Describe how to determine the average rate of change between x=1 and x=3 for the function f(x)=3x3+1 include the avg. roc. in the answer

The average rate of change of the function in the interval between and , is 39.

The rate of change of a function represents how much does a function change on the y axis (meaning vertically) when we move along the x axis (meaning horizontally). This rate of change (which for simplicity, we will call ROC from now on) can be computed in 2 ways:

Our problem corresponds to that of the discrete case, and it gets the name "average ROC" because it doesn't represent the correct ROC of the function (the correct ROC is given by its derivative), rather it gives an average of it over the whole interval. Having explained this, we just need to compute the points of the function at the extremes of the interval, so:

Now we just need to compute the slope of a line that passes through points and . The slope then will be given by:

Slope, Rate of change, Derivative, Average

Average Rate of Change = 39

Step-by-step explanation:

The formula to find the average rate of change is

here x₂=3 and x₁ = 1

We are given: f(x) = 3x^3+1

finding f(3) = 3(3)^3 +1

f(3) = 3(27)+1

f(3) = 81+2

f(3) = 82

now finding f(1):

f(1) = 3(1)^3 +1

f(1) = 3(1)+1

f(1) = 4

Now, putting values of f(3)=82 and f(1) = 4 in the formula:

Average Rate of Change = 39

x= -3 and

step-by-step explanation:

to find the roots of a cubic, use division or factoring to find the factors then set equal to 0. as this is a difference of cubes, it factors into

where a is the cube root of .

here   this is 27 and the cube root of 27 is 3. so we write using the form:

.

set each factor to 0 and solve for x.

x-3=0

x=3

the quadratic expression does not factor and must be solved using the quadratic formula.

where a = 1, b=3, and c=9

this cubic has two complex roots.