Question

For the function y = 2x^2(a) Find the average rate of change of y with respect to r over the interval [1,4](b) Find the instantaneous rate of change of y with respect to a at the value x = 1.Average Rate of Change= ___Instantaneous Rate of Change at x = 1: _____

Answer 1

Given the function

$y=2x^2$

The formula to find the average rate of change of y with respect to x in [a,b]

$\frac{Change\text{ in y}}{Change\text{ in x}}=\frac{f(b)-f(a)}{b-a}$

Here, a = 1 and b = 4.

$\begin{gathered} \text{Average rate of change =}\frac{f(4)-f(1)}{4-1} \\ =\frac{2\cdot4^2-2\cdot1^2}{3} \\ =\frac{30}{3} \\ =10 \end{gathered}$

The instantaneous rate of change is the slope of the tangent line at x = 1.

$\begin{gathered} \text{Slope}=\frac{d}{dx}(2x^2) \\ =2\cdot2x \\ =4x \end{gathered}$

Slope at x = 1 is

$4\cdot1=4$

So, the instantaneous rate of change is 4.