# MATH 2301-102 Fall 2018 Calculus I Recitation 6 Week 9

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

1. Compute the following limits. If you use the Squeeze theorem or L'Hopital's rule, then show that their assumptions are satisfied.
1. $$\displaystyle\lim_{x\rightarrow \infty}x^2 e^{-x}$$
2. $$\displaystyle\lim_{x\rightarrow \infty}\sin(x^2) e^{-x}$$
1. On the interval $$x\in[-1,1]$$, sketch the graph of a function $$f$$ that is even (meaning $$f(x)=f(-x)$$).
2. On the interval $$x\in[-1,1]$$, sketch the graph of a function $$f$$ that is odd (meaning $$f(x)=-f(-x)$$).
3. On the interval $$x\in[-1,1]$$, sketch the graph of a function $$f$$ that has period $$1/2$$ (meaning $$f(x)=f(x+1/2)$$).
2. 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 $$(4,3)$$.