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Given the function
h(x)=x^(2)+7x+5, determine the average rate of change of the function over the interval
-6 <= x <= 1.
Answer:

Given the function $h(x)=x^{2}+7 x+5$ , determine the average rate of change of the function over the interval $-6 \leq x \leq 1$ . $\newline$ Answer:

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Find derivatives using implicit differentiation

Full solution

Q. Given the function

h(x)=x^{2}+7 x+5

, determine the average rate of change of the function over the interval

-6 \leq x \leq 1

.

\newline

Answer:

Calculate Function Value at $-6$ : To find the average rate of change of the function $h(x) = x^2 + 7x + 5$ over the interval from $x = -6$ to $x = 1$ , we need to calculate the difference in the function values at these points and divide by the difference in the $x$ -values.
Calculate Function Value at $1$ : First, we calculate the function value at $x = -6$ : $h(-6) = (-6)^2 + 7*(-6) + 5$ .
Find Difference in Function Values: Performing the calculation: $h(-6) = 36 - 42 + 5 = -1$ .
Find Difference in X-Values: Next, we calculate the function value at $x = 1$ : $h(1) = (1)^2 + 7*(1) + 5$ .
Calculate Average Rate of Change: Performing the calculation: $h(1) = 1 + 7 + 5 = 13$ .
Calculate Average Rate of Change: Performing the calculation: $h(1) = 1 + 7 + 5 = 13$ .Now, we find the difference in the function values: $h(1) - h(-6) = 13 - (-1) = 14$ .
Calculate Average Rate of Change: Performing the calculation: $h(1) = 1 + 7 + 5 = 13$ .Now, we find the difference in the function values: $h(1) - h(-6) = 13 - (-1) = 14$ .We also find the difference in the $x$ -values: $1 - (-6) = 1 + 6 = 7$ .
Calculate Average Rate of Change: Performing the calculation: $h(1) = 1 + 7 + 5 = 13$ .Now, we find the difference in the function values: $h(1) - h(-6) = 13 - (-1) = 14$ .We also find the difference in the x-values: $1 - (-6) = 1 + 6 = 7$ .Finally, we calculate the average rate of change by dividing the difference in function values by the difference in x-values: Average rate of change = $(h(1) - h(-6)) / (1 - (-6))$ .
Calculate Average Rate of Change: Performing the calculation: $h(1) = 1 + 7 + 5 = 13$ .Now, we find the difference in the function values: $h(1) - h(-6) = 13 - (-1) = 14$ .We also find the difference in the x-values: $1 - (-6) = 1 + 6 = 7$ .Finally, we calculate the average rate of change by dividing the difference in function values by the difference in x-values: Average rate of change = $(h(1) - h(-6)) / (1 - (-6))$ .Performing the calculation: Average rate of change = $14 / 7 = 2$ .

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