Question

Find the average rate of change of $f\left(x\right)=x^2+3x+1$from 1 to 2. Use this result to find the equation of the secant line containing$(1,f(1\left)\right),(2,(f\left(2\right))$

Short answer

The average rate of change is $6$.

The equation of secant line is$y=6x-1$

Step-by-step solution

Given information

We are given a$f\left(x\right)=x^2+3x+1$

Find f(1),f(2)

We substitute the values in the function we get

$$\begin{gathered} f\left(x\right)=x^2+3x+1 \\ f\left(1\right)=1+3+1 \\ f\left(1\right)=5 \\ also \\ f\left(2\right)=2^2+3\left(2\right)+1 \\ f\left(2\right)=4+6+1 \\ f\left(2\right)=11 \end{gathered}$$

Find average rate of change

Rate of change

$$\begin{gathered} =\frac{f\left(2\right)-f\left(1\right)}{2-1} \\ =\frac{11-5}{2-1} \\ =6 \end{gathered}$$

Now to find the equation of secant line

The equation of line can be given as

$y-y_1=m(x-x_1)$

Substitute

$$\begin{gathered} m=6 \\ (x_1,y_1)=(1,5) \end{gathered}$$

We get,

$$\begin{gathered} y-5=6(x-1) \\ y-5=6x-6 \\ y=6x-1 \end{gathered}$$

Conclusion

The average rate of change is $6$.

The equation of secant line is$y=6x-1$.