See the coversheet for instructions and the point value of each problem.
- Sketch the graph of a function \(f\) for which:
- It has a removable discontinuity at \(x=1\),
- \(\lim_{x\rightarrow 2^-}f(x)=+\infty\),
- \(\lim_{x\rightarrow 2^+}f(x)=-1\),
- \(\lim_{x\rightarrow +\infty}f(x)=3\),
- \(f'(x)>0\) everywhere it exists.
- Let \(f(x)=\sqrt{x}-2\).
- State the definition of the derivative as a limit.
- Using this definition, compute \(f'(x)\).
- Find the equation for the tangent line at \(x=9\).
- Graph \(f(x)\) and the tangent line.
- Find values for \(m\) and \(b\) so that
\(\displaystyle f(x)=
\begin{cases}
x^2 & \text{if \(x\le -2\)}\\
mx+b & \text{if \(x> -2\)}
\end{cases}\) is differentiable at \(x=-2\).
- Compute the following limits. If you use the squeeze theorem,
then indicate the two functions that you are using to squeeze.
- \(\displaystyle\lim_{x\rightarrow \infty}x\sin(x^{-1})\)
- \(\displaystyle\lim_{x\rightarrow \infty}\frac{\sin(x)}{x}\)
- \(\displaystyle\lim_{x\rightarrow \infty}x\cos(x^{-1})\)
- \(\displaystyle\lim_{x\rightarrow \infty} (x-x^2)\)
- \(\displaystyle\lim_{x\rightarrow \infty} (x-\sin(x))\)
Last modified: Thu Sep 14 18:58:06 UTC 2017