To find the average rate of change of the function [tex]\( f(x) = -2x^2 + 4 \)[/tex] over the interval [tex]\((1, 1.1)\)[/tex], we follow these steps:
1. Evaluate the function at the endpoints of the interval:
- At [tex]\( x = 1 \)[/tex]:
[tex]\[
f(1) = -2(1)^2 + 4 = -2 \cdot 1 + 4 = -2 + 4 = 2
\][/tex]
- At [tex]\( x = 1.1 \)[/tex]:
[tex]\[
f(1.1) = -2(1.1)^2 + 4 = -2 \cdot (1.21) + 4 = -2.42 + 4 = 1.58
\][/tex]
Note that [tex]\( f(1.1) \)[/tex] is approximately [tex]\( 1.5799999999999996 \)[/tex].
2. Use these values to calculate the average rate of change:
The average rate of change of a function over an interval [tex]\([a, b]\)[/tex] is given by the formula:
[tex]\[
\frac{f(b) - f(a)}{b - a}
\][/tex]
3. Substitute the appropriate values into the formula:
Here, [tex]\( a = 1 \)[/tex] and [tex]\( b = 1.1 \)[/tex]:
[tex]\[
\frac{f(1.1) - f(1)}{1.1 - 1} = \frac{1.5799999999999996 - 2}{0.1}
\][/tex]
4. Perform the arithmetic:
- Calculate the difference in the function values:
[tex]\[
1.5799999999999996 - 2 = -0.4200000000000004
\][/tex]
- Divide by the difference in [tex]\( x \)[/tex]-values ([tex]\(0.1\)[/tex]):
[tex]\[
\frac{-0.4200000000000004}{0.1} = -4.2
\][/tex]
Therefore, the average rate of change of the function [tex]\( f(x) = -2x^2 + 4 \)[/tex] over the interval [tex]\( (1, 1.1) \)[/tex] is:
[tex]\[ \boxed{-4.2} \][/tex]
