For the function y=4x^2 Find the average rate of change of y with respect to x over the interval [2,4]. And find the instantaneous rate of change of y with respect to x at the value x=2.

Question

For the function y=4x^2

Find the average rate of change of y with respect to x over the interval [2,4].

And find the instantaneous rate of change of y with respect to x at the value x=2.

Answer 1

Given: y = 4x^2

Find:

Average ROC on [2,4]

Instantaneous ROC at x = 2

Solution:

Average ROC = slope between the two end points

Average ROC = (change in y) / (change in x)

Average ROC = [ 4(4)^2 - 4(2)^2] / [ 4 - 2 ]

Average ROC = [ 64 - 16 ] / [ 2 ]

Average ROC = [ 48 ] / [ 2 ]

Average ROC = 24

Instantaneous ROC = slope of tangent line at x = 2

We need the derivative of the function for this.

y = 4x^2

y' = (4)(2)x

y' = 8x

At x = 2, slope of tangent line is 8(2) = 16

So, Instantaneous ROC at x=2 is 16