1) Determine the average rate of change of $$ f(x) $$ and $$ g(x) $$ over the interval $$ [-2,0] $$. Which function move at a faster rate? $$ \begin{array}{l}g(x)=3(x-2)=9 \ \ f(x)=3 x+3\end{array} $$

Question

UpStudy AI · Accepted Answer

To determine the average rate of change of the functions $$ f(x) $$ and $$ g(x) $$ over the interval $$ [-2, 0] $$, we will follow these steps:

1. **Calculate the values of $$ f(x) $$ and $$ g(x) $$ at the endpoints of the interval.**
2. **Use the formula for the average rate of change:**
   $$
   \text{Average Rate of Change} = \frac{f(b) - f(a)}{b - a}
   $$
   where $$ [a, b] $$ is the interval.

3. **Compare the average rates of change of both functions.**

### Step 1: Calculate $$ f(-2) $$, $$ f(0) $$, $$ g(-2) $$, and $$ g(0) $$

- For $$ f(x) = 3x + 3 $$:
  - $$ f(-2) = 3(-2) + 3 = -6 + 3 = -3 $$
  - $$ f(0) = 3(0) + 3 = 0 + 3 = 3 $$

- For $$ g(x) = 3(x - 2) $$:
  - $$ g(-2) = 3(-2 - 2) = 3(-4) = -12 $$
  - $$ g(0) = 3(0 - 2) = 3(-2) = -6 $$

### Step 2: Calculate the average rates of change

- For $$ f(x) $$:
$$
\text{Average Rate of Change of } f = \frac{f(0) - f(-2)}{0 - (-2)} = \frac{3 - (-3)}{0 + 2} = \frac{3 + 3}{2} = \frac{6}{2} = 3
$$

- For $$ g(x) $$:
$$
\text{Average Rate of Change of } g = \frac{g(0) - g(-2)}{0 - (-2)} = \frac{-6 - (-12)}{0 + 2} = \frac{-6 + 12}{2} = \frac{6}{2} = 3
$$

### Step 3: Compare the average rates of change

Both functions $$ f(x) $$ and $$ g(x) $$ have the same average rate of change of $$ 3 $$ over the interval $$ [-2, 0] $$. Therefore, neither function moves at a faster rate; they both move at the same rate.
Both functions $$ f(x) $$ and $$ g(x) $$ have the same average rate of change of 3 over the interval $$[-2, 0]$$.

Determine the average rate of change of and over the interval . Which function move
at a faster rate?

Solución de inteligencia artificial de Upstudy

Responder

Solución

Step 1: Calculate , , , and

Step 2: Calculate the average rates of change

Step 3: Compare the average rates of change

Extra Insights

preguntas relacionadas

Latest Calculus Questions

Determine the average rate of change of and over the interval . Which function move at a faster rate?