9.2.1How can I model motion?

Parametric Equations

9-46.

BATTLING ROBOTS, QUALIFICATION ROUND

Your team is entering a Battling Robot competition being held later this month. Only the best robots are allowed into the competition, so your robot must first qualify.

In this qualification round, your team scores points for how closely your robot follows a particular path around the gym floor. Since the judges also care about how well you can control the speed of your robot, points are awarded for being at certain locations along the path at the times given in the table below. The point $\left(0, 0\right)$ represents the corner of the gym.

Time (sec)	$x$ (ft)	$y$ (ft)
$0$	$40$	$50$
$5$	$30$	$50$
$10$	$40$	$50$
$15$	$50$	$50$
$20$	$60$	$40$
$25$	$70$	$30$
$30$	$80$	$20$
$35$	$80$	$20$
$40$	$80$	$20$
$45$	$75$	$20$
$50$	$65$	$20$
$55$	$45$	$20$
$60$	$15$	$20$

First quadrant, x axis labeled feet, scaled in tens from 0 to 90, y axis labeled feet, scaled in tens from 0 to 50, with highlighted point, labeled t = 0, @ (40, comma 50).

Use a sheet of graph paper with suitable scales to show the path of your robot. An example of how you might start your graph is shown above. Be sure to note the time at each of the points.
Write down at least three observations about how the robot travels during the qualifying round of the contest.

Good work on the qualification round!
You will return to the Battling Robots in Chapter 11.

9-47.

Suppose you flick a nickel off the edge of your desk and, at the same time, another student drops a second nickel from the same height. Do you think the two nickels will hit the floor at the same time? In other words, will the horizontal motion due to flicking the first nickel affect the rate of fall? Discuss briefly in your study team and be prepared to share your conclusions with the class.

9-48.

The Interceptors football team is creating a new pass play. A favorite strategy is to have the quarterback, Artfish L. Turf, throw the ball as far as possible and then have the famous wide receiver Pete Moss catch the ball. Unfortunately, Pete missed practice and only has the coach’s notes to work from.

Coach’s Notes:

Artfish L. Turf will throw the football at $t = 3$ seconds. The ball’s horizontal position $x(t)$ and the height $y(t)$ (measured in feet) is shown below. The ball will be caught $6$ feet above the ground by Pete Moss.

$x(t) = 40t - 120$

$y(t) = -16t^2 + 144t - 282$

Pete is wondering where he will need to run to catch the ball for this play. Help him by completing the table he started below for $3 ≤ t ≤ 7$ .
Time (seconds)
$x(t) = 40t - 120$
$y(t) = -16t^2 + 144t - 282$
$3.0$
$0$
$6$
$3.5$
$20$
$26$
$4.0$
$40$
$38$
$…$
Graph the position of the ball using the data from your table. Let the $x$ -axis represent the horizontal position and the $y$ -axis represent the height of the ball. What type of curve do you get?
Give Pete advice on where he needs to be to catch the ball. When should he be there? How do you know?
Describe another way that you could have found the answers to part (c).
Having carefully studied the coach’s notes, Pete is considering two different ways to run to catch the ball. They are shown in the strategies below. Will either strategy enable Pete to catch the ball?
Strategy A
$x(t) = 3t^2 + 4t$
$y(t) = 6$
Strategy B
$x(t) = t^2 + 12t$
$y(t) = 6$
Apparently Pete has no idea how to be in the right place at the right time! Create a route for him to run so that he will be able to catch the football, thereby helping his team win their next game.

9-49.

Without a calculator, determine the absolute minimum of $y = x^{2/3}$ over the interval $−3 ≤ x ≤ 3$ . Justify your answer. Then check your solution on your graphing calculator. Homework Help ✎

Compute without a calculator

9-50.

Using summation notation, write a Riemann sum to represent the area under $f(x) = 2\cos^2(x) - x$ from $-\frac { \pi } { 2 }≤ x ≤\frac { \pi } { 2 }$ using six left endpoint rectangles. Homework Help ✎

9-51.

No calculator! Determine the area between $x = y^2$ and $x = y + 2$ over the interval $0 ≤ y ≤ 2$ . Homework Help ✎

Compute without a calculator

9-52.

The graph at right shows $y = f^{\prime\prime}(x)$ . Use it to answer the following questions. Homework Help ✎

Over what interval(s) is $f$ concave up?

Estimate $f^{\prime\prime\prime}(5)$ .

At what $x$ -value(s) does $f$ have a point of inflection?

At what $x$ -value(s) does $f^\prime$ have a minimum value?

Continuous curve labeled, f double prime of x, coming from left just below x axis, turning at the following approximate points, (0, comma 0), (1.5, comma negative 0.25), (3, comma 0.5), (5, comma negative 1), continuing up & right, passing through the x axis at 2, 4, & 6.

9-53.

No calculator! Determine all point(s) of intersection of $y = 3\cos(x)$ and $y = 2\cos^2(x) + 1$ over the interval $0 ≤ x ≤ 2π$ . Homework Help ✎

Compute without a calculator

9-54.

What is the difference between velocity and speed? Give an example that demonstrates the difference. Homework Help ✎

9-55.

Examine the integrals below. Consider the multiple tools available for integrating and use the best strategy for each part. Evaluate each integral and briefly describe your method. Homework Help ✎

$\int x ^ { 3 } \operatorname { ln } ( x ) d x$

$\int ( 2 x - 1 ) e ^ { x } d x$

$\int \frac { x ^ { 2 } - 4 x + 4 } { x + 2 } d x$

$\int \frac { x + 1 } { x ( x ^ { 2 } + 1 ) } d x$

9-56.

No calculator! A particle’s position as function of time is defined as $x(t) = 2t$ and $y(t) = \sqrt { t + 3 }+ 1$ . Create a table for $-3 ≤ t ≤ 3$ and carefully graph the situation. Homework Help ✎

Compute without a calculator

9-57.

For each infinite series below, if the series converges, calculate the sum. Homework Help ✎

$16 − 4 + 1 −\frac { 1 } { 4 }+...$

$\frac { 1 } { 9 } + \frac { 1 } { 3 } + 1 + 3 + \ldots$

Time (seconds)	$x(t) = 40t - 120$	$y(t) = -16t^2 + 144t - 282$
$3.0$	$0$	$6$
$3.5$	$20$	$26$
$4.0$	$40$	$38$
$…$

Calculus Third Edition

Coach’s Notes: