Q. 8 - Precalculus Enhanced with Graphing Utilities

Question

For the function $f\left(x\right)=3x^2-2x+4$, find the average rate

of change of f  from $3$ to $4$.

Step-by-Step Solution

Answer

The average rate of change of $f\left(x\right)=3x^2-2x+4$ form $3$ to $4$ is $19$

Step 1. Given data

The given function is

$f\left(x\right)=3x^2-2x+4$

Step 2. The average rate of change

The average rate of change function is

$$\begin{gathered} \frac{∆y}{∆x}=\frac{f\left(b\right)-f\left(a\right)}{b-a} \\ \frac{∆y}{∆x}=\frac{3b^2-2b+4-(3a^2-2a+4)}{b-a} \\ \frac{∆y}{∆x}=\frac{3(b^2-a^2)-2(b-a)}{b-a} \\ \frac{∆y}{∆x}=\frac{3(b-a)(b+a)-2(b-a)}{b-a} \\ \frac{∆y}{∆x}=3(b+a)-2 \end{gathered}$$

Step 3. substitute values

substitute $3$ for a and $4$ for b in the average rate of change of function

$$\begin{gathered} \frac{∆y}{∆x}=3(a+b)-2 \\ \frac{∆y}{∆x}=3(3+4)-2 \\ \frac{∆y}{∆x}=3\left(7\right)-2 \\ \frac{∆y}{∆x}=21-2 \\ \frac{∆y}{∆x}=19 \end{gathered}$$