Calculus I

Lab 08

Directions

Go to https://student.desmos.com/ and enter the code RVMCG8 . Your work saves automatically. You may use your book and notes, and this lab may require some scratch work. Not all questions are the same, so read each question carefully. Complete the following (there are corresponding slides for each problem):

Problem 0

Enter in your name, I need to know who to give the grades to.

Problem 1

Let \(f(x) = 3x^4-8x^3+6x^2+1\).

Find the domain of \(f\), and graph \(f\).
Label all zeros of \(f\) on your graph.
Label all transition points (inflection points and local extrema points) of \(f\) on your graph.
On your graph, color regions where \(f\) is concave up in one color, and regions where \(f\) is concave down in another color.
State any asymptotic behavior of \(f\).

Problem 2

Let \(g(x) = \sin x + \cos x\) with domain \([0,2\pi]\).

Graph \(g\) on its domain.
Label all zeros of \(g\) on your graph.
Label all transition points of \(g\) on your graph.
On your graph, color intervals where \(g\) is increasing in one color, and intervals where \(g\) is decreasing in another color.
In a text cell, state any intervals where \(g\) is concave up, and any intervals where \(g\) is concave down.

Problem 3

Let \(h(x) = xe^{-x^2}\).

State the domain of \(h\), and graph \(h(x)\).
Label all zeros of \(h\) on your graph.
Label all transition points of \(h\) on your graph.
On your graph, color regions where \(h\) is concave up in one color, and regions where \(h\) is concave down in another color.
State any asymptotic behavior of \(h\).

Old Exam Question

The above is based off an old exam question, given below. You don't need to do this for the lab, but it might be helpful for the lab and preparing for the third exam.

Let \(h(x) = xe^{-x^2}\).

(5 points) Find all critical numbers of \(h\).
(5 points) Find all intervals on which \(h\) is increasing and all intervals on which \(h\) is decreasing.
(5 points) Use the first derivative test to determine whether each critical number yields a local minimum, local maximum, or neither. Give the values of any local extrema.
(5 points) Apply the second derivative test to \(h\) or state that it fails.
(5 points) Find all intervals on which \(h\) is concave up, and all intervals on which \(h\) is concave down.
(5 points) Find all inflection points of \(h\).
(5 points) Use L'Hopital's rule to find \(\displaystyle \lim_{x \to \infty} h(x)\) and \(\displaystyle \lim_{x \to -\infty} h(x)\).
(5 points) Find the absolute minimum and maximum value of \(h\) on the interval \([0,2]\).
(5 points) Sketch a graph of \(h\).