2.3.1Can I approximate velocity?

Ramp Lab

2-102.

SUDDEN IMPACT, Part One

One morning, because Ms. Dietiker was late to school, she decided to park in front of school. Unfortunately, in the rush, she forgot to engage her parking brake!! As the car rolled down the hill, students dove left and right to avoid being struck. The car rolled unimpeded until it reached the bottom of the hill, where it slammed into a building and stopped.

When filing for damages with her insurance company, Ms. Dietiker discovers that she will be ineligible for coverage if her car is traveling above the speed limit ( $55$ mph) at the instant of impact. She needs your help! How fast was her car traveling when it hit the building?

What information is needed in order to answer this question?
What assumptions should be made to simplify the problem?

Note: We will begin with an investigation of an object rolling down a hill. Ms. Dietiker’s problem will be solved in the next lesson.

2-103.

RAMP LAB

Using a simplified ramp and a ball, measure the distance the ball travels over time. Take measurements for time $t = 0, 1, 2, 3, 4,$ and $5$ seconds. Collect data as accurately as possible.

Plot the data on your graphing calculator. Decide with your team what curve will best fit this data and then use your calculator to determine a function of best fit.
On graph paper, create a table of data and carefully plot the points. Use the points to sketch your curve of best fit. Choose an appropriate scale for the axes.
Examine the graph of your data. What happens to the ball’s velocity as time increases? Why?
The ball did not travel at a constant speed throughout the experiment. How does the shape of the graph support this statement?
Since the ball does not travel at a constant speed, use the graph to decide when the ball is traveling slowest and fastest. What happens to the curve when the ball is traveling faster? Slower?
Estimate how fast the ball is traveling at $t = 4$ seconds. Discuss with your team how to best and most accurately do this. Prepare a presentation to explain how you decided to determine this velocity.

2-104.

During his $30$ -minute trip to work, Mr. Molinari travels with a velocity of $v(t) = 35 + 30t$ where $t$ is measured in hours and $v(t)$ is measured in miles per hour. Homework Help ✎

Sketch a graph of this scenario.
What are his maximum and minimum rates?
How far from work does Mr. Molinari live?

2-105.

Without your graphing calculator, determine if each of the following functions is even, odd, or neither. Explain how you determined your choice. Then, check your answer with your graphing calculator. Homework Help ✎

Compute without a calculator

$y=\cos(3x)$

$y = |2x|$

2-106.

Given $f (x) = 2x + 1$ and $g(x) = \sqrt { x - 3 } + 2$ , write equations for the following functions. Homework Help ✎

$f^{−1} (x)$

$g^{−1} (x)$

$g^{−1} (f(x))$

2-107.

Write a complete slope statement from left to right for the curve drawn below. Homework Help ✎

Increasing curve starting at lower left, rising about 1 sixth up & 1 eighth right, staying level for another 2 eighths right, increasing over the next 2 eighths, rising 4 sixths, staying level over 2 eighths, then rising the last sixth & running the last eighth to the upper right corner.

2-108.

Given: $\left | x ^ { 2 } - 1 \right |$ 2-108 HW eTool. Homework Help ✎

Rewrite $h$ as a piecewise-defined function.
Using set notation, state the domain and range of $h$ .
Estimate the area under the curve for $−1\le x\le3$ by any method.
Write a Riemann sum to approximate the area under the curve for $−1\le x\le3$ using $24$ rectangles of equal width. Then evaluate the sum.

2-109.

The sigma notation expressions below represents Riemann sums that calculate the area under the curve of a function, $f$ , for $a\le x\le b$ using $n$ rectangles of equal width. For each summation, determine the values of $a, b,$ and $n$ . Homework Help ✎

$\displaystyle\sum _ { i = 0 } ^ { 17 } \frac { 1 } { 3 } f ( - 6 + \frac { 1 } { 3 } i )$

$\displaystyle\sum _ { i = 0 } ^ { 9 } \frac { 1 } { 10 } f ( 4 + \frac { 1 } { 10 } i )$

2-110.

Lena loves limits because they help her visualize the graphs of complicated looking functions. For example, by evaluating the limit as as $x \rightarrow \infty$ , she can determine if a rational function such as $f(x) = \frac { p ( x ) } { r ( x ) }$ will have a horizontal asymptote or not. Homework Help ✎

Evaluate each limit below and explain to Lena how she can determine $\lim\limits _ { x \rightarrow \infty } \frac { p ( x ) } { r ( x ) }$ without graphing. (Be careful! The expressions below are all slightly different.)

$\lim\limits _ { x \rightarrow \infty } ( \frac { x ^ { 2 } - 7 x + 6 } { x ^ { 3 } + 9 x - 2 } )$

$\lim\limits _ { x \rightarrow \infty } ( \frac { x ^ { 2 } - 7 x + 6 } { x ^ { 2 } + 9 x - 2 } )$

$\lim\limits _ { x \rightarrow \infty } ( \frac { x ^ { 3 } - 7 x + 6 } { x ^ { 2 } + 9 x - 2 } )$

2-110s.

Jasmine is trying to determine $\lim\limits _ { x \rightarrow 0 } \frac { \operatorname { sin } ( x ) } { x }$ by squeezing it between two other functions. Her work is shown below. Why will this attempt not work?

$\lim\limits _ { x \rightarrow 0 } \frac { \operatorname { sin } ( x ) } { x }$

$- 1 ≤ \sin ( x ) ≤ 1$

$- \frac { 1 } { x } \leq \frac { \operatorname { sin } ( x ) } { x } \leq \frac { 1 } { x }$

2-111.

Given $f (x) = 2x +1$ and $g(x) = \sqrt { x - 3 } + 2$ , write expressions for the following functions and state the domain of each composition. Homework Help ✎

$h(x) = g( f(x))$

$k(x) = f( f(x))$

Calculus Third Edition

2.3.1Can I approximate velocity?

The Squeeze Theorem