MATH 2301-102 and -103 Fall 2017 Calculus I Recitation 4 Week 4

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

  1. Sketch the graph of a function \(f\) for which:
    1. It has a removable discontinuity at \(x=1\),
    2. \(\lim_{x\rightarrow 2^-}f(x)=+\infty\),
    3. \(\lim_{x\rightarrow 2^+}f(x)=-1\),
    4. \(\lim_{x\rightarrow +\infty}f(x)=3\),
    5. \(f'(x)>0\) everywhere it exists.
  2. Let \(f(x)=\sqrt{x}-2\).
    1. State the definition of the derivative as a limit.
    2. Using this definition, compute \(f'(x)\).
    3. Find the equation for the tangent line at \(x=9\).
    4. Graph \(f(x)\) and the tangent line.
  3. 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 \(x=-2\).
  4. Compute the following limits. If you use the squeeze theorem, then indicate the two functions that you are using to squeeze.
    1. \(\displaystyle\lim_{x\rightarrow \infty}x\sin(x^{-1})\)
    2. \(\displaystyle\lim_{x\rightarrow \infty}\frac{\sin(x)}{x}\)
    3. \(\displaystyle\lim_{x\rightarrow \infty}x\cos(x^{-1})\)
    4. \(\displaystyle\lim_{x\rightarrow \infty} (x-x^2)\)
    5. \(\displaystyle\lim_{x\rightarrow \infty} (x-\sin(x))\)

Last modified: Thu Sep 14 18:58:06 UTC 2017