Question

suppose that ( f(x)=3 x^{2} ).
(a) what is the average rate of change of ( f(x) ) over each of the following intervals: 3 to 4,3 to 3.5,3 and to 3.1?
(b) what is the (instantaneous) rate of change of ( f(x) ) when ( x = 3 )?
the average rate of change of ( f(x) ) over the interval 3 to 3.5 is 19.5
(simplify your answer.)
which expression can be used to find the average rate of change of ( f(x) ) over the interval 3 to 3.1?
the average rate of change of ( f(x) ) over the interval 3 to 3.1 18.3
(simplify your answer.)
(b) to compute the (instantaneous) rate of change of ( f(x) ) when ( x = 3 ), first compute ( f(x) ).
( f(x)=square )

Explanation:

Step1: Differentiate \( f(x) = 3x^{2} \)

Using the power rule \( (x^{n})^\prime=nx^{n - 1} \), for \( y = ax^{n}\), \(y^\prime=anx^{n-1}\).
For \( f(x)=3x^{2}\), \(f^\prime(x)=3\times2x=6x\)

Answer:

\(6x\)