Chapter 2, Problem 8CT

(a)

To determine

To calculate:An average rate of change for the function $f(x)=3x^2-3x+4$ from $3\text{to}4$ .

Answer to Problem 8CT

Solution:

Average rate of change of function $f(x)=3x^2-3x+4$ is $18$ .

Explanation of Solution

Given information:

The function is $f(x)=3x^2-3x+4$ .

Formula used:

The average rate of change of function $y=f\left(x\right)$ from $a$ to $b$ is defined as:

Average rate of change $=\frac{\Delta y}{\Delta x}=\frac{f\left(b\right)-f\left(a\right)}{b-a}$ where $a$ and $b$, $a\ne b$, are in the domain of $f\left(x\right)$ .

Calculation:

The average rate of change of $f(x)=3x^2-3x+4$ from $3\text{to}4$ is

Average rate of change $=\frac{f(4)-f(3)}{4-3}$ .

Substitute $x=3$ to find $f(3)$ in $f(x)=3x^2-3x+4$ .

  $\Rightarrow f(3)=3{(3)}^2-3(3)+4$

  $\Rightarrow f(3)=27-9+4=22$

  $\Rightarrow f(3)=22$ .

Substitute $x=4$ to find $f(4)$ in $f(x)=3x^2-3x+4$ .

  $\Rightarrow f(4)=3{(4)}^2-3(4)+4$

  $\Rightarrow f(4)=48-12+4$ .

  $\Rightarrow f(4)=40$ .

Thus, the average rate of change of function is $=\frac{40-22}{4-3}=\frac{18}{1}$ .

Therefore, the average rate of change is $18$ .

(b)

To determine

An equation of secant line of function $f(x)=3x^2-3x+4$ from $3\text{to}4$ .

Answer to Problem 8CT

Solution:

The equation of secant line of $f(x)=3x^2-3x+4$ is $y=18x-32$ .

Explanation of Solution

Given information:

The function is $f(x)=3x^2-3x+4$ .

Slope of the secant line of function $f(x)=3x^2-3x+4$ is same as average rate of change of function $f(x)=3x^2-3x+4$ from $3\text{to}4$ .

From part (a), average rate of change is $18$ .

Therefore, $m_{\sec }=18$ .

One point on a secant line is $(3,f(3))=(3,22)$ .

The slope intercept formula is $y-y_1=m(x-x_1)$ .

Substitute $x_1=3,y_1=22$ and $m=m_{\sec }=18$ in slope intercept equation.

Therefore,

  $y-22=18(x-3)$ .

Simplify above equation,

  $\Rightarrow y-22=18x-54$ .

Add $22$ on both sides,

  $\Rightarrow y-22+22=18x-54+22$ .

Simplify above equation,

  $\Rightarrow y=18x-32$ .

Thus, the equation of secant line is $y=18x-32$ .