Bright Chambers
11/14/2023 · Middle School
Rates of Change Assignment 1. Determine the average rate of change of \( y \) in the function \( y=2 x^{3}+7 x^{2}+2 x-3 \) over the interval [3, 5] [5 marks]
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Step-by-step Solution
To determine the average rate of change of the function \( y = 2x^3 + 7x^2 + 2x - 3 \) over the interval \([3, 5]\), we can use the formula for the average rate of change:
\[
\text{Average Rate of Change} = \frac{f(b) - f(a)}{b - a}
\]
where \( a = 3 \) and \( b = 5 \).
First, we need to calculate \( f(3) \) and \( f(5) \).
1. **Calculate \( f(3) \)**:
\[
f(3) = 2(3)^3 + 7(3)^2 + 2(3) - 3
\]
\[
= 2(27) + 7(9) + 6 - 3
\]
\[
= 54 + 63 + 6 - 3
\]
\[
= 120
\]
2. **Calculate \( f(5) \)**:
\[
f(5) = 2(5)^3 + 7(5)^2 + 2(5) - 3
\]
\[
= 2(125) + 7(25) + 10 - 3
\]
\[
= 250 + 175 + 10 - 3
\]
\[
= 432
\]
Now we can substitute \( f(3) \) and \( f(5) \) into the average rate of change formula:
\[
\text{Average Rate of Change} = \frac{f(5) - f(3)}{5 - 3}
\]
\[
= \frac{432 - 120}{5 - 3}
\]
\[
= \frac{312}{2}
\]
\[
= 156
\]
Thus, the average rate of change of \( y \) over the interval \([3, 5]\) is \( \boxed{156} \).
Quick Answer
The average rate of change of \( y \) over the interval [3, 5] is 156.
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