help me solve this | 8 parts remaining\nuse the formula for instantaneous rate of change, approximating the…

help me solve this | 8 parts remaining\nuse 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.\nf(x)=3x^x; x = 3\nuse the formula for instantaneous rate of change.\nlim_{h→0} (f(x + h)-f(x))/h\nsubstitute the value of x in the formula and rewrite using the given function.\nlim_{h→0} (f(3 + h)-f(3))/h=lim_{h→0} (3(3 + h)^(3 + h)-3□)/h

help me solve this | 8 parts remaining\nuse 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.\nf(x)=3x^x; x = 3\nuse the formula for instantaneous rate of change.\nlim_{h→0} (f(x + h)-f(x))/h\nsubstitute the value of x in the formula and rewrite using the given function.\nlim_{h→0} (f(3 + h)-f(3))/h=lim_{h→0} (3(3 + h)^(3 + h)-3□)/h

Answer

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\rightarrow0}\frac{f(x + h)-f(x)}{h}).

Step2: Substitute (x = 3) into the formula

We get (\lim_{h\rightarrow0}\frac{f(3 + h)-f(3)}{h}=\lim_{h\rightarrow0}\frac{3(3 + h)^{(3 + h)}-3\times3^{3}}{h}). First, simplify (3\times3^{3}=3^{4}=81). So we have (\lim_{h\rightarrow0}\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)