A baseball is shot from a cannon so that its height $h$ (in feet) above the ground at time $t$ (in seconds) is given by $h(t) = -16t^2 + 128t +4$.

Determine the speed of the ball when it hits the ground. Use 2-decimal place accuracy.

Determine the speed of the ball when it reaches its highest point.

The table below shows the populations over time of two species that occupy the same area: rabbits and wolves. Counts are taken at the beginning of each year.

Calculate the average increase in the wolf population over the first 4 years.

Calculate the average decrease in the rabbit population during the 3 years that start with the beginning of the 19th year.

Approximately what was the rate of change (wolves per year) of the wolf population at the beginning of year 20?

Indicate the longest time period over which both species simultaneously have a positive growth rate.

Indicate the longest period over which one species has a positive growth rate while the other has a negative growth rate.

When is the first time that the rabbits have zero growth rate? What is happening with the other species and why might this be happening?

The following table is for a function $y = {b^x}$.

Sketch a reasonable graph for the function.

Estimate the value of $b$. Show your work or explain how you arrived at your solution.

The graph of a function $f(x) = a \cdot {b^x}$ is shown below. As accurately as possible estimate the slope of the curve at $x = 3$ and at $x = 6$.

Let $g(x) = 2 \cdot ({0.5^x})$. Calculate $\frac{{g(2 + h) - g(2)}}{h}$, where $h = 0.001$. What does this calculation tell you about the function $g$? Can you come up with an even better estimate in this situation?

A bacterium population has been decreasing exponentially. Twenty hours ago the population was 1 million. Ten hours later the population had dropped to half a million. Suppose the bacterium population continues to drop off at the same rate.

Write an equation that models the bacterium population.

What is the percentage hourly decay rate of the population? Use 2-decimal place accuracy.

Exactly ten hours ago, what was the population decay rate (in bacteria per hour)? Use 3-decimal place accuracy.

Five hours ago would the population decay rate (in bacteria per hour) have been smaller (in magnitude) than your answer to Part c, larger than it, or effectively the same? Answer this question without doing any calculations. Explain how you arrived at your answer.

In how many hours will the population drop to one bacterium? Use 3-decimal place accuracy.

On a mathematics quiz the following problem was presented:

Calculate $\frac{{{2^{2.2001}} - {2^{2.2}}}}{{.0001}}$ using 3-decimal place accuracy and explain the meaning of this number.

In response to this question, Jessica wrote, “3.185; It’s the approximate slope of the curve $y = {2^x}$ at $y = 2.2$. David wrote, “3.185; It’s the approximate slope of the curve $y = {2^x}$ at $x = 2.2001$.” Meredith wrote, “3.185; It’s the approximate slope of the curve $y = {2^x}$ at $x = 2.20005$.” How would you grade their responses?

Tony wishes to estimate the slope of the tangent line to the curve $y = 12{(1.3)^x}$ at $x = - 1.25$. He wants to write a fraction that will complete this task and have $0.0001$ in the denominator. Write a fraction that will work for him. Now, write another fraction that will work. Write one more.

Peter says that $\frac{0}{0}$ is equal to 17! Here’s his argument: another way of writing $5 \times 6 = 30$ is to say that $\frac{{30}}{6} = 5$. Therefore we can say that another way of writing $17 \times 0 = 0$ is that $\frac{0}{0} = 17$. Do you agree with Peter, or not? Explain!

Robert and Allan each throw a baseball at the same time. The height (in feet) of Robert’s ball above the ground after t seconds is given by $R(t) = - 16{t^2} + 50t + 5$. The formula for Allan’s ball is $A(t) = - 16{t^2} + 60t + 4$.

What is the average (vertical) speed of Robert’s ball during the first second?

Estimate the speed of Allan’s ball at 1 second.

Is there a time when both balls have the same speed?

Let $f(x) = \sqrt {25 - {x^2}} $.

Explain why the graph of $f$ is a semicircle. What is the radius of the semicircle?

First find the slope of $f$ at the point $(4,3)$. Then find an equation for the line tangent to the graph going through this point.

Find an equation for a line perpendicular to the tangent line, also passing through the point (4,3). Before you do, would you predict that this line crosses the $x$-axis to left of (0,0), to the right of (0,0), or that it goes through (0,0)?

Let $f(x) = \sqrt x $ and $g(x) = 4 - 4({0.5^x})$.

Kiran was calculating the slope of f at x and thought he could express his answer in two equivalent ways: $\frac{{\sqrt {x + .001} - \sqrt x }}{{.001}}$ and $\frac{{\sqrt x + \sqrt {.001} - \sqrt x }}{{.001}}$. Give two explanations for why these expressions are not equivalent: one involving algebra and the other involving slope.

Kiran then calculated the slope of $g$ at $x$ and wrote $\frac{4-4(0.5^{x+.001}) - (4 - 4(0.5^x))}{.001}$. Show how this expression can be rewritten as $\frac{4(0.5^{x+.001})(1-4(0.5^x))}{.001}$.

As $x$ increases, what happens to the slope of $f$? of $g$?

Kiran argues that, given what’s happening in Part c, both functions should be doing essentially the same thing as $x$ goes toward infinity. What do you think?

Speed and velocity are not synonyms. “What?!” you say. Velocity is speed with direction. For example, if you go forward with a velocity of 5 yd/sec, stop, and then go backwards with a speed of 3 yd/sec, your velocity while going backwards was -3 yd/sec, where the negative merely indicates that you are going in the direction opposite to the positive direction.

Let $h(t) = - 16{t^2} + 60t + 4$ describe the height, in feet, of a ball above the ground at t seconds. Estimate the velocity of the ball at $t = 1$ second and at $t = 3$ seconds.

Acceleration is defined to be the change in velocity over the change in time. For the ball in Problem 25, estimate its acceleration at $t = 1$ second.

Let $f(x) = {2^x}$.

Complete the table below using 3-decimal place accuracy.

On a coordinate axis system plot the ordered pairs from the completed table in Part a. Draw a smooth curve through these points.

Estimate the slope of the curve you drew in Part b at $x = 0.5$. What does this number tell you about the original curve $f$ at $x = 0.5$?

Don’t use a calculator for this problem.

Simplify $\frac{{\sin 135^\circ + \cos 135^\circ }}{{\tan 135^\circ }}$.

Subtract $\frac{{x + 3}}{6} - \frac{{x - 5}}{8}$.

Simplify $\frac{{{{({x^2}y)}^{ - 2}}{x^3}z}}{{{x^4}{{(y{z^2})}^3}}}$.

Solve for $x$ if $|x - 8| \le 16$.

Solve for $x$: $x = \frac{1}{{x + 1}} + \frac{3}{{2x + 2}}$