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.

Further explanation

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:

For the continuous case, in which the ROC is calculated at a point and its value corresponds to that of the derivative of the function at that point.For the discrete case, in which the ROC is calculated over an interval, and it's value corresponds to that of the slope of a line which passes through points on the function at the extremes of the interval.

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:

Learn moreSlope of a line: of a function: of the slope:

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

Of the integers between 1 and 700 (inclusive), 350 of them are even:



100 of them are exact multiples of 7:



Half of these are multiples of 14:



This means the number of distinct integers in the given set that are multiples of 2 or 7 is