Question

The average rate of change of

g(x)=8x 4 + 2/x2 over the interval [−1,4].

a. 136
b. 112
c. 84 d. 64

Answer 1

The average rate of change of
`\(g(x) = 8x^4 + (2)/(x^2)\)` over the interval
`\([-1,4]\)` is 112 (Option B).

Step-by-step explanation:

The average rate of change of a function over an interval
`\([a, b]\)` is given by the formula
`\((f(b) - f(a))/(b - a)\).` Applying this formula to the function
`\(g(x) = 8x^4 + (2)/(x^2)\` ) over the interval
`\([-1,4]\),` we calculate
`\(g(4)\)` and
`\(g(-1)\),` and then use the formula to find the average rate of change. The calculation results in
`\((g(4) - g(-1))/(4 - (-1)) = 112\),` confirming Option B as the correct answer.

To compute
`\(g(4)\)` and \
`(g(-1)\),` substitute these values into the expression for
`\(g(x)\). For \(g(4)\),` we get
`\(8 * 4^4 + (2)/(4^2) = 2048 + (2)/(16)\).` For \(g(-1)\), we get
`\(8 * (-1)^4 + (2)/((-1)^2) = 8 + 2\).` Plugging these values into the average rate of change formula, we obtain
`\((2048 + (2)/(16) - (8 + 2))/(4 - (-1)) = 112\).`

In summary, the average rate of change is determined by evaluating the function at the endpoints of the given interval and applying the formula for average rate of change. The calculated value of 112 represents the average rate of change of the function
`\(g(x)\)` over the interval
`\([-1,4]\),` validating Option B as the correct answer.