Questions: Find the average rate of change of h(x)=x^2+3x+8 from x=2 to x=6. Simplify your answer as much as possible.

Transcript text: Find the average rate of change of $h(x)=x^{2}+3 x+8$ from $x=2$ to $x=6$. Simplify your answer as much as possible.

Solution

Solution Steps

Step 1: Calculate the value of the function at the starting and ending points

Given the function $h(x) = x^2 + 3x + 8$, calculate $h(a)$ and $h(b)$ by substituting $x=a$ and $x=b$ respectively.

$h(a) = 1_(2)^2 + 3_(2)^1 + 8*(2)^0 = 18$
$h(b) = 1_(6)^2 + 3_(6)^1 + 8*(6)^0 = 62$

Step 2: Compute the difference in function values and in $x$ values

$\Delta h = h(b) - h(a) = 62 - 18 = 44$
$\Delta x = b - a = 6 - 2 = 4$

Step 3: Calculate the average rate of change

The average rate of change of the function over the interval from $x=a$ to $x=b$ is given by: \[\text{Average Rate of Change} = \frac{\Delta h}{\Delta x} = \frac{44}{4} = 11\]

Final Answer:

The average rate of change of the function over the interval from $x=2$ to $x=6$ is 11.

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