Exam 1 -- Lab portion 
(1/3 of total exam grade)

MAT 229, Spring 2025

Name:

Instructions:

In this portion of the exam, you will study a function, called “f”, defined in the cell below:

f[x_]:=-36 x-5 (x^3)+(x^5)

“Show your work”: document your thoughts or any issues encountered in the notebook (using comments or text cells).

If you get stuck, don’t panic. Ask me a question.

At the end of the hour, please email your final notebook to me at longa@nku.edu. I should be able to execute your file and see the solutions clearly.

Featured commands:

Plot (Options: PlotRange, PlotStyle)

Solve (Options: Real)

ListPlot (Options: PlotStyle)

Show

Integrate

Exercises:

Plot f on the interval [-3.25,3.25] (5 pts).

Find the local extrema of f by solving for where the derivative of f is equal to 0. Numerical approximations are fine, and you should ignore any imaginary or complex roots (8 points). Create a list “extrema” of the pairs {x,f(x)} (2 points)

Find the inflection points of f by solving for where the second derivative is equal to 0. Numerical approximations are fine, and you should ignore any imaginary or complex roots (8 points). Create a list “inflection” of the pairs {x,f(x)} (2 points)

Define the local linearization to f  at x=(1/2)  (whose graph is the tangent line when x=(1/2)).  Call it locallinear. (10 points)

Define a function called g, which is the antiderivative of f  whose value is 0 at x=0 .  (10 points)

Provide four graphs:

a. A graph of f and its first and second derivatives on the interval [-3.25, 3.25] (8 points). Make the graph of f Blue and the two derivatives Dashed (2 points).

b. A graph of f with points indicating the locations of the extrema and the inflection points  on the interval [-3.25, 3.25] (8 points). Make the extreme points Red and the inflection points Blue (2 points). Extra Credit: make the points Large for 3 points.

c. A graph of the difference between f and locallinear on the interval [0,1]  (8 points).  Comment on how well the local linearization approximates f on this interval (2 points).

d. A graph of f and its antiderivative g on the interval [-3.25, 3.25] (8 points). Make the range [-200, 100] (2 points).