Question

find the average rate of change of k(x) = x² over the interval 7, 10. write your answer as an integer, fraction, or decimal rounded to the nearest tenth. simplify any fractions.

Explanation:

Step1: Recall the formula for average rate of change

The average rate of change of a function \( k(x) \) over the interval \([a, b]\) is given by \(\frac{k(b)-k(a)}{b - a}\). Here, \( a = 7 \), \( b = 10 \) and \( k(x)=x^{2}\).

Step2: Calculate \( k(7) \) and \( k(10) \)

First, find \( k(7) \): substitute \( x = 7 \) into \( k(x)=x^{2} \), so \( k(7)=7^{2}=49 \).
Then, find \( k(10) \): substitute \( x = 10 \) into \( k(x)=x^{2} \), so \( k(10)=10^{2}=100 \).

Step3: Substitute into the average rate of change formula

Now, substitute \( k(7) = 49 \), \( k(10)=100 \), \( a = 7 \) and \( b = 10 \) into the formula \(\frac{k(b)-k(a)}{b - a}\).
We get \(\frac{100 - 49}{10 - 7}=\frac{51}{3}\).

Step4: Simplify the fraction

Simplify \(\frac{51}{3}\), which equals \( 17 \).

Answer:

\( 17 \)