2.3.2How fast was the car traveling on impact?

Sudden Impact

2-112.

SUDDEN IMPACT, Part Two

Ms. Dietiker still needs to determine the velocity her car was traveling down the hill when it slammed into the building so she can fill out her insurance report. Since she did not see the accident herself, she had to collect information from the students walking up the steep hill to school. (Fortunately, some of them were her calculus students who know the importance of collecting good data!)

Here were their statements:

Eric:	“I noticed the car start to roll at 7:54 and $46$ seconds.”
Joshua:	“It almost killed me at 7:55 and $20$ seconds.”
Lisa:	“It practically ran over me $7$ seconds after it passed Kirt.”
Samantha:	“I’ll mark the time 7:54 and $57$ seconds forever in my memory as the moment I nearly died.”
Kirt:	“I noticed it took twice as long for the car to reach me as it did to reach Samantha.”

Determine at what time and distance the car passes each person. Then plot the data on a distance vs. time graph.
These data points can be connected with a smooth “best fit” curve; describe the shape of this curve.
According to the curve of best fit, how long did it take for Ms. Dietiker’s car to slam into the building?
Determine how fast the car was traveling when it slammed into the building. Will Ms. Dietiker be able to collect on her insurance policy (see problem 2-102)?
Approximately when is the car traveling the fastest? How does the shape of the graph help you answer this question?

Hill with 5 people placed from top to bottom, bottom has car hitting building, labels between people from top to bottom: 142, 427, 421, 367, 403 to crashed car.

Note: Distances are in feet.
Drawing is not to scale.

2-113.

After Marlayna was playing with her graphing calculator, Sandy picked it up and saw the graph at right. Knowing her functions well, Sandy figures that Marlayna was examining a line. However, when Sandy looks at the function screen, the only equation in the calculator is $y=\sin(x)$ ! Now Sandy is very confused... this does not look like any trigonometric graph she has ever seen before.

Increasing line, passing through the origin, each axis is unscaled.

What happened? Can you recreate this graph of $y=\sin(x)$ on your graphing calculator?
Find three more functions that appear to be linear when zooming in on the calculator at $x = 0$ .
Marlayna has a conjecture. She thinks that all functions will appear linear when “zooming in” on the calculator at $\it x = 0$ . Can you find a counterexample to Marlayna’s conjecture?

2-114.

Sketch a function f that satisfies all of the following conditions: 2-114 HW eToo. Homework Help ✎

$D = [2, \infty), R = (0, 6]$

$\lim\limits_ { x \rightarrow \infty } f ( x ) = 0$

$f (2) = 3$

$f (4) = 6$

2-115.

While running in a straight line to class, Steven’s distance from class (in meters) was recorded on the graph at right. Homework Help ✎

Estimate his velocity (in meters per second) at $t = 0, 1,$ and $3$ seconds.
Did Steven ever stop and turn around? If so, when? How does the graph show this?
Approximate the interval(s) of time when Steven was headed toward class.

First quadrant, x axis labeled time, seconds, y axis labeled, distance, meters, curve starting at (0, comma 5), turning at the following approximate points: down at (0.5, comma 5.1), up at (2.5, comma 1.5), down at (3.75, comma 2.5), ending at (5, comma 0).

2-116.

A local fast-food restaurant records data on the rate customers enter their establishment during a typical lunch hour. Using this data in the table below, predict the total number of customers served during this $30$ -minute period. Homework Help ✎

Time (min)	$0$	$4$	$9$	$15$	$19$	$26$	$30$
Rate (cust/min)	$12$	$13$	$17$	$23$	$19$	$14$	$6$

2-117.

If $h(x) = \frac { 4 } { 5 }x^3 − 2x + 5$ use sigma notation to write Riemann sums to approximate the area under the curve for $10\le x\le15$ using $10, 20,$ and $100$ left endpoint rectangles of equal width. Then, use your calculator to determine these approximations. What happens as the number of rectangles increases? Homework Help ✎

2-118.

Evaluate the following limits. Homework Help ✎

$\lim\limits _ { x \rightarrow 3 } \large\frac { x ^ { 2 } - 9 } { x - 3 }$

$\lim\limits _ { x \rightarrow \infty } \sqrt { \frac { 16 x ^ { 2 } - 1 } { 4 x ^ { 2 } - 1 } }$

$\lim\limits _ { x \rightarrow \infty } \large\frac { 2 ^ { - x } } { 2 ^ { x } }$

$\lim\limits _ { x \rightarrow \infty } \large\frac { 2 x ^ { 3 } - x - 1 } { 12 - x - x ^ { 2 } }$

$\lim\limits _ { x \rightarrow \infty } \operatorname { sin } ( x )$

$\lim\limits _ { x \rightarrow \infty } \large\frac { \operatorname { sin } ( x ) } { x }$

2-118s.

Jasmine is making another attempt to determine $\lim\limits_{x\rightarrow 0 }\frac{\sin (x)}{x}$ using the Squeeze Theorem. Her work is shown below. Did she meet all of the conditions of the Squeeze Theorem? Explain how you know.

On an interval around $x=0$ , I noticed that $y=\frac{\sin(x)}{x}$ was between $y=\cos(x)$ and $y=1$ .

Since $\cos(x)≤ \frac{\sin(x)}{x}≤1$ and $\lim\limits_{x\rightarrow 0 }\cos (x)=\lim\limits_{x\rightarrow 0 }1=1$ , then $\lim\limits_{x\rightarrow 0 }\frac{\sin (x)}{x}=1$ .

Black Horizontal line at y = 1, gray curve, coming through (negative 2 pi, comma 0), turning down at (0, comma 1), passing through the point (pi, comma 0), turning up (3 halves pi, comma negative 0.2), passing through (2 pi, comma 0), black curve, coming through (negative 2 pi, comma negative 1), turning at (0, comma 1), & at (pi, comma negative 1), passing through (2 pi, comma 1).

2-119.

When the balloon in problem 2-81 reached $500$ feet, it popped and started to fall back towards the ground. The height of the balloon as it falls can be modeled by the function $h(t)=-16t^2+500$ , where $t$ is the number of seconds since the time it popped and $h(t)$ is the height of the balloon (in feet) above the ground. Homework Help ✎

According to the model, when will the balloon hit the ground?
Approximate the balloon’s velocity at $t = 5$ seconds.

2-120.

For each part below, draw a graph of a function that meets the given conditions, if possible. If such a function is not possible, explain why not. Homework Help ✎

$g$ is discontinuous at $x = a$ , but $\lim\limits _ { x \rightarrow a } g ( x )$ exists.
$g$ is continuous at $x = a$ , but $\lim\limits _ { x \rightarrow a } g ( x )$ does not exist.
$g$ is discontinuous at $x = a$ , and $\lim\limits _ { x \rightarrow a } g ( x )$ does not exist.

2-121.

You have found (or approximated) the volume of a rotated flag with various shapes. What if the axis of rotation (the “pole”) is not attached directly to the flag?

In the graph at right, there is a gap between the flag and the x-axis. Decide what will occur when this flag is rotated about the $x$ -axis. Draw a sketch of the result and calculate the volume of the resulting solid. To help you visualize this, use the 2-121 eToo. Homework Help ✎

First quadrant, with shaded rectangle with vertices as follows: (1, comma 1), (1, comma 2), (4, comma 2), & (4, comma 1).

Calculus Third Edition