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Math Topics
Applied Calculus
Functions, Limits and Rates of Change
1
Introduction to Functions and Relations
2
Algebra and Composition with Functions
3
Slope, Rates of Change, and Linear Functions
4
Introduction to Limits
Find each limit for the function below \[P(x)= \begin{cases} 20x-100 & 0 \leq x \leq 150 \\ 20x-250 & 150< x \leq 350 \end{cases}\]
\(\displaystyle\lim_{x\to 145} P(x)\)
\(\displaystyle\lim_{x\to 150} P(x)\)
\(\displaystyle\lim_{x\to 155} P(x)\)
Each of the following limits result in the indeterminate form \(\frac{\infty}{\infty}\) upon direct substitution. Use the Dominant Term Property to evaluate each one.
\(\displaystyle\lim_{x\to\infty}\dfrac{3x}{x^2+1}\)
\(\displaystyle\lim_{x\to\infty}\dfrac{2x+1}{x-3}\)
\(\displaystyle\lim_{x\to\infty}\dfrac{x^2}{2x-4}\)
5
Functions and Continuity
Discuss the continuity of the function at \(x=2\).
\(f(x)=\dfrac{x^2-x-2}{x-2}\)
\(g(x)=\begin{cases} \dfrac{x^2-x-2}{x-2} & \text{if } x\neq 2\\ 1 & \text{if } x=2 \end{cases}\)
\(h(x)=\begin{cases} \dfrac{x^2-x-2}{x-2} & \text{if } x\neq 2\\ 3 & \text{if } x=2 \end{cases}\)
Discuss the continuity of the function at \(x=4\).
\(f(x)=\begin{cases} \dfrac{1}{2}x-3 & \text{if } x\leq 4\\ -x+3 & \text{if } x>4 \end{cases}\)
\(g(x)=\begin{cases} -\dfrac{1}{2}x+4 & \text{if } x\leq 4\\ 3x-9 & \text{if } x>4 \end{cases}\)
\(h(x)=\begin{cases} 2x-1 & \text{if } x< 4\\ \dfrac{1}{4}x+5 & \text{if } x>4 \end{cases}\)
6
Average and Instantaneous Rates of Change
The function relating to the bookstore’s profit \(P\) and selling price \(x\) is
\(P(x)=-x^2+13x-22\) for \(2\leq x \leq 11\)
Find the average rate of change of the profit with respect to the selling price if the selling price changes from \(\$3\) to \(\$6\).
A magazine publisher has determined that the cost \(C\) (in dollars) associated with \(x\) number of half-page articles is given by the function \[C(x)=-5x^2+200x \quad 0\leq x \leq 20\]
If the magazine is currently running \(12\) half-page articles, what is the expected increase in cost if it were to run \(13\)?
If the magazine is currently running \(15\) half-page articles, what is the expected increase in cost if it were to run \(16\)?
The distance of the ball above Kendra’s hand is given by the function \[s(t)=32t-16t^2 \quad \text{for} \; 0\leq t\leq2\] Find the instantaneous rate of change for the height at \(t=\displaystyle\frac{1}{4}\) second and \(t=1\) second.