1.4.1Can you walk the walk?

Distance and Velocity

1-133.

Shant and Kier each took a $10$ second walk and wrote descriptions of their experiences.

Shant’s Walk: Start close to the motion detector. Begin walking at a very fast rate and gradually slow down to a stop. Then increase your speed gradually until you are walking as quickly as when you started.

Kier’s Walk: Start close to the motion detector. Begin walking at a very slow rate and gradually speed up to the fastest speed you can achieve. Then gradually slow down to your starting speed.

Complete the tasks below in any order:

Sketch a distance vs. time graph of each scenario
Walk the walk.
Compare the two walks. Is there a relationship between the graphs of the two scenarios?

1-134.

THE SLOPE WALK, Part Two Your teacher will provide you with a model.

During her $15$ second Slope Walk, Diamonique walked in front of a motion detector and the graph at right appeared.

If the graph represents her distance traveled, describe her motion.
1. State the time interval when Diamonique was walking toward the motion detector. When did she walk away from the motion detector? When did she change directions? Justify your answers.
2. On what interval(s) of time was Diamonique standing still? Justify your answer.
3. Estimate a time when Diamonique was walking fastest. Justify your answer.
4. Estimate a time when Diamonique was slowing down. Justify your answer.
Now assume that the graph represents velocity vs. time—in other words, let the $y$ -axis represent velocity. Similarly describe her motion.

1-135.

FUNDAMENTALLY THE SAME

Fredo (short for Alfredo) and Frieda were both given the responsibility to collect data for a foot race. After watching the event, the Fredo and Frieda gave the coach graphs of their data, shown below.

Fredo’s Graph

First quadrant with segment between the origin & the point (18, comma 90).

Frieda’s

First quadrant with horizontal segment between the points (0, comma 5), & (18, comma 5).

The coach is dismayed because the graphs are different. However, Fredo and Frieda know that they watched the same race and that their data is confirmed in the other’s graph. Your goal is to help them convince their coach that the graphs represent the same race.

To help the coach understand the graphs, label each graph with the appropriate (and reasonable) units.
Explain how Frieda’s graph confirms Fredo’s and how Fredo’s graph confirms Frieda’s.

1-136.

Ellen rode her bike one morning to visit some friends. Her distance from home is documented in the graph at right.

Estimate her rate (velocity) in miles per hour for times $t = 7$ , $32$ , and $45$ minutes. Describe your strategy.
How is Ellen’s velocity related to the graph of her distance from home?
What is her total distance traveled?

Your teacher will provide you with a model.

1-137.

As a team, summarize what you learned about the relationship between distance and velocity from problem 1-135. Be prepared to share your statements with the class.

1-138.

A portion of the graph of $y = f(x)$ is given at right: 1-138 HW eTool. Homework Help ✎

Sketch the rest of the graph if $f$ is even and calculate the area under the curve for $−3 ≤ x ≤ 3$ .
Sketch the rest of the graph if $f$ is odd and calculate the area under the curve for $−3 ≤ x ≤ 3$ .

Continuous linear Piecewise, starting at the origin, turning horizontal at (1, comma 1), continuing to the right.

1-139.

What value of $a$ will make $h$ continuous? Homework Help ✎

$h ( x ) = \left\{ \begin{array} { l l } { \sqrt { x + 2 } - 1 } & { \text { for } x < 2 } \\ { a ( x + 1 ) ^ { 2 } } & { \text { for } x \geq 2 } \end{array} \right.$

1-140.

The graph of $y = f(x)$ is sketched at right. Homework Help ✎

Sketch $y = f(−x)$ .
Sketch $y = −f(x)$ .
Sketch $y = f(f(x))$ .

Upward V shape graph, vertex at the point (0 comma negative 2), passing through the points (negative 2, comma 0), & (2, comma 0).

1-141.

State the domain of each of the following functions. Note: The functions mentioned in parts (c) and (d) refer to those in parts (a) and (b). Homework Help ✎

$f(x)=\frac{1}{x+2}$
$g(x)=\sqrt{x-4}$
$h(x) = f(g(x))$
$k(x) = g(f(x))$

1-142.

For each function, use algebra to identify all holes and asymptotes. Homework Help ✎

$f(x)=\frac{x-3}{x^2+4x-21}$
$g(x)=\frac{x^{4/3}}{x^2-2x}$

1-143.

Velocity is only one example of a rate of change. Name at least two other familiar rates that you encounter in your daily life. Homework Help ✎

1-144.

Let $f$ be a function whose finite differences grow by $4$ . What kind of function can $f$ be? Give two examples. Homework Help ✎

1-145.

Cynthia began to draw midpoint rectangles to approximate the area under the curve for $−2 ≤ x ≤ 4$ given $f(x) = \frac { 1 } { 4 }x^3 −\frac { 1 } { 2 }x^2 − x + 3$ . Trace Cynthia’s graph and finish drawing the remaining four midpoint rectangles. Then, compute the estimated area. Homework Help ✎

Cubic curve, coming from lower left, turning in second & first quadrant, continuing up & right, & 2 vertical shaded bars, bottom edges on x axis, each with width of 1, starting at x = negative 2, with midpoint of top edge of each bar, on the curve.

1-146.

Each of of the functions below has one or more holes and/or asymptotes. Graph the functions on your graphing calculator and write a complete set of approach statements for each function. Homework Help ✎

$f(x) =\frac { 2 ^ { x } } { x }$
$f(x)=\frac{2^x-1}{x}$

Calculus Third Edition