Questions: Compare the rate of change for the function f(x)=x^3-2x^2+x+1 over the intervals [0,2] and [2,4]

Transcript text: Compare the rate of change for the function $f(x)=x^{3}-2 x^{2}+x+1$ over the intervals $[0,2]$ and $[2,4]$

Solution

Solution Steps

To compare the rate of change of the function $ f(x) = x^3 - 2x^2 + x + 1 $ over the intervals $[0,2]$ and $[2,4]$, we need to calculate the average rate of change for each interval. The average rate of change of a function $ f(x) $ over an interval $[a, b]$ is given by the formula $\frac{f(b) - f(a)}{b - a}$. We will apply this formula to both intervals and compare the results.

Step 1: Calculate the Average Rate of Change over $[0, 2]$

To find the average rate of change of the function $ f(x) = x^3 - 2x^2 + x + 1 $ over the interval $[0, 2]$, we use the formula:

\[ \text{Average Rate of Change} = \frac{f(2) - f(0)}{2 - 0} \]

Calculating $ f(2) $ and $ f(0) $:

\[ f(2) = 2^3 - 2(2^2) + 2 + 1 = 8 - 8 + 2 + 1 = 3 \] \[ f(0) = 0^3 - 2(0^2) + 0 + 1 = 1 \]

Thus, the average rate of change over $[0, 2]$ is:

\[ \frac{3 - 1}{2 - 0} = \frac{2}{2} = 1.0 \]

Step 2: Calculate the Average Rate of Change over $[2, 4]$

Next, we calculate the average rate of change over the interval $[2, 4]$ using the same formula:

\[ \text{Average Rate of Change} = \frac{f(4) - f(2)}{4 - 2} \]

Calculating $ f(4) $:

\[ f(4) = 4^3 - 2(4^2) + 4 + 1 = 64 - 32 + 4 + 1 = 37 \]

Thus, the average rate of change over $[2, 4]$ is:

\[ \frac{37 - 3}{4 - 2} = \frac{34}{2} = 17.0 \]

Step 3: Compare the Average Rates of Change

We have found the following average rates of change:

Over $[0, 2]$: $ 1.0 $
Over $[2, 4]$: $ 17.0 $

Since $ 17.0 > 1.0 $, the average rate of change is much greater over the interval $[2, 4]$.

Final Answer

The average rate of change over the interval $[0, 2]$ is $ 1.0 $, the average rate of change over the interval $[2, 4]$ is $ 17.0 $, and therefore, the average rate of change is much greater over the interval $\boxed{[2, 4]}$.

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