See the coversheet for instructions and the point value of each problem.
-
- For \(f(x)=x^3\), compute \(\displaystyle \lim_{h\rightarrow
0}\frac{f(x+h)-f(x)}{h}=\)
- For \(f(x)= (x+1)^{-1}\), compute
\(\displaystyle \lim_{h\rightarrow 0}\frac{f(x+h)-f(x)}{h}=\)
- Compute
\(\displaystyle \lim_{z\rightarrow x^{-}} \frac{z^2-x^2}{|x-z|}=\)
-
Sketch the graph of a single function that is continuous except:
- at 1, where it has a removable discontinuity;
- at 2, where it has a jump discontinuity but is continuous from the right; and
- at 3, where it has an infinite discontinuity.
- 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.
-
- State the Intermediate Value Theorem. Identify what are
its assumptions (hypotheses) and what are its conclusions.
- 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