# MATH 2301-102 and -103 Fall 2017 Calculus I Recitation 9 Week 10

See the coversheet for instructions and the point value of each problem.

1. 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. For the function $$\displaystyle f(x)= \frac{\sqrt{1-x^2}}{x}$$:
1. Find the domain.
2. Find the intercepts.
3. Determine any symmetries.
4. Find any asymptotes.
5. Find the intervals on which $$f$$ is increasing or decreasing.
6. Find the local maximum and minimum values of $$f$$.
7. Find the intervals of concavity and the inflection points.
8. Use the information above to sketch the graph.
[This is section 4.4 #21; you can check your answer in the back of the book, but you need to show how to get to the answer.]
3. For the function $$\displaystyle f(x)= \frac{1}{2}x -\sin(x)$$ on the interval $$0 < x < 3\pi$$:
1. Determine any symmetries.
2. Find any vertical 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.
[This is section 4.4 #31; you can check your answer in the back of the book, but you need to show how to get to the answer.]