MATH 2301-102 and -103 Fall 2017 Calculus I Recitation 3 Week 3

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

    1. For \(f(x)=x^3\), compute \(\displaystyle \lim_{h\rightarrow 0}\frac{f(x+h)-f(x)}{h}=\)
    2. For \(f(x)= (x+1)^{-1}\), compute \(\displaystyle \lim_{h\rightarrow 0}\frac{f(x+h)-f(x)}{h}=\)
    3. Compute \(\displaystyle \lim_{z\rightarrow x^{-}} \frac{z^2-x^2}{|x-z|}=\)
  1. Sketch the graph of a single function that is continuous except:
    1. at 1, where it has a removable discontinuity;
    2. at 2, where it has a jump discontinuity but is continuous from the right; and
    3. at 3, where it has an infinite discontinuity.
  2. Let \[ f(x)= \begin{cases} {\displaystyle\frac{2x^2-x-15}{x-3}} & \text{if \(x \lt 3\)}\\ kx-1 & \text{if \(x \ge 3\)} \end{cases}\, . \] Determine the value of \(k\) that will make the function \(f\) continuous at 3, or explain why no value of \(k\) will work.
    1. State the Intermediate Value Theorem. Identify what are its assumptions (hypotheses) and what are its conclusions.
    2. Use the Intermediate Value Theorem to show that the equation \(x^2 =\cos(x)\) has a solution.

Last modified: Thu Sep 7 18:06:20 UTC 2017