Question

sections 1.3 - 1.4
score: 3/21 answered: 3/21
question 4
find the average rate of change of ( g(x)=8 x^{2}+\frac{6}{x^{4}} ) on the interval (-1,4)

Explanation:

Step1: Recall the formula for average rate of change

The average rate of change of a function \(y = g(x)\) over the interval \([a,b]\) is \(\frac{g(b)-g(a)}{b - a}\). Here \(a=-1\), \(b = 4\), and \(g(x)=8x^{2}+\frac{6}{x^{4}}\).

Step2: Calculate \(g(-1)\)

Substitute \(x=-1\) into \(g(x)\):
\(g(-1)=8(-1)^{2}+\frac{6}{(-1)^{4}}=8 + 6=14\).

Step3: Calculate \(g(4)\)

Substitute \(x = 4\) into \(g(x)\):
\(g(4)=8\times(4)^{2}+\frac{6}{(4)^{4}}=8\times16+\frac{6}{256}=128+\frac{3}{128}=\frac{128\times128 + 3}{128}=\frac{16384+3}{128}=\frac{16387}{128}\).

Step4: Calculate the average rate of change

\(\frac{g(4)-g(-1)}{4-(-1)}=\frac{\frac{16387}{128}-14}{5}=\frac{\frac{16387-14\times128}{128}}{5}=\frac{\frac{16387 - 1792}{128}}{5}=\frac{\frac{14595}{128}}{5}=\frac{14595}{128\times5}=\frac{2919}{128}=22.8046875\).

Answer:

\(22.8046875\)