the police department has a machine that makes a graph for each car that drives on a certain road, graphing…

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).\n1. 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?\n2. 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.\n3. 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.\n4. determine whether the following statements are true and give an explanation or counterexample.\na. the value of lim(x→3) (x² - 9)/(x - 3) does not exist.\nb. the value of lim(x→a) f(x) can always be found by computing f(a).\nc. the value of lim(x→a) f(x) does not exist if f(a) is undefined.

Answer

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, C

Let $s_A(t)$, $s_B(t)$, and $s_C(t)$ be the position - functions of 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, C

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

Step4: Choose a smaller interval (e.g., $[2,2.1]$)

For cars A, B, C: $v_{A3}=\frac{s_A(2.1)-s_A(2)}{2.1 - 2}=10(s_A(2.1)-s_A(2))$, $v_{B3}=\frac{s_B(2.1)-s_B(2)}{2.1 - 2}=10(s_B(2.1)-s_B(2))$, $v_{C3}=\frac{s_C(2.1)-s_C(2)}{2.1 - 2}=10(s_C(2.1)-s_C(2))$. We can approximate the instantaneous velocity $v_A(2)\approx v_{A3}$, $v_B(2)\approx v_{B3}$, $v_C(2)\approx v_{C3}$ using the results from smaller and smaller intervals.

Step5: Create a position - function graph

A position - function $s(t)$ with a relatively linear behavior over the interval $[2,3]$ and non - linear behavior over $[2,2.5]$ can be a graph where the function has a constant slope over $[2,3]$ and a changing slope over $[2,2.5]$. For example, a piece - wise linear function where $s(t)=mt + c$ for $t\in[2,3]$ and a different linear or non - linear function for $t\in[2,2.5]$.

Step6: Analyze limit statements

a. Analyze $\lim_{x\rightarrow3}\frac{x^{2}-9}{x - 3}$

We can factor the numerator: $x^{2}-9=(x + 3)(x - 3)$. Then $\lim_{x\rightarrow3}\frac{x^{2}-9}{x - 3}=\lim_{x\rightarrow3}\frac{(x + 3)(x - 3)}{x - 3}=\lim_{x\rightarrow3}(x + 3)=6$. So the statement "The value of $\lim_{x\rightarrow3}\frac{x^{2}-9}{x - 3}$ does not exist" is false.

b. Analyze $\lim_{x\rightarrow a}f(x)$ and $f(a)$

The value of $\lim_{x\rightarrow a}f(x)$ cannot always be found by computing $f(a)$. For example, consider $f(x)=\frac{x^{2}-9}{x - 3}$ at $a = 3$. $f(3)$ is undefined, but $\lim_{x\rightarrow3}f(x)=6$. So the statement "The value of $\lim_{x\rightarrow a}f(x)$ can always be found by computing $f(a)$" is false.

c. Analyze $\lim_{x\rightarrow a}f(x)$ when $f(a)$ is undefined

The value of $\lim_{x\rightarrow a}f(x)$ can exist even if $f(a)$ is undefined. As in the example of $f(x)=\frac{x^{2}-9}{x - 3}$ at $a = 3$, $f(3)$ is undefined but $\lim_{x\rightarrow3}f(x)=6$. So the statement "The value of $\lim_{x\rightarrow a}f(x)$ does not exist if $f(a)$ is undefined" is false.

Answer:

For cars A, B, C, calculate average velocities as shown in Step 2 and Step 3. The average velocity over $[2,2.5]$ is closer to the instantaneous velocity at $t = 2$ minutes.
For cars A, B, C, calculate average velocity over $[2,2.1]$ as shown in Step 4 and approximate the instantaneous velocity at $t = 2$ minutes.
Create a graph with a relatively linear behavior over $[2,3]$ and non - linear behavior over $[2,2.5]$.

a. False. $\lim_{x\rightarrow3}\frac{x^{2}-9}{x - 3}=6$. 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 $x = 3$.