# How do you find the instantaneous rate of change of g(t)=3t^2+6 at t=4?

Oct 20, 2016

Compute the first derivative and evaluate it at $t = 4$

$g ' \left(4\right) = 24$

#### Explanation:

Compute the first derivative:

$g ' \left(t\right) = 6 t$

Evaluate it at $t = 4$

$g ' \left(4\right) = 24$

Oct 20, 2016

It depends on what you have in your mathematical toolbox.

#### Explanation:

If you have learned the power rule, constant multiple rule and derivative of a constant, you can quickly find the derivative of $g$.

$g ' \left(t\right) = 3 \left(2 {x}^{1}\right) + 0 = 6 t$.

To find the instantaneous rate of change at a particular value of $t$, evaluate the derivative at that value of $t$.

At $t = 4$ the instantaneous rate of change is $g ' \left(4\right) = 6 \left(4\right) = 24$.

If you are using a definition then it depends on the particular definition you are using.

There are several ways to express the definition.

One way of expressing it is to give:

The rate of change of $g$ with respect to $t$ at $t = 4$ is

${\lim}_{t \rightarrow 4} \frac{g \left(t\right) - g \left(4\right)}{t - 4}$.

Another is

The rate of change of $g$ with respect to $t$ at $t = 4$ is

${\lim}_{h \rightarrow 0} \frac{g \left(4 + h\right) - g \left(4\right)}{h}$.

Still another is

The rate of change of $g$ with respect to $t$ at $t$ is

${\lim}_{h \rightarrow 0} \frac{g \left(t + h\right) - g \left(t\right)}{h}$.

(After we find this, we evaluate at $t = 4$.

Here is the work for the first definition above.

${\lim}_{t \rightarrow 4} \frac{g \left(t\right) - g \left(4\right)}{t - 4} = {\lim}_{t \rightarrow 4} \frac{\left[3 {t}^{2} + 6\right] - \left[3 {\left(4\right)}^{2} + 6\right]}{t - 4}$ (Observe that is we substitute $t = 4$, we get the indeterminate form $\frac{0}{0}$.)

$= {\lim}_{t \rightarrow 4} \frac{3 {t}^{2} + 6 - 48 - 6}{t - 4}$ $\text{ }$ (Still $\frac{0}{0}$)

$= {\lim}_{t \rightarrow 4} \frac{3 {t}^{2} - 48}{t - 4}$

$= {\lim}_{t \rightarrow 4} \frac{3 \left({t}^{2} - 16\right)}{t - 4}$

 = lim_(trarr4) (3(t+4)(t-4)))/(t-4)

$= {\lim}_{t \rightarrow 4} 3 \left(t + 4\right)$

$= 3 \left(4 + 4\right) = 24$