1.4.2Where is average velocity on a position graph?

Average Velocity on a Position Graph

1-147.

WEARY VERONICA, Part One

An exhausted Veronica produced the distance graph shown below when walking the Slope Walk. Afterwards, her team bombarded her with questions, to which she tiredly replied, “It’s shown here in the graph.”

Examine the graph carefully to determine the answers to her teammate’s questions.

Your teacher will provide you with a model.

Answer the following questions from Veronica’s teammates. Justify your answers. The motion detector was set to measure distance in meters and time in seconds.
1. How much time did your walk take?
2. How far did you travel overall?
3. How far from your starting place did you end up?
4. Did you ever stop? If so, when?
5. Did you only walk in one direction?
Explain why the answers to questions (ii) and (iii) are not the same.

1-148.

WEARY VERONICA, Part Two

While looking at the graph, Veronica’s teammate point out that she could have saved her energy and walked from her starting place directly to her ending place instead.

Your teacher will provide you with a model.

On the Lesson 1.4.2 Resource Page, locate Veronica’s graph. Using color, draw what the motion detector would have shown if she had walked directly from her starting position to her ending position at a constant rate while taking the same amount of time.
What would Veronica's velocity have been had she taken this direct route? This rate is referred to as her average velocity.
Explain the relationship between the graph of Veronica’s direct route and her average velocity so that it makes sense to an Algebra I student.

Math Notes

Initial Position, Final Position, Displacement, and Total Distance

Suppose that an object is in motion, and its position is a function of time with domain $[t_{0}, t_{n}]$ .

The initial position is the object’s location at time $t_{0}$ .

The object’s final position is its location at time $t_{n}$ .

The displacement of the object is the distance between the final position and the initial position.

The total distance an object travels is the total of all its motions both forward and back.

For example, suppose a bug begins at $x = 3$ on the number line, crawls forward $2$ units and then backward $4$ units. Its initial position is $3$ , its final position is $1$ , its displacement is $–2$ units (because it ended up at a lower number than where it started), and its total distance traveled is $6$ units.

1-149.

With your team (or with the whole class), create a new Slope Walk graph that has an average velocity of $−\text{m/sec}$ . The graph may not be linear. Sketch a graph of the data on the blank set of axes on the resource page. Use a contrasting color to draw a direct path (secant line) from the starting position to the ending position.

With this new graph, answer the questions asked by Veronica’s teammates in problem
Create another Slope Walk and sketch another non-linear graph that has an average velocity of $0$ feet per second. Once again, answer the questions in problem

1-150.

Poor Agnalia! Her motion detector produced the graph below. “My calculator is broken,” she cries jumping up and down, “This graph is physically impossible!” “No it’s not,” says Amanda, “It’s just a piecewise-defined function.” Who is correct, Agnalia or Amanda?

TI-83 graph, x axis labeled, t of s, y axis labeled, d of m, with 2 horizontal segments, left, half way up, between approximate x values, 0 & 4, right, almost to top, between approximate x values, 4 & 11.

1-151.

The table below represents the position of a bug crawling along the $x$ -axis of centimeter graph paper. Time is given in seconds.

Time (s)	$0$	$2$	$5$	$6$	$8$	$10$	$11$	$15$	$16$
Position	$(0, 0)$	$(2, 0)$	$(4, 0)$	$(6, 0)$	$(9, 0)$	$(8, 0)$	$(4, 0)$	$(4, 0)$	$(8, 0)$

Describe the bug’s motion. Does it always crawl in the same direction? Is its velocity constant?
Compute the bug’s average velocity over the following intervals. Use correct units.
1. $0 < t < 16$
2. $0 < t < 2$
3. $8 < t < 11$
4. $5 < t < 15$
For $0 < t < 15$ , will there be a time that the bug is at $(5, 0)$ ? Explain.
For $0 < t < 16$ , will there be a time at which the bug’s average velocity is the same as its actual velocity? Explain.

1-152.

The shaded region at right represents a quarter circle combined with a right triangle “flag.” Homework Help ✎

Imagine rotating this flag about its “pole” and describe the resulting three-dimensional figure. Draw a picture of this figure on your paper. To help you visualize this, use the 1-152 eTool.
Calculate the volume of the rotated flag.

Adjacent angles forming a linear pair, with line segment labeled pole, shaded left angle, labeled 30 degrees, top ray of angle labeled 10.

1-153.

For $f(x) = 3x \cos(x)$ , approximate the area under the curve for $0≤x≤\frac { \pi } { 2 }$ using two different methods. If the actual area is approximately $1.712$ square units, which of your methods was most accurate? Analyze why that particular method was more accurate. Homework Help ✎

1-154.

Compute without a calculator If $\sin(x) =\frac { 1 } { 2 }$ and $0 ≤ x ≤ \frac { \pi } { 2 }$ , then without a calculator evaluate: Homework Help ✎

$\cos\left(x\right)$

$\tan(x)$

$\sec(x)$

$\csc(x)$

1-155.

After Theo used the motion detector, he used his distance-time graph to determine the following properties of his motion. However, he has lost a copy of his graph. Help him re-create a possible graph of his motion. Homework Help ✎
Details:

His average velocity was $0.5$ feet per second.
He turned around twice.
He started while standing $3$ feet from the motion detector and began to walk away from it at $t = 0$ .
He walked a total of $9$ feet during the $10$ -second interval.

1-156.

Write the equation of the line parallel to $9y - 4x = 12$ through the point $(6, –7)$ . Write the equation in graphing form as shown in the Math Notes box preceding problem 1-8. Homework Help ✎

1-157.

DO YOU KNOW THE WAY TO SAN JOSE?

Salima and Karim were driving from Sacramento to San Jose. Salima kept track of their rate as Karim drove. At right is a graph of their rate during the trip. Homework Help ✎

What is the driving distance between Sacramento and San Jose?
What was Karim’s average speed?

1-158.

Without using a calculator, determine the exact value of each of the following trig expressions. Compute without a calculator Homework Help ✎

$\sin(\frac { 5 \pi } { 6 })$

$\cos(\frac { - 3 \pi } { 4 })$

$\tan(\frac { \pi } { 3 })$

$\sec(\frac { 5 \pi } { 3 })$

1-159.

The height of a right circular cone is twice the radius. If the height of the cone is $h$ , write an expression for the volume of the cone using only $h$ . Homework Help ✎

1-160.

The function $y = x^3 + 1$ is graphed at right, along with four left endpoint rectangles which approximate the area under the curve from $x = -2$ to $x = 2$ . Homework Help ✎

Why does it look like there are only three rectangles?
Recall that area under the $x$ -axis is negative, while area above the $x$ -axis is positive. Approximate the area under the curve for $−2 ≤ x ≤ 2$ using these four rectangles.

Increasing cubic curve, y = x cubed + 1, & 3 shaded vertical rectangular bars of width 1, first bar, top left vertex at (negative 2, comma. 0), & bottom left vertex at (negative 2, comma negative 7), second bar, bottom left vertex at the origin, top left vertex at (0, comma 1), third bar, bottom let vertex at (1, comma 0), top left vertex at (1, comma 1).

1-161.

The parabola $y = -(x - 3)^2 + 4$ is graphed below. Use four trapezoids of equal width to approximate the area under the parabola for $1 ≤ x ≤ 5$ . Is this area an overestimate or an underestimate of the true area under the parabola? Explore this using the Estimating Area Under a Curve eTool. Homework Help ✎

Downward parabola, vertex at the point (3, comma 4).

Calculus Third Edition

Initial Position, Final Position, Displacement, and Total Distance