Question

practising

1. for the function $f(x)=6x^{2}-4$, estimate the instantaneous rate of change for the given values of $x$.

a) $x = - 2$ b) $x = 0$ c) $x = 4$ d) $x = 8$

Explanation:

Step1: Find the derivative of the function

The derivative of $f(x)=6x^{2}-4$ using the power - rule $(x^n)' = nx^{n - 1}$ is $f'(x)=12x$.

Step2: Calculate the instantaneous rate of change for $x=-2$

Substitute $x = - 2$ into $f'(x)$: $f'(-2)=12\times(-2)=-24$.

Step3: Calculate the instantaneous rate of change for $x = 0$

Substitute $x = 0$ into $f'(x)$: $f'(0)=12\times0 = 0$.

Step4: Calculate the instantaneous rate of change for $x = 4$

Substitute $x = 4$ into $f'(x)$: $f'(4)=12\times4=48$.

Step5: Calculate the instantaneous rate of change for $x = 8$

Substitute $x = 8$ into $f'(x)$: $f'(8)=12\times8 = 96$.

Answer:

a) -24
b) 0
c) 48
d) 96