Question

1. given ( f(x) = x^2 - 4x + 7 )

a. find the average rate of change of the function on the interval (-1,1).

b. describe the relationship between the rate of change and the behavior of the function on this interval.

c. how would you describe the rate of change at ( x = 2 )?

Explanation:

Part A

Step1: Recall the average rate of change formula

The average rate of change of a function \( f(x) \) on the interval \([a, b]\) is given by \(\frac{f(b)-f(a)}{b - a}\). Here, \( a=-1 \) and \( b = 1 \), and \( f(x)=x^{2}-4x + 7 \).

Step2: Calculate \( f(-1) \)

Substitute \( x=-1 \) into \( f(x) \):
\[

$$\begin{align*} f(-1)&=(-1)^{2}-4(-1)+7\\ &=1 + 4+7\\ &=12 \end{align*}$$

\]

Step3: Calculate \( f(1) \)

Substitute \( x = 1 \) into \( f(x) \):
\[

$$\begin{align*} f(1)&=(1)^{2}-4(1)+7\\ &=1-4 + 7\\ &=4 \end{align*}$$

\]

Step4: Calculate the average rate of change

Using the formula \(\frac{f(1)-f(-1)}{1-(-1)}\):
\[

$$\begin{align*} \frac{f(1)-f(-1)}{1-(-1)}&=\frac{4 - 12}{1 + 1}\\ &=\frac{-8}{2}\\ &=-4 \end{align*}$$

\]

The average rate of change we found is \(-4\) (negative). For a quadratic function \( f(x)=x^{2}-4x + 7 \), the graph is a parabola opening upwards (since the coefficient of \( x^{2} \) is positive). The vertex of the parabola \( y = ax^{2}+bx + c \) is at \( x=-\frac{b}{2a}\), here \( x =-\frac{-4}{2\times1}=2 \). On the interval \([-1,1]\), which is to the left of the vertex \( x = 2 \), the function is decreasing (because as \( x \) increases from \(-1\) to \( 1 \), the \( y \)-values decrease from \( 12 \) to \( 4 \)). A negative average rate of change indicates that the function is decreasing on the interval \([-1,1]\), so the rate of change (negative) is consistent with the function's behavior of decreasing on this interval.

Step1: Recall the concept of instantaneous rate of change

The rate of change at a point \( x = a \) for a function \( f(x) \) is the instantaneous rate of change, which is given by the derivative of the function at \( x = a \), i.e., \( f^{\prime}(a) \).

Step2: Find the derivative of \( f(x) \)

Given \( f(x)=x^{2}-4x + 7 \), using the power rule \((x^{n})^\prime=nx^{n - 1}\), the derivative \( f^{\prime}(x) \) is:
\[
f^{\prime}(x)=2x-4
\]

Step3: Evaluate the derivative at \( x = 2 \)

Substitute \( x = 2 \) into \( f^{\prime}(x) \):
\[
f^{\prime}(2)=2(2)-4=4 - 4=0
\]

Answer:

The average rate of change on \([-1,1]\) is \(-4\).

Part B