Question

help me solve this | 8 parts remaining
use the formula for instantaneous rate of change, approximating the limit by using smaller and smaller values of h, to find the instantaneous rate of change for the given function at the given value.
f(x)=3x^x; x = 3
use the formula for instantaneous rate of change.
lim_{h→0} (f(x + h)-f(x))/h
substitute the value of x in the formula and rewrite using the given function.
lim_{h→0} (f(3 + h)-f(3))/h=lim_{h→0} (3(3 + h)^(3 + h)-3□)/h

Explanation:

Step1: Recall the function and x - value

The function is \(f(x)=3x^{x}\) and \(x = 3\). The formula for the instantaneous rate of change is \(\lim_{h
ightarrow0}\frac{f(x + h)-f(x)}{h}\).

Step2: Substitute \(x = 3\) into the formula

We get \(\lim_{h
ightarrow0}\frac{f(3 + h)-f(3)}{h}=\lim_{h
ightarrow0}\frac{3(3 + h)^{(3 + h)}-3\times3^{3}}{h}\). First, simplify \(3\times3^{3}=3^{4}=81\). So we have \(\lim_{h
ightarrow0}\frac{3(3 + h)^{(3 + h)}-81}{h}\).
To approximate the limit, we can use smaller and smaller values of \(h\).
Let's use a calculator to evaluate the difference - quotient for small values of \(h\).
If \(h = 0.01\):
\[

$$\begin{align*} \frac{3(3 + 0.01)^{(3+0.01)}-81}{0.01}&=\frac{3\times3.01^{3.01}-81}{0.01}\\ 3.01^{3.01}&=3.01^{3}\times3.01^{0.01}\approx27.2703\times1.00697 = 27.4587\\ 3\times27.4587-81&=82.3761 - 81=1.3761\\ \frac{1.3761}{0.01}&=137.61 \end{align*}$$

\]
If \(h = 0.001\):
\[

$$\begin{align*} \frac{3(3 + 0.001)^{(3+0.001)}-81}{0.001}&=\frac{3\times3.001^{3.001}-81}{0.001}\\ 3.001^{3.001}&=3.001^{3}\times3.001^{0.001}\approx27.027\times1.000693=27.0457\\ 3\times27.0457-81&=81.1371 - 81 = 0.1371\\ \frac{0.1371}{0.001}&=137.1 \end{align*}$$

\]
If \(h=- 0.001\):
\[

$$\begin{align*} \frac{3(3-0.001)^{(3 - 0.001)}-81}{-0.001}&=\frac{3\times2.999^{2.999}-81}{-0.001}\\ 2.999^{2.999}&=2.999^{3}\times2.999^{-0.001}\approx26.973\times0.999307 = 26.953\\ 3\times26.953-81&=80.859 - 81=-0.141\\ \frac{-0.141}{-0.001}&=141 \end{align*}$$

\]
As \(h\) approaches \(0\), the instantaneous rate of change is approximately \(137.1\).

Answer:

The instantaneous rate of change of \(y = f(x)=3x^{x}\) at \(x = 3\) is approximately \(137.1\)