Question

1. determine the average value of the following function over the stated closed intervals. state with reasons whether the function is increasing or decreasing over the given closed interval. over the closed interval 1, 3. 4. determine the average rate of change of the function f(x)=2x + 5 over the closed interval -2, 4.

Explanation:

3.

Step1: Recall average - value formula

The average value of a function $y = f(x)$ over the interval $[a,b]$ is given by $f_{avg}=\frac{1}{b - a}\int_{a}^{b}f(x)dx$. Since we have a graph, we can estimate the area under the curve over $[1,3]$ using geometric shapes. The region under the curve from $x = 1$ to $x = 3$ is a part of a parabola. We can also use the fact that if we assume the function is a quadratic $y=ax^{2}+bx + c$. From the graph, we can see that the function is symmetric about $x = 1$. The area under the curve from $x = 1$ to $x = 3$ can be thought of as a combination of a triangle - like shape. The function values at $x = 1$ is $y = 1$ and at $x=3$ is $y=- 3$.
The area $A$ under the curve from $x = 1$ to $x = 3$ can be approximated by the formula for the area of a trapezoid $A=\frac{1}{2}(y_1 + y_2)(x_2 - x_1)$, where $y_1 = 1,y_2=-3,x_1 = 1,x_2 = 3$. So $A=\frac{1}{2}(1+( - 3))(3 - 1)=\frac{1}{2}(1 - 3)\times2=-2$.
The average value $f_{avg}=\frac{1}{3 - 1}\times A=\frac{-2}{2}=-1$.
To determine if the function is increasing or decreasing:
The slope of the tangent line to the function at any point gives the rate of change of the function. For $x\in[1,3]$, as $x$ increases from $1$ to $3$, the $y$ - values of the function decrease. So the function is decreasing on the interval $[1,3]$ because the slope of the secant line (and the tangent lines in the interval) is negative.

Step2: State the results

The average value of the function over the interval $[1,3]$ is $-1$ and the function is decreasing on $[1,3]$.

Step1: Recall average - rate - of - change formula

The average rate of change of a function $y = f(x)$ over the interval $[a,b]$ is given by $\frac{f(b)-f(a)}{b - a}$.
For the function $f(x)=2x + 5$ and the interval $[a=-2,b = 4]$, first find $f(-2)$ and $f(4)$.
$f(-2)=2\times(-2)+5=-4 + 5=1$.
$f(4)=2\times4+5=8 + 5=13$.

Step2: Calculate the average rate of change

Substitute into the formula: $\frac{f(4)-f(-2)}{4-(-2)}=\frac{13 - 1}{4+2}=\frac{12}{6}=2$.

Answer:

The average value of the function over the interval $[1,3]$ is $-1$. The function is decreasing on the interval $[1,3]$ because as $x$ increases from $1$ to $3$, the $y$ - values of the function decrease.

4.