MATH 2301-100 Spring 2022 Calculus I Recitation 6 Week 8

Work in a group of at most 4. Explain to those who do not understand. Ask questions if you do not understand.

1. (25 points) Sketch the graph of a single function that has all of the following properties:
   1. Continuous and differentiable everywhere except at \(x=-3\), where it has a vertical asymptote.
   2. A horizontal asymptote at \(y=1\).
   3. An \(x\)-intercept at \(x=-2\).
   4. A \(y\)-intercept at \(y=4\).
   5. \(f'(x) \gt 0\) on the intervals \((-\infty,-3)\) and \((-3,2)\).
   6. \(f'(x) \lt 0\) on the interval \((2,\infty)\).
   7. \(f''(x) \gt 0\) on the intervals \((-\infty,-3)\) and \((4,\infty)\).
   8. \(f''(x) \lt 0\) on the interval \((-3,4)\).
   9. \(f'(2)=0\).
   10. An inflection point at \((4,3)\).
2. (25 points) For the function \(\displaystyle f(x)= \frac{1}{3}x^3-\frac{1}{2}x^2-2x+9\):
   1. Find the domain.
   2. Find any asymptotes.
   3. Find the intervals on which \(f\) is increasing or decreasing.
   4. Find the local maximum and minimum values of \(f\).
   5. Find the intervals of concavity and the inflection points.
   6. Use the information above to sketch the graph.
   [You can check your answer with sage, but you need to show how to get to the answer.]
3. (25 points) For the function \(\displaystyle f(x)= \frac{\sqrt{1-x^2}}{x}\):
   1. Find the domain.
   2. Find any asymptotes.
   3. Find the intervals on which \(f\) is increasing or decreasing.
   4. Find the local maximum and minimum values of \(f\).
   5. Find the intervals of concavity and the inflection points.
   6. Use the information above to sketch the graph.
   [You can check your answer with sage, but you need to show how to get to the answer.]
4. (25 points) For the function \(\displaystyle f(x)= \exp(-x^2)\):
   1. Find the domain.
   2. Find any asymptotes.
   3. Find the intervals on which \(f\) is increasing or decreasing.
   4. Find the local maximum and minimum values of \(f\).
   5. Find the intervals of concavity and the inflection points.
   6. Use the information above to sketch the graph.
   [You can check your answer with sage, but you need to show how to get to the answer.]

Last modified: Thu Feb 24 20:00:06 UTC 2022