Question

the police department has a machine that makes a graph for each car that drives on a certain road, graphing the position of the car (in miles) as a function of time (in minutes).

1. suppose the six graphs from the week 1 group project are graphs that this machine recorded. for cars (a), (b), and (c) from week 1, calculate the average velocity on the interval 2,3, i.e. between 2 and 3 minutes. then calculate the average velocity over 2,2.5. which calculation seems closer to the instantaneous velocity at 2 minutes?
2. one way of estimating the instantaneous velocity at 2 minutes is by calculating average velocities over 2,2 + h, and choosing smaller and smaller values for the interval length h. you started this process already in question #1. (h was 1 for the interval 2,3 and.5 for the interval 2,2.5.) continue to carry out this strategy for one more smaller interval on cars (a), (b), and (c). use the results from questions #1 and #2 to approximate what the instantaneous velocity at 2 minutes is for each car.
3. create a graph of a position function where the average velocity over 2,3 is a better estimate for the instantaneous velocity at 2 minutes than the average velocity over 2,2.5.
4. determine whether the following statements are true and give an explanation or counterexample.

a. the value of lim_x→3 (x² - 9)/(x - 3) does not exist.
b. the value of lim_x→a f(x) can always be found by computing f(a).
c. the value of lim_x→a f(x) does not exist if f(a) is undefined.

Explanation:

Step1: Recall average - velocity formula

The average velocity $v_{avg}$ over the interval $[a,b]$ for a position - function $s(t)$ is given by $v_{avg}=\frac{s(b)-s(a)}{b - a}$.

Step2: Calculate average velocity for interval $[2,3]$ for cars (A), (B), and (C)

Let $s_A(t)$, $s_B(t)$, and $s_C(t)$ be the position - functions for cars (A), (B), and (C) respectively. Then $v_{A1}=\frac{s_A(3)-s_A(2)}{3 - 2}=s_A(3)-s_A(2)$, $v_{B1}=\frac{s_B(3)-s_B(2)}{3 - 2}=s_B(3)-s_B(2)$, $v_{C1}=\frac{s_C(3)-s_C(2)}{3 - 2}=s_C(3)-s_C(2)$.

Step3: Calculate average velocity for interval $[2,2.5]$ for cars (A), (B), and (C)

$v_{A2}=\frac{s_A(2.5)-s_A(2)}{2.5 - 2}=\frac{s_A(2.5)-s_A(2)}{0.5}=2(s_A(2.5)-s_A(2))$, $v_{B2}=\frac{s_B(2.5)-s_B(2)}{2.5 - 2}=\frac{s_B(2.5)-s_B(2)}{0.5}=2(s_B(2.5)-s_B(2))$, $v_{C2}=\frac{s_C(2.5)-s_C(2)}{2.5 - 2}=\frac{s_C(2.5)-s_C(2)}{0.5}=2(s_C(2.5)-s_C(2))$.
The average velocity over $[2,2.5]$ is closer to the instantaneous velocity at $t = 2$ because the interval is smaller.

Step4: For question 2

Choose a smaller interval, say $[2,2.1]$. Calculate $v_{A3}=\frac{s_A(2.1)-s_A(2)}{2.1 - 2}=10(s_A(2.1)-s_A(2))$, $v_{B3}=\frac{s_B(2.1)-s_B(2)}{2.1 - 2}=10(s_B(2.1)-s_B(2))$, $v_{C3}=\frac{s_C(2.1)-s_C(2)}{2.1 - 2}=10(s_C(2.1)-s_C(2))$. Approximate the instantaneous velocity at $t = 2$ by using the values from the smaller - interval calculations.

Step5: For question 3

A graph of a position - function $s(t)$ where the function is relatively linear over $[2,3]$ and non - linear over $[2,2.5]$ would satisfy the condition. For example, a piece - wise linear function where the slope is constant over $[2,3]$ and changes over $[2,2.5]$.

Step6: For question 4a

Simplify $\lim_{x
ightarrow3}\frac{x^{2}-9}{x - 3}=\lim_{x
ightarrow3}\frac{(x + 3)(x - 3)}{x - 3}=\lim_{x
ightarrow3}(x + 3)=6$. So the statement "The value of $\lim_{x
ightarrow3}\frac{x^{2}-9}{x - 3}$ does not exist" is false.

Step7: For question 4b

The value of $\lim_{x
ightarrow a}f(x)$ cannot always be found by computing $f(a)$. For example, for $f(x)=\frac{x^{2}-9}{x - 3}$ at $a = 3$, $f(3)$ is undefined but $\lim_{x
ightarrow3}f(x)=6$. So the statement is false.

Step8: For question 4c

The value of $\lim_{x
ightarrow a}f(x)$ can exist even if $f(a)$ is undefined. As in the previous example of $f(x)=\frac{x^{2}-9}{x - 3}$ at $a = 3$, $\lim_{x
ightarrow3}f(x)=6$ while $f(3)$ is undefined. So the statement is false.

Answer:

1. Average velocities for cars (A), (B), and (C) over $[2,3]$ and $[2,2.5]$ are calculated as above. The $[2,2.5]$ calculation is closer to the instantaneous velocity at $t = 2$.
2. Approximate instantaneous velocities for cars (A), (B), and (C) at $t = 2$ using the smaller - interval $[2,2.1]$ as above.
3. A piece - wise linear position - function with appropriate slopes can be created.

4.

• a. False. $\lim_{x

ightarrow3}\frac{x^{2}-9}{x - 3}=6$.

• b. False. Counter - example: $f(x)=\frac{x^{2}-9}{x - 3}$ at $x = 3$.
• c. False. Counter - example: $f(x)=\frac{x^{2}-9}{x - 3}$ at $x = 3$.