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Applied Calculus
Functions, Limits and Rates of Change

1
Introduction to Functions and Relations

welcome
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We have two main topics to cover in this course.  To get ready for those two topics we need to review a little algebra.  Then we need to develop some new material with what are called limits.

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Mr. McKeague cc
Mr. McKeague
Problem  2
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Make a table and graph for \(y=7.5x\) for \(0\leq x \leq 40\).

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Stefanie cc
Stefanie
Betsy cc
Betsy
Preston cc
Preston
Edwin cc espanol spanish
Edwin
Problem  3

State the domain and range for \(y=7.5x\), \(0\leq x \leq 40\).

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Stefanie cc
Stefanie
Betsy cc
Betsy
Preston cc
Preston
Problem  4
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Use the equation \(h=32t-16t^2\) for \(0\leq t \leq 2\) to construct a table to give the height at quarter-second intervals, then graph the function.

Choose instructor to watch:
Stefanie cc
Stefanie
Betsy cc
Betsy
Preston cc
Preston
Edwin cc espanol spanish
Edwin
Problem  5

The table shows the prices of used Ford Mustangs in the local paper. Figure 7 is a scatter diagram of those cars. Why is the data not a function?

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Aaron cc
Aaron
Stefanie cc
Stefanie
CJ cc
CJ
Problem  6
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Graph \(x=y^2\)

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Stefanie cc
Stefanie
Betsy cc
Betsy
CJ cc
CJ
Edwin cc espanol spanish
Edwin
Problem  7
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Graph \(y=\displaystyle\frac{1}{x}\)

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Mr. McKeague cc
Mr. McKeague
Stefanie cc
Stefanie
Preston cc
Preston
Edwin cc espanol spanish
Edwin
Problem  8
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Graph \(y=\sqrt{x}\) and \(y=\sqrt[3]{x}\)

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Katherine cc
Katherine
Betsy cc
Betsy
CJ cc
CJ
Julieta cc espanol spanish
Julieta
Problem  9
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If \(f(x)=7.5x\), find \(f(0)\), \(f(10)\), \(f(20)\).

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Molly S. cc
Molly S.
Betsy cc
Betsy
Octabio cc
Octabio
Edwin cc espanol spanish
Edwin
Problem  10
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Is \(f(x)=4x-1\) and \(g(x)=x^2+2\), then find \(f(x)\) and \(g(x)\) when \(x=5\), \(-2\), \(0\), \(z\), \(a\), and \(a+3\).

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Betsy cc
Betsy
Preston cc
Preston
Julieta cc espanol spanish
Julieta
Problem  11
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If it takes Lorena \(t\) minutes to run a mile, then her average speed \(s\), in miles per hour, is given by the formula \[s(t)=\frac{60}{t} \text{ for } t>0\] Find \(s(10)\) and \(s(8)\), and then explain what they mean.

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Octabio cc
Octabio
Betsy cc
Betsy
Preston cc
Preston
Gordon cc espanol spanish
Gordon
Problem  12
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A balloon has the shape of a sphere with a radius of \(3\) inches. Use the following formulas to find the volume and surface area of the balloon. \[V(r)=\frac{4}{3}\pi r^3 \qquad S(r)=4\pi r^2\]

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Octabio
Octabio
Aaron cc
Aaron
Stefanie cc
Stefanie
Octabio cc espanol spanish
Octabio
Mini Lecture

Mini Lecture
Are these functions?

  1. \(\{(7,-1), (3,-1), (7,4)\}\)

  2. \(\{(4,1), (1,4), (-1,-4)\}\)

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Mr. McKeague cc
Mr. McKeague

2
Algebra and Composition with Functions

Problem  1
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Let \(f(x)=4x-3\), \(g(x)=4x^2-7x+3\), and \(h(x)=x-1\). Find \(f+g\), \(fh\), \(fg\), and \(\dfrac{g}{f}\).

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Molly S. cc
Molly S.
Betsy cc
Betsy
Preston cc
Preston
Octabio cc espanol spanish
Octabio
Problem  2
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Let \(f(x)=4x-3\), \(g(x)=4x^2-7x+3\), and \(h(x)=x-1\). Find \((f+g)(2)\), \((fh)(-1)\), \((fg)(0)\), and \(\left(\dfrac{g}{f}\right)(5)\).

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Molly S. cc
Molly S.
Betsy cc
Betsy
Preston cc
Preston
Edwin cc espanol spanish
Edwin
Problem  3
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Use the equation \(x=1300-100p\) to determine the price that should be charged for a weekly revenue of \(\$4000\).

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Mr. Damarest cc
Mr. Damarest
Molly S. cc
Molly S.
Logan cc
Logan
Octabio cc espanol spanish
Octabio
Problem  4
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If \(f(x)=x+5\) and \(g(x)=x^2-2x\), find \((f \circ g)(x)\) and \((g \circ f)(x)\)

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Molly S. cc
Molly S.
Betsy cc
Betsy
Preston cc
Preston
Octabio cc espanol spanish
Octabio

3
Slope, Rates of Change, and Linear Functions

Problem  1
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Find the slope of \(y=2x-3\)

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Stefanie cc
Stefanie
Betsy cc
Betsy
Molly S. cc
Molly S.
Edwin cc espanol spanish
Edwin
Problem  2
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Find the slope through \((-2,1)\) and \((5,-4)\).

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Betsy cc
Betsy
Preston cc
Preston
Molly S. cc
Molly S.
Edwin cc espanol spanish
Edwin
Problem  3
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Find the slope of the line containing \((3, -1)\) and \((3,\, 4)\)

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Stefanie cc
Stefanie
Betsy cc
Betsy
CJ cc
CJ
Cynthia cc espanol spanish
Cynthia
Problem  4
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Use the function \(h(t)=32t-16t^2\) for \(0\leq t \leq 2\) to find the average rate of change from \(\dfrac{1}{4}\) to \(1\) second.

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Mr. Damarest cc
Mr. Damarest
Molly S. cc
Molly S.
Octabio cc
Octabio
Octabio cc espanol spanish
Octabio
Problem  5

Use the function \(V(t)=125\cdot 2^\frac{t}{5}\) for \(t\geq 0\) to find the average rate of change in the value of the painting from \(t=0\) to \(t=10\).

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Mr. Damarest cc
Mr. Damarest
Molly S. cc
Molly S.
Octabio cc
Octabio
Octabio cc espanol spanish
Octabio
Problem  6

Use the formula \(S(r)=4\pi r^2\) for \(r\geq 0\) to find the average rate of change for a radius of \(1\) to \(3\) inches.

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Mr. Damarest cc
Mr. Damarest
Molly S. cc
Molly S.
Octabio cc
Octabio
Octabio cc espanol spanish
Octabio
Problem  7
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If \(f(x)=3x-5\), find \(\dfrac{f(x_2)-f(x_1)}{x_2-x_1}\)

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Mr. Damarest cc
Mr. Damarest
Molly S. cc
Molly S.
Octabio cc
Octabio
Octabio cc espanol spanish
Octabio
Problem  8
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If \(f(x)=x^2-4\), find and simplify \(\dfrac{f(x_2)-f(x_1)}{x_2-x_1}\)

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Mr. Damarest cc
Mr. Damarest
Molly S. cc
Molly S.
Octabio cc
Octabio
Octabio cc espanol spanish
Octabio
Problem  9
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Find the equation and graph the line with slope \(-\dfrac{4}{3}\) and \(y\)-intercept \(5\).

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Betsy cc
Betsy
Preston cc
Preston
Molly S. cc
Molly S.
Octabio cc espanol spanish
Octabio
Problem  10
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Give the slope and \(y\)-intercept for the line \(2x-3y=5\)

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Octabio cc
Octabio
Betsy cc
Betsy
Gordon cc
Gordon
Cynthia cc espanol spanish
Cynthia
Problem  11

Graph the linear function \(f(x)=-\dfrac{2}{3}x+2\) using the slope and \(y\)-intercept.

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Octabio cc
Octabio
Stefanie cc
Stefanie
CJ cc
CJ
Edwin cc espanol spanish
Edwin
Problem  12
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Find the equation of the line with slope \(-2\) that contains the point \((-4,\,3)\).

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Mr. McKeague cc
Mr. McKeague
Preston cc
Preston
Betsy cc
Betsy
Edwin cc espanol spanish
Edwin
Problem  13
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Find the equation of the line that passes through \((-3,\, 3)\) and \((3, -1)\).

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Betsy cc
Betsy
Preston cc
Preston
Octabio cc
Octabio
Octabio cc espanol spanish
Octabio
Problem  14
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Graph the function defined by \(f(x)=\begin{cases} x+1 & \text{if }\, x\leq 1\\ 3 & \text{if }\, x> 1 \end{cases}\)

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Molly S. cc
Molly S.
Gordon cc
Gordon
Octabio cc
Octabio
Octabio cc espanol spanish
Octabio
Problem  15

Use the function \( P(x)=\begin{cases} 20x-100 & 0\leq x \leq 150\\ 20x-250 & 150\leq x \leq 350 \end{cases} \) and its graph to find

  1. The domain

  2. The range

  3. \(P(145)\)

  4. \(P(150)\)

  5. \(P(155)\)

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Octabio cc
Octabio
Lauren cc
Lauren
Octabio cc espanol spanish
Octabio

4
Introduction to Limits

Take 5
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An Introduction to Limits

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Mr. McKeague cc
Mr. McKeague
Problem  2
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Find the limit, if it exists, as \(x\) approaches \(2\), of the function \[f(x)=\frac{3x^2-2x-8}{x-2}\]

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Mr. McKeague cc
Mr. McKeague
Molly S. cc
Molly S.
Octabio cc
Octabio
Octabio cc espanol spanish
Octabio
Problem  3
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Find each limit for the function below \[P(x)= \begin{cases} 20x-100 & 0 \leq x \leq 150 \\ 20x-250 & 150< x \leq 350 \end{cases}\]

  1. \(\displaystyle\lim_{x\to 145} P(x)\)

  2. \(\displaystyle\lim_{x\to 150} P(x)\)

  3. \(\displaystyle\lim_{x\to 155} P(x)\)

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Mr. Damarest cc
Mr. Damarest
Molly S. cc
Molly S.
Octabio cc
Octabio
Octabio cc espanol spanish
Octabio
Problem  4
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If \(f(x)=4\), find \(\displaystyle\lim_{x\to 7}f(x)\).

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Molly S. cc
Molly S.
Octabio cc
Octabio
Octabio cc espanol spanish
Octabio
Problem  5
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If \(f(x)=3x^2-5x+1\), find \(\displaystyle\lim_{x\to 2} f(x)\).

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Mr. McKeague cc
Mr. McKeague
Molly S. cc
Molly S.
Octabio cc
Octabio
Octabio cc espanol spanish
Octabio
Problem  6
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If \(f(x)=\dfrac{x^2-3}{x+5}\), find \(\displaystyle\lim_{x\to -2}f(x)\).

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Mr. McKeague cc
Mr. McKeague
Molly S. cc
Molly S.
Octabio cc
Octabio
Octabio cc espanol spanish
Octabio
Problem  7
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Find \(\displaystyle\lim_{x\to -1}\left(4x^2-2x+\dfrac{5}{x}\right)\)

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Mr. McKeague cc
Mr. McKeague
Molly S. cc
Molly S.
Octabio cc
Octabio
Octabio cc espanol spanish
Octabio
Problem  8
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Find

  1. \(\displaystyle\lim_{x\to 1}5x^2\)

  2. \(\displaystyle\lim_{x\to 1}\left[(2x-3)(x^2+1)\right]\)

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Mr. McKeague cc
Mr. McKeague
Molly S. cc
Molly S.
Octabio cc
Octabio
Octabio cc espanol spanish
Octabio
Problem  9
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Find \(\displaystyle\lim_{x\to -2}\dfrac{3x-2}{x+1}\)

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Mr. Damarest cc
Mr. Damarest
Octabio cc
Octabio
Molly S. cc
Molly S.
Octabio cc espanol spanish
Octabio
Problem  10
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Find \(\displaystyle\lim_{x\to \frac{1}{5}}(5x-3)^5\)

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Mr. McKeague cc
Mr. McKeague
Octabio cc
Octabio
Molly S. cc
Molly S.
Octabio cc espanol spanish
Octabio
Problem  11
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Find \(\displaystyle\lim_{x\to 8} \sqrt[3]{x^2}\)

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Mr. McKeague cc
Mr. McKeague
Octabio cc
Octabio
Molly S. cc
Molly S.
Octabio cc espanol spanish
Octabio
Problem  12
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Find each limit for the function below: \[f(x)=\frac{1}{(x-3)^2}\]

  1. \(\displaystyle\lim_{x\to 2}f(x)\)

  2. \(\displaystyle\lim_{x\to 3}f(x)\)

  3. \(\displaystyle\lim_{x\to 5}f(x)\)

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Mr. Damarest cc
Mr. Damarest
Octabio cc
Octabio
Molly S. cc
Molly S.
Problem  13
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Find \(\displaystyle\lim_{x\to -1}\dfrac{x^2-x-3}{x+1}\)

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Mr. Damarest cc
Mr. Damarest
Octabio cc
Octabio
Molly S. cc
Molly S.
Octabio cc espanol spanish
Octabio
Problem  14
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Find \(\displaystyle\lim_{x\to 2}\dfrac{x^2-x-2}{x-2}\)

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Mr. Damarest cc
Mr. Damarest
Octabio cc
Octabio
Molly S. cc
Molly S.
Octabio cc espanol spanish
Octabio
Problem  15
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Find \(\displaystyle\lim_{x\to 4}\dfrac{\sqrt{x}-2}{x-4}\)

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Mr. Damarest cc
Mr. Damarest
Octabio cc
Octabio
Molly S. cc
Molly S.
Octabio cc espanol spanish
Octabio
Problem  16
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Divide the numerator and denominator of the rational expression by \(x^2\) to help evaluate this limit. \[\lim_{x\to\infty}\frac{5x^2+7x-2}{8x^2-3x+1}\]

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Mr. Damarest cc
Mr. Damarest
Octabio cc
Octabio
Molly S. cc
Molly S.
Octabio cc espanol spanish
Octabio
Problem  17
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Find \(\displaystyle\lim_{x\to\infty}\dfrac{8x^3+7x-2}{5x^2+3x+11}\)

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Mr. Damarest cc
Mr. Damarest
Octabio cc
Octabio
Molly S. cc
Molly S.
Octabio cc espanol spanish
Octabio
Problem  18
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Find \(\displaystyle\lim_{x\to\infty}\dfrac{7x-3}{2x^2+5}\)

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Mr. Damarest cc
Mr. Damarest
Octabio cc
Octabio
Molly S. cc
Molly S.
Octabio cc espanol spanish
Octabio
Problem  19

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.

  1. \(\displaystyle\lim_{x\to\infty}\dfrac{3x}{x^2+1}\)

  2. \(\displaystyle\lim_{x\to\infty}\dfrac{2x+1}{x-3}\)

  3. \(\displaystyle\lim_{x\to\infty}\dfrac{x^2}{2x-4}\)

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Mr. Damarest cc
Mr. Damarest
Octabio cc
Octabio
Molly S. cc
Molly S.
Octabio cc espanol spanish
Octabio

5
Functions and Continuity

Problem  1
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Discuss the continuity of the function at \(x=2\).

  1. \(f(x)=\dfrac{x^2-x-2}{x-2}\)

  2. \(g(x)=\begin{cases} \dfrac{x^2-x-2}{x-2} & \text{if } x\neq 2\\ 1 & \text{if } x=2 \end{cases}\)

  3. \(h(x)=\begin{cases} \dfrac{x^2-x-2}{x-2} & \text{if } x\neq 2\\ 3 & \text{if } x=2 \end{cases}\)

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Octabio cc
Octabio
Molly S. cc
Molly S.
Octabio cc espanol spanish
Octabio
Problem  2
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Discuss the continuity of the function at \(x=4\).

  1. \(f(x)=\begin{cases} \dfrac{1}{2}x-3 & \text{if } x\leq 4\\ -x+3 & \text{if } x>4 \end{cases}\)

  2. \(g(x)=\begin{cases} -\dfrac{1}{2}x+4 & \text{if } x\leq 4\\ 3x-9 & \text{if } x>4 \end{cases}\)

  3. \(h(x)=\begin{cases} 2x-1 & \text{if } x< 4\\ \dfrac{1}{4}x+5 & \text{if } x>4 \end{cases}\)

Choose instructor to watch:
Octabio cc
Octabio
Molly S. cc
Molly S.
Octabio cc espanol spanish
Octabio
Problem  3
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Determine the continuity of the function at each of the given points. \(P(x)=\begin{cases} 20x-100 & 0\leq x \leq 150\\ 20x-250 & 150 \leq x \leq 350 \end{cases}\)

  1. \(x=145\)

  2. \(x=150\)

  3. \(x=155\)

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Mr. Damarest cc
Mr. Damarest
Octabio cc
Octabio
Lauren cc
Lauren
Octabio cc espanol spanish
Octabio
Problem  4

Make a statement about the continuity of the function \(A(s)\) at \(s=c\) and give a possible interpretation to the point \(s=c\).

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Octabio cc
Octabio
Molly S. cc
Molly S.
Octabio cc espanol spanish
Octabio
Problem  5
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Discuss the continuity of each function over the real numbers. Use interval notation to show where each is continuous.

  1. \(f(x)=x^2+3x-1\)

  2. \(g(x)=\dfrac{x+2}{x^2-7x+12}\)

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Mr. Damarest cc
Mr. Damarest
Octabio cc
Octabio
Molly S. cc
Molly S.
Octabio cc espanol spanish
Octabio

6
Average and Instantaneous Rates of Change

Problem  1
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For the function \(f(x)=3\sqrt{2x}+1\), find the average rate of change with respect to \(x\) on the interval \([2, 8]\).

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Mr. Damarest cc
Mr. Damarest
Octabio cc
Octabio
Molly S. cc
Molly S.
Octabio cc espanol spanish
Octabio
Problem  2
practice icon

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 \(\$6\).

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Octabio cc
Octabio
Molly S. cc
Molly S.
Octabio cc espanol spanish
Octabio
Problem  3
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For the function \(f(x)=5x+8\), find the difference quotient \[\frac{f(x+h)-f(x)}{h}\]

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Mr. Damarest cc
Mr. Damarest
Octabio cc
Octabio
Molly S. cc
Molly S.
Octabio cc espanol spanish
Octabio
Problem  4
practice icon

For the function \(f(x)=-x^2+13x-22\), find and simplify the difference quotient \[\frac{f(x+h)-f(x)}{h}\]

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Octabio cc
Octabio
Molly S. cc
Molly S.
Octabio cc espanol spanish
Octabio
Problem  5
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Find the instantaneous rate of change of the function \[f(x)=3x^2+7x-2\]

  1. \(x=1\)

  2. \(x=2\)

  3. \(x=-4\)

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Mr. Damarest cc
Mr. Damarest
Octabio cc
Octabio
Molly S. cc
Molly S.
Octabio cc espanol spanish
Octabio
Problem  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 \[C(x)=-5x^2+200x \quad 0\leq x \leq 20\]

  1. If the magazine is currently running \(12\) half-page articles, what is the expected increase in cost if it were to run \(13\)?

  2. If the magazine is currently running \(15\) half-page articles, what is the expected increase in cost if it were to run \(16\)?

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Octabio cc
Octabio
Molly S. cc
Molly S.
Octabio cc espanol spanish
Octabio
Problem  7
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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.

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Mr. Damarest cc
Mr. Damarest
Octabio cc
Octabio
Molly S. cc
Molly S.
Octabio cc espanol spanish
Octabio