Bot GPT 4-o mini To determine the average rate of change of the function \( f(x) = x^2 + 9x + 18 \) over the given intervals, we can use the formula for the average rate of change between two points \( a \) and \( b \):
\[ \text{Average Rate of Change} = \frac{f(b) - f(a)}{b - a} \]
Now, let's calculate the average rate of change for each of the given options.
A. Interval: \( 1 \leq x \leq 3 \)
- \( f(1) = 1^2 + 9(1) + 18 = 1 + 9 + 18 = 28 \)
- \( f(3) = 3^2 + 9(3) + 18 = 9 + 27 + 18 = 54 \)
\[ \text{Average Rate of Change} = \frac{f(3) - f(1)}{3 - 1} = \frac{54 - 28}{2} = \frac{26}{2} = 13 \]
B. Interval: \( 1 \leq x \leq 5 \)
- \( f(5) = 5^2 + 9(5) + 18 = 25 + 45 + 18 = 88 \)
\[ \text{Average Rate of Change} = \frac{f(5) - f(1)}{5 - 1} = \frac{88 - 28}{4} = \frac{60}{4} = 15 \]
C. Interval: \( 1 \leq x \leq 7 \)
- \( f(7) = 7^2 + 9(7) + 18 = 49 + 63 + 18 = 130 \)
\[ \text{Average Rate of Change} = \frac{f(7) - f(1)}{7 - 1} = \frac{130 - 28}{6} = \frac{102}{6} = 17 \]
D. Interval: \( 2 \leq x \leq 6 \)
- \( f(2) = 2^2 + 9(2) + 18 = 4 + 18 + 18 = 40 \)
- \( f(6) = 6^2 + 9(6) + 18 = 36 + 54 + 18 = 108 \)
\[ \text{Average Rate of Change} = \frac{f(6) - f(2)}{6 - 2} = \frac{108 - 40}{4} = \frac{68}{4} = 17 \]
E. Interval: \( 2 \leq x \leq 8 \)
- \( f(8) = 8^2 + 9(8) + 18 = 64 + 72 + 18 = 154 \)
\[ \text{Average Rate of Change} = \frac{f(8) - f(2)}{8 - 2} = \frac{154 - 40}{6} = \frac{114}{6} = 19 \]
Summary of Results:
- A: 13
- B: 15
- C: 17
- D: 17
- E: 19
From the calculations, the correct statements are:
- B: The rate of change is 15 over the interval 1 ≤ x ≤ 5.
- C: The rate of change is 17 over the interval 2 ≤ x ≤ 6.
Therefore, the correct answers are B and D.
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