See the coversheet for instructions and the point value of each problem.
- Compute the following limits. If you use the Squeeze
theorem or L'Hopital's rule, then show that their assumptions are satisfied.
- \(\displaystyle\lim_{x\rightarrow \infty}x^2 e^{-x}\)
- \(\displaystyle\lim_{x\rightarrow \infty}\sin(x^2) e^{-x}\)
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- On the interval \(x\in[-1,1]\), sketch the graph of a function \(f\) that is even (meaning \(f(x)=f(-x)\)).
- On the interval \(x\in[-1,1]\), sketch the graph of a function \(f\) that is odd (meaning \(f(x)=-f(-x)\)).
- On the interval \(x\in[-1,1]\), sketch the graph of a function \(f\) that has period \(1/2\) (meaning \(f(x)=f(x+1/2)\)).
- Sketch the graph of a single function that has all of the
following properties:
- Continuous and differentiable everywhere except at \(x=-3\),
where it has a vertical asymptote.
- A horizontal asymptote at \(y=1\).
- An \(x\)-intercept at \(x=-2\).
- A \(y\)-intercept at \(y=4\).
- \(f'(x) \gt 0\) on the intervals \((-\infty,-3)\) and \((-3,2)\).
- \(f'(x) \lt 0\) on the interval \((2,\infty)\).
- \(f''(x) \gt 0\) on the intervals \((-\infty,-3)\) and \((4,\infty)\).
- \(f''(x) \lt 0\) on the interval \((-3,4)\).
- \(f'(2)=0\).
- An inflection point at \((4,3)\).
Last modified: Thu Oct 18 17:33:39 UTC 2018