What is the average rate of change for this function!?!??

Question

On the indicated intervals. . .

1) f(x) = x^3 - 2x; [0, 4]



Can you also clarify what rate of change is?? Thanks

Answer 1

Rate of change = [change in f(x)] / [change in x]

Average rate of change = (f(4)-f(0))/(4-0)
= ((4^3-2*4)-(0^3-2*0))/(4-0)
= (56-0)/4
= 14

Answer 2

There are several definitions of average. The most common is to take a bunch of observations and add together the values observed and divide by the number of observations. In the problem you have describe there are an infinite number of possible points between zero and four. It is not possible to utilize that many observations so a sample might be chosen or an advance mathematical discipline called differential calculus might be used or you might just calculate the function at each end of the range of X-Values and subtract the result associated with the beginning of the range from the result associated with the end of the range. Then divide the difference by the difference in X-values.
In this case the result would be ((64-8)-0)/(4-0)=14
The rate of change is comparable to the speed of a car moving down the road. In the case of the car the distance traveled is a function of time. At 60 miles an hour the car will travel one mile each minute.
The rate of change is the change in the value of a function in a unit change of the independent variable.
In the case of the 60 mile per hour car the independent variable is time and the distance traveled is dependent on the amount of time the car travels.
In your problem the independent variable is "x" and the value of the function is dependent on what the value of "x" is.

Answer 3

df(x)/dx = 3x^2 -2 .then subistitute by any point in the interval [0, 4]
the rate of change of a differentiable function at any point represents the slope of the tangent to the cuve of the function at this point i.e. the tan(theta) where theta is the angle which the tangent makes with the positive direction of the X- axis

Answer 4

Rate of change = (f(x2) - f(x1)) / (x2 - x1)
The instantaneous rate of change is 3x^2 - 2
The average rate of change of this function over the interval [0,4] is (64 - 8 - 0) / 4 = 14