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.

Make a table and graph for \(y=7.5x\) for \(0\leq x \leq 40\).

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.

Graph \(x=y^2\)

Graph \(y=\displaystyle\frac{1}{x}\)

Graph \(y=\sqrt{x}\) and \(y=\sqrt[3]{x}\)

If \(f(x)=7.5x\), find \(f(0)\), \(f(10)\), \(f(20)\).

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 \(a+3\).

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.

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. \[V(r)=\frac{4}{3}\pi r^3 \qquad S(r)=4\pi r^2\]

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

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 \(\left(\dfrac{g}{f}\right)(5)\).

Use the equation \(x=1300-100p\) to determine the price that should be charged for a weekly revenue of \(\$4000\).

If \(f(x)=x+5\) and \(g(x)=x^2-2x\), find \((f \circ g)(x)\) and \((g \circ f)(x)\)

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

Find the slope through \((-2,1)\) and \((5,-4)\).

Find the slope of the line containing \((3, -1)\) and \((3,\, 4)\)

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.

If \(f(x)=3x-5\), find \(\dfrac{f(x_2)-f(x_1)}{x_2-x_1}\)

If \(f(x)=x^2-4\), find and simplify \(\dfrac{f(x_2)-f(x_1)}{x_2-x_1}\)

Find the equation and graph the line with slope \(-\dfrac{4}{3}\) and \(y\)-intercept \(5\).

Give the slope and \(y\)-intercept for the line \(2x-3y=5\)

Find the equation of the line with slope \(-2\) that contains the point \((-4,\,3)\).

Find the equation of the line that passes through \((-3,\, 3)\) and \((3, -1)\).

Graph the function defined by \(f(x)=\begin{cases} x+1 & \text{if }\, x\leq 1\\ 3 & \text{if }\, x> 1 \end{cases}\)

An Introduction to Limits

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

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

If \(f(x)=4\), find \(\displaystyle\lim_{x\to 7}f(x)\).

If \(f(x)=3x^2-5x+1\), find \(\displaystyle\lim_{x\to 2} f(x)\).

If \(f(x)=\dfrac{x^2-3}{x+5}\), find \(\displaystyle\lim_{x\to -2}f(x)\).

Find \(\displaystyle\lim_{x\to -1}\left(4x^2-2x+\dfrac{5}{x}\right)\)

Find

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

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

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

Find \(\displaystyle\lim_{x\to \frac{1}{5}}(5x-3)^5\)

Find \(\displaystyle\lim_{x\to 8} \sqrt[3]{x^2}\)

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

Find \(\displaystyle\lim_{x\to -1}\dfrac{x^2-x-3}{x+1}\)

Find \(\displaystyle\lim_{x\to 2}\dfrac{x^2-x-2}{x-2}\)

Find \(\displaystyle\lim_{x\to 4}\dfrac{\sqrt{x}-2}{x-4}\)

Divide the numerator and denominator of the rational expression by \(x^2\) to help evaluate this limit. \[\lim_{x\to\infty}\frac{5x^2+7x-2}{8x^2-3x+1}\]

Find \(\displaystyle\lim_{x\to\infty}\dfrac{8x^3+7x-2}{5x^2+3x+11}\)

Find \(\displaystyle\lim_{x\to\infty}\dfrac{7x-3}{2x^2+5}\)

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

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

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. \(x=155\)

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

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

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 \(\$6\).

For the function \(f(x)=5x+8\), find the difference quotient \[\frac{f(x+h)-f(x)}{h}\]

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

Find the instantaneous rate of change of the function \[f(x)=3x^2+7x-2\]

1. \(x=1\)

2. \(x=2\)

3. \(x=-4\)

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.