4.3.1The Case of the Heavy-Footed Teacher

Fast Times: Parts One & Two

4-106.

You will be working on the following problem over the next few days. Evidence will be presented by the prosecution that your team will need to verify or refute. Be careful with your analysis; the judge is less than patient with inaccuracies in arguments!

FAST TIMES
The Case of the Heavy-Footed Teacher

Your calculus teacher is in major trouble with the police department from a remote county. The police have issued tickets with very large fines for traveling at excessive speeds during a return trip from the mountains. In addition, if the tickets are upheld, your teacher’s license will be suspended and insurance canceled! (Driving is one of the ways your teacher releases stress from teaching and without this release, upcoming tests are sure to become unbearably difficult.) The prosecution has gathered extensive data to support their case. Your task is to defend your teacher against these accusations.

FAST TIMES Cast of Characters:

Judge Ironhand: Traffic court judge for $23$ years. Has a reputation of being hard on sloppy evidence. Holds the record for fines levied on contempt charges.

Jerry Verigreen: The assistant District Attorney who is prosecuting the case. Trying to impress upon his boss that he can be tough on crime. This is his first case.

Officer Tightman: The officer that issued the two tickets. Known to reach the monthly ticket quota during the first week of every month.

Patrol Officer Rongway: Highway Patrol pilot who is in charge of identifying speeders between the towns of Accelerton and Geprime.

Inspector Knoclew: In charge of analyzing traffic data for the police department.

The Defendant: Your teacher. Known for high speeds and hard tests. In desperate need of your help

Council for the Defense: That’s you!

4-107.

FAST TIMES, Part One—The Prosecution Presents Their Case

Prosecution: “Judge Ironhand, through extensive data collection we have solid evidence that the defendant was using excessive speeds while traveling through our tranquil law-abiding community. Through the outstanding work of our police department, we will be able to bring justice to the defendant who clearly has no respect for our laws.

As my first witness I would like to call the officer who issued the traffic citations.

Officer Tightman, what evidence do you have that first suggested the defendant was using excessive speed?”

Officer Tightman: “The defendant claims to have departed Accelerton at 3:20 p.m. A receipt from the service station in Geprime shows that the defendant arrived at 5:38 p.m.

Even with our liberal laws allowing speeds of $65$ mph, this is clearly too short of time to travel safely the distance shown at right in miles.”

Judge Ironhand: “Council for the defense, how do you respond?”

Your Task: Respond to the Judge. Is Officer Tightman correct in his claim? Justify your answer. Does this prove your teacher’s guilt or innocence?

Curve representing a road, labeled Prosecution Exhibit A, with the following labels & distances between points on the curve, in order starting at Accelerton, 12 to Boundless, 15 to Calctown, 21 to D'exdete, 26 to Efofex, 10 to Functionville, & 16 to Geprime.

4-108.

FAST TIMES, Part Two - Compelling Evidence

Prosecution: “Although I will acknowledge the mathematical prowess of my esteemed colleague, we have further evidence to suggest the defendant was traveling at an excessive speed on multiple occasions.

Using the information from our roadside cameras, we have found the following function for the distance (in miles) traveled by the defendant between Accelerton and D’exdete. I would like to now present Exhibit B, shown below, to demonstrate the first infraction.”

First quadrant, labeled Prosecution exhibit B, x axis labeled time, hours, y axis labeled distance, miles, continuous curve starting at the origin, passing through highlighted point labeled, Boundless, changing from concave up to concave down at highlighted point labeled, Calctown, ending at highlighted point labeled, D'exdete.

$s(t)=-93.75t^3+93.75t^2+45t$

Your Task: Confirm or refute the prosecution’s claim by determining the maximum velocity in the interval.

4-109.

Examine the following integrals. Consider the multiple tools available for evaluating integrals and use the best strategy for each part. Evaluate the definite integrals and state the strategies that you used. For the indefinite integrals, find the antiderivative function, if you can. Homework Help ✎

$\int _ { - \pi } ^ { \pi } \operatorname { cos } ( x ) d x$
$\int ( 5 \sqrt { y } - \operatorname { sin } ( y ) ) d y$
$\int _ { 1 } ^ { 2.7183 } \frac { 1 } { x } d x$
$\int ( \operatorname { sin } ^ { 2 } ( x ) + \operatorname { cos } ^ { 2 } ( x ) ) d x$
$\int \frac { e ^ { x } } { x } d x$

4-110.

The three graphs below each tell a story. Examine each graph carefully and then complete each part below. Pay close attention to the labeling of the $x$ - and $y$ -axes. Homework Help ✎

The graph of $y = s\left(t\right)$ involves a bicycle. Use a complete sentence to describe the physical meaning of $s^\prime\left(6\right)$ . Include units of measure.

First quadrant, x axis labeled time, hours, y axis labeled distance from home, miles, downward parabola vertex at the approximate point (3.5, comma 6.5), passing through the origin, & passing through the x axis after x = 7.

The graph of $y = r\left(t\right)$ involves an employee of a calculator company. Use a complete sentence to $\int _ { 0 } ^ { 5 } r ( t ) d t$ . Include units of measure.

First quadrant, x axis labeled time, hours, y axis labeled rate, sales per hour, increasing curve starting at the origin, concave down, passing through the approximate points (2, comma 3.5), & (6, comma 5).

The graph of $y = w\left(t\right)$ involves a babysitter. Use a complete sentence to describe the physical meaning of $\int _ { 4 } ^ { 10 } w ( i ) d i$ . Include units of measure.

First quadrant, x axis labeled time of day, P M, y axis labeled Hourly wage, $ per hour, increasing line starting at the origin, passing through the approximate points (2, comma 3), & (4, comma 7).

4-111.

The graph at right shows the velocity (in miles per hour) of a car during a road trip. At time $t = 0$ , the car was on the Golden Gate Bridge heading north. Homework Help ✎

Write a function for $v\left(t\right)$ .
How far north has the car traveled at $t = 3$ hours? At $t = 4$ hours?
Explain what happened to the car for $3 ≤ t ≤ 5$ hours.
Set up an integral to represent the displacement from $0 ≤ t ≤ 6$ .
Set up an integral to represent the total distance from $0 ≤ t ≤ 6$ .

Continuous linear piecewise, starting at (0, comma 20), turning at approximate point (4.5, comma negative 10), ending at (6, comma 20).

4-112.

During this course, you have studied the connection between distance and velocity. Homework Help ✎

What is the relationship among position, velocity, and the Fundamental Theorem of Calculus?
Considering your answer from part (a), write a function to represent the car’s position, $s\left(t\right)$ , from problem 4-111.

4-113.

Without graphing, analytically determine where the function $f\left(x\right)=$ $x^3-7x^2+15x-2$ is increasing. Check your answer with a graph. Homework Help ✎

4-114.

Differentiate the following functions. Determine if each function is differentiable for all real values of $x$ . Homework Help ✎

$y=\sin\left(x-3\right)$
$f ( x ) = \left\{ \begin{array} { c c } { 4 - x ^ { 2 } } & { \text { for } x < 1 } \\ { ( x - 1 ) ^ { 3 } + 3 } & { \text { for } x \geq 1 } \end{array} \right.$

4-115.

Write a Riemann sum to approximate $\int _ { 0 } ^ { 2 } 6 ^ { x } d x$ using $n$ rectangles. Homework Help ✎

4-116.

A function has a derivative of $f^\prime\left(x\right) = 6x^{2} + 12x – 7$ . Homework Help ✎

If $f\left(0\right) = 0$ , what was the original function?
If $f\left(–2\right) = 25$ , what was the original function?
Describe how you found the constant of integration in parts (a) and (b).

Calculus Third Edition