Question

find the average rate of change of k(x) = -3x⁵ + 5 over the interval 0, 2. write your answer as an integer, fraction, or decimal rounded to the nearest tenth. simplify any fractions.

Explanation:

Step1: Recall the average rate of change formula

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, \( a = 0 \) and \( b = 2 \), and \( k(x)=-3x^{5}+5 \).

Step2: Calculate \( k(0) \)

Substitute \( x = 0 \) into \( k(x) \):
\( k(0)=-3(0)^{5}+5 = 0 + 5=5 \)

Step3: Calculate \( k(2) \)

Substitute \( x = 2 \) into \( k(x) \):
\( k(2)=-3(2)^{5}+5=-3(32)+5=-96 + 5=-91 \)

Step4: Apply the average rate of change formula

Using the formula \(\frac{k(2)-k(0)}{2 - 0}\), substitute the values of \( k(2) \) and \( k(0) \):
\(\frac{-91 - 5}{2-0}=\frac{-96}{2}=-48\)

Answer:

\(-48\)