1.4.3How do I find average velocity on a velocity graph?

Average Velocity on a Velocity Graph

1-162.

DIJIN'S TRAVELS

Dijin is late to his calculus class again! However, he is in no hurry. Below is a graph of the velocity (in meters per minute) Dijin travels during the first $4$ minutes of his journey.

First quadrant linear piecewise, starting at (0, comma 20), going right until (1, comma 20), turning down until (2, comma 0), turning right until (3, comma 0), turning up & ending at, (4, comma 10).

Estimate Dijin’s velocity at $t=3.5$ minutes.
What are the appropriate units for the area under the curve? Why?
What does the area under a velocity curve represent?
How far does Dijin travel during the first $4$ minutes of his journey?
What is Dijin’s average velocity during the first $4$ minutes?
Sketch Dijin’s graph on your paper. On the same set of axes, sketch the line $y=\text {avg}$ velocity (from part (e)) and shade the area under the average velocity curve. What shape is this?
How many times did Dijin travel at his average velocity during the first $4$ minutes of his journey? Justify your answer.
If Dijin’s initial position was $100$ meters away from his calculus classroom and he continues to travel at the same rate he was traveling when $t = 4$ , sketch a graph of his velocity. How many minutes will it take him to get to class?

1-163.

Due to eye-fatigue, a person’s reading rate decreases after $2$ hours. The rate (in words per hour) of a certain reader is represented by the piecewise-defined function below.

$\text{reading rate} = \left\{ \begin{array} { l l } { 6000 } & { \text { for } \quad 0 \leq t \leq 2 \ \text{hrs} } \\ { 7000 - 500 t } & { \text { for } } \qquad \ \ \ { t > 2 \ \text{hrs} } \end{array} \right.$

Sketch a graph of this reading rate model.
According to this model, how many words can this person read in $5$ hours? How did you obtain your answer?
What is the average number of words per hour this person reads over a five-hour interval?

1-164.

Summarize your method for determining how much something has accumulated (or changed) given its rate of change. For example, given a rate such as how fast a person types, how can you calculate the number of words typed? Explain why your method works.

1-165.

TRIP TO BAJA

Ms. D is on an awesome cruise aboard the pleasure boat Suluclac. One morning, at 8:00 a.m., she was informed that the boat was traveling at a rate of $39$ miles per hour.

She created the table below and recorded the speed of the ship at $15$ -minute intervals.

Time (hrs)	8:00	8:15	8:30	8:45	9:00	9:15	9:30	9:45
Speed (mph)	$39$	$34$	$26$	$23$	$15$	$18$	$26$	$27$

Draw a speed-time graph and estimate the distance traveled between 8 a.m. and 10 a.m. Make your estimate as accurate as possible.
If the Asia-bound pleasure boat left from Catalina Island, which is $22$ miles from Los Angeles Harbor, approximately how far from LA was the ship at 10 a.m.?

1-166.

Theo left his Slope Walk graph at home. Fortunately, while he had the graph, he determined the properties shown below. Help him re-create a possible distance-time graph of his motion.

Details:

Theo walked in one direction the entire time.
His average velocity during the first half of the walk was $5$ feet per second. His average velocity for the second half was only $3$ feet per second.
He started while standing $2$ feet from the motion detector and began to walk away from it at $t=0$ . He stopped when he was $26$ feet from the motion detector.

1-167.

Oil is leaking out of a car at a rate of $y = 0.2^{t}$ liters/hour for $0 ≤ t ≤ T$ . Homework Help ✎

Sketch a graph of this situation. Label the units on the axes.
Shade the area under the curve for $0 ≤ t ≤ 1$ . What does the shaded area represent? What are the units?
Use two left endpoint rectangles to approximate the amount of oil that has accumulated during the first hour.

1-168.

Write the equation of the line that passes through the point $(3, –2)$ with a slope of $7$ . Write your answer in point-slope form. Homework Help ✎

1-169.

Revisit your rates of change from problem 1-143. Decide what measurement will be determined if the areas under graphs of these rates are calculated. Homework Help ✎

1-170.

Use a quadratic function and an absolute value function to write a piecewise-defined function that will produce the graph shown below. Homework Help ✎

Continuous piecewise, left piece, upward parabola, vertex at the point (negative 2, comma negative 3), ending at the point (2, comma 7), right piece, upward V shape, starting at (2, comma 7), vertex at (4,, comma 0), continuing up & right.

1-171.

Write a complete set of approach statements for $y = \frac { ( 2 x + 1 ) ( 2 - x ) } { 2 x + 1 }$ . Homework Help ✎

1-172.

A flag is defined by the region between the $x$ -axis and the function listed below. Sketch the flag and calculate the volume of the solid generated by rotating the bounded region about the $x$ -axis. To help you visualize this, use the 1-172 eTool. Homework Help ✎
$f ( x ) = \left\{ \begin{array} { c c } { 2 } & { \text { for } 0 \leq x \leq 2 } \\ { 4 - x } & { \text { for } 2 < x \leq 4 } \end{array} \right.$

1-173.

If $f(x)=\frac{x^2-2x-3}{x-3}$ and $g(x)=-x+2$ , write an equation for $h(x)=f(g(x))$ . Homework Help ✎

Graph $y=h(x)$ and write its domain in set notation.
Write an equation the end-behavior function of $h$ .

1-174.

A focus of this course will be determining maximum and minimum values of a function on a given interval. However, you already have the skills to do this for certain functions. On your paper, sketch $y = 2 \sin(3x)$ for $\frac { 2 \pi } { 3 }$ . Determine the points at which the function is a maximum and a minimum. Homework Help ✎

1-175.

TEST DRIVE

Sarah wants to buy a new car and is deciding between two models. She has convinced both car dealerships to allow her to test drive each car for $4$ hours as long as she returns with a full tank of gas. In order to test the performance of both vehicles, she kept track of her velocity during her test drive every half hour. The results of her test drives are shown below.

Coronado Speedster

D.O. Sensation

Sarah finds that both cars performed exceptionally well and she will be very happy with either one. She decides to make her final decision based on the gas mileage of each model. Her test drive of the Coronado Speedster used $7.955$ gallons of gas and her test drive of the D.O. Sensation used $8.542$ gallons. Which car should she choose? Homework Help ✎

Calculus Third Edition