Question

Given the function h(x) = -x^2 + 7x + 19, determine the average rate of change of the function over the interval 1 < x < 10.

a) 2
b) 4
c) -2
d) -4

Answer 1

The average rate of change of the function h(x) = -x² + 7x + 19 on the interval 1 < x < 10 is found by evaluating the function at x=1 and x=10, and then using the difference quotient. The calculation yields an average rate of change of -4.

Step-by-step explanation:

The question asks to determine the average rate of change of the function h(x) = -x² + 7x + 19 over the interval 1 < x < 10. The average rate of change in the context of a function is analogous to the slope of the secant line that connects two points on the graph of the function.

To calculate this, we will use the values of the function at the endpoints of the interval: h(1) and h(10).

First, evaluate the function at both points:

• h(1) = -(1)² + 7(1) + 19 = 25
• h(10) = -(10)² + 7(10) + 19 = -100 + 70 + 19 = -11

Next, apply the formula for the average rate of change:

Average rate of change = (f(b) - f(a)) / (b - a)

In this case, it will be:

((-11) - 25) / (10 - 1) = -36 / 9 = -4

The average rate of change of the function over the interval is -4, which corresponds to option d.