1.1.1How are speed and distance related?

Applying Rates and Distance

Some of the situations you will encounter in calculus will seem like problems you have solved in previous courses. In this course, you will learn new ways to approach and find solutions to geometric and algebraic questions with dazzling precision, and in more complex situations. For that reason, it is important that your precalculus skills are well honed.

It is equally important that you are open to productive discourse as you proceed through these chapters. Many of the problems in this text have multiple solution paths. By effectively collaborating with your team (exploring strategies, justifying your ideas, and taking mathematical risks), your algebraic and geometric knowledge will deepen as your understanding of calculus develops.

1-1.

FREEWAY FATALITIES

Some people think that the number of freeway accidents can be reduced if cars and trucks were prevented from speeding. In fact, in some countries, every truck is legally required to have a special device (called tachographs) on its wheels which records the truck’s speed at all times.

A graph showing the speed of a truck in miles per hour over a six-hour period is shown below. Estimate the total distance the truck traveled during this time. Then explain how you could get more accurate estimates using the same graph.
The graph below shows the distance traveled by a different truck over an eight-hour period. Make and justify as many statements as you can about the truck’s speed at various times.
Look back at your work for both graphs. The answers you got related to the geometry of each graph.
For instance, in the first graph, confirm that the truck traveled about $27$ miles from 6:00 a.m. to 6:40 a.m. and $53$ miles from 7 a.m. to 8 a.m. What do $27$ and $53$ represent geometrically in the first graph?
In the second graph, confirm that from 8 a.m. to 9 a.m., the speed of the truck is $50$ mph. What does this $50$ represent geometrically about the second graph?

General Team Roles

Resource Manager
Responsible for getting your team’s materials and asking the teacher questions.

Get supplies for your team and make sure that your team cleans up.
Help your team decide when it needs outside help. Assist in creating team questions for the teacher. Call the teacher over for team questions.

Facilitator
Responsible for keeping your team working together.

Get your team started by having someone read the task out loud.
Check that everyone understands what to work on.
Make sure that each team member has shared his or her ideas.
Make sure no one is getting left out or left behind. Make sure each person has time to write his or her answer before your team moves on.

Recorder/Reporter
Responsible for verifying that your team is writing justifications and explanations.

Make sure that each team member can see the work your team is discussing.
Make sure that your team agrees about how to explain and justify your answers, and that everyone understands your team’s answer.
Make sure that each member of your team is able to share ideas.

Task Manager
Responsible for facilitating an effective, participating team.

Help keep your team on task, talking about math, and respecting each other’s right to learn.
Keep track of time if you have been given a time limit and make sure your team is making progress at an appropriate pace.
Make sure that no one talks to others outside your team.

1-2.

Calculate the volume of each of the following solids. Homework Help ✎

1-3.

In this course, a “flag” is defined as a geometric region attached to a line segment (its “pole”). An example is shown at right. To help you visualize this, use the 1-3 HW eTool Homework Help ✎

Imagine rotating the flag about its pole and describe the resulting three-dimensional figure. Draw a picture of this figure on your paper.

Calculate the volume of the rotated flag.

Vertical segment, labeled pole, with rectangle, labeled flag, whose left edge, shares most of the upper part of the segment, left edge of rectangle, labeled 8, bottom edge labeled, 6.

1-4.

Examine the graph the function $f (x) = 5 - x$ at right. Homework Help ✎

Calculate the area of the shaded region.

Notice that the line dips below the $x$ -axis when $x > 5$ . When you are asked to calculate the “area under a curve” this refers to the region between the curve and the $x$ -axis. Any area below the $x$ -axis is considered to be negative. Calculate the area under the curve for $0 ≤ x ≤ 7$ .

Determine the value of k such that the area under the curve for $0 ≤ x ≤ k$ is $0$ . Show how you obtained your answer clearly and completely.

1-5.

Sketch the function $g(x) =\sqrt { 16 - x ^ { 2 } }$ . Homework Help ✎

State the domain and range of $g$ .
Use geometry to calculate the area under the curve for $0 ≤ x ≤ 4$ .

Now calculate the area under the curve for $–4 ≤ x ≤ 4$ .

What is the relationship between the answers to parts (b) and (c)?

1-6.

A car travels $50$ miles per hour for two hours and $40$ miles per hour for one hour. Homework Help ✎

Sketch a graph of velocity vs. time. Label the axes with units.

Fill out the table below for the distance vs. time.

	Time (hours)	$0.5$	$1$	$1.5$	$2$	$2.5$	$3$
	Distance (miles)

Sketch a graph of distance vs. time. Label the axes with units.

1-7.

TRANSLATING FUNCTIONS Homework Help ✎ Compute without a calculator

Graph the function $y=\frac{2}{3}x^2$ . On the same set of axes graph a translation of the function that is shifted $1$ unit to the right and $5$ units down. Write the equation of the translated function.
Does the same strategy work for $y =\frac { 2 } { 3 }x$ ? Write an equation that will shift $y =\frac { 2 } { 3 }x$ , $1$ unit to the right and $5$ units down.
Compare the graphs of $y = -\frac { 1 } { 2 }x$ and $y = -\frac { 1 } { 2 }(x + 2) + 3$ . Describe their similarities and differences.
Explain how you know that the graph of $y = -9(x + 1) - 6$ goes through the point $(-1, -6)$ and has a slope of $–9$ .
Sketch the graph of $y = 5(x - 2) - 1$ .

1-8.

Write the equation of the line through the point $(–5, –2)$ with a slope of $–3$ in graphing form using the method developed in problem 1-7. Refer to the Math Notes box for help. Homework Help ✎

1-9.

Now you know two general equations used to write the equation of a line:

$y=mx+b$ and $y=m(x-h)+k$

Under what circumstances is each equation easier to use? For parts (a) through (c) below, determine which method is best to use with the given information. Then, write the equation of the line. Homework Help ✎

$m=−\frac{2}{5}$ and passes through $(−6,2)$
$m=3$ and $b=-6$
passes through $(2,8)$ and $(1,3)$

1-10.

For each function sketched below, sketch $y=f(-x)$ and compare it with the original graph. Then describe its symmetry. Homework Help ✎

a. Damped Periodic curve, with approximate turning points at, (negative 3.5, comma 0.1), (negative 3, comma negative 0.3), (negative 2, comma 0.6), (negative 1, comma negative 1), (0, comma 4), (1, comma negative 1), (2, comma 0.6), (3, comma negative 0.3), (3.5, comma 01), with the curves continuing to open up on each side of the vertex (0, comma 4).

b. Continuous curve, starting in lower left, with approximate turning points at (negative 2.2, comma 2.2), negative 1, comma negative 0.2), (1, comma 0.2), (2.2, comma negative 2.2), continuing up & right, crossing at x axis at approximate x values, negative 3.2, negative 1.2, 0, 1.2, & 3.2.

c. EVEN AND ODD FUNCTIONS—INFORMALLY

A function that is symmetric with respect to the y-axis, like the one in part (a) above, is called an even function. A function that is symmetric with respect to the origin, like the one in part (b), is called an odd function.

Sketch examples of even and odd functions. Include how you can test whether a function is even or odd. Then list some famous even/odd functions that you have studied in a previous course, and the symmetries associated with even and odd functions.

Prism $V = Bh$		Cylinder $V = πr^{2}h$		Sphere $V=\frac{4}{3}\pi r^3$
	Pyramid $V=\frac{1}{3}Bh$		Cone $V=\frac{1}{3}\pi r^2h$

Calculus Third Edition

1.1.1How are speed and distance related?

General Team Roles

Volumes of Standard Solids

The Equation of a Line