitted from the graphs and that different scales were used to generate the two graphs. Resist any temptation to use your calculator; use of your calculator totally obviates the point of the exercise.

Figure9.1.1mystery curve 1
Figure9.1.2mystery curve 2

The potential functions are \begin{align*} \fe{f_1}{x}&=\frac{1}{(x-2)^{\sfrac{10}{7}}}+C_1&\fe{f_2}{x}&=\frac{1}{(x-2)^{\sfrac{2}{7}}}+C_2\\ \fe{f_3}{x}&=(x-2)^{\sfrac{2}{7}}+C_3&\fe{f_4}{x}&=(x-2)^{\sfrac{10}{7}}+C_4 \end{align*} where \(C_1,C_2,C_3,C_4\) are unknown constants.