Question

How do you find the average or mean slope of the function \[f\left( x \right)=3{{x}^{3}}-2x\] on the interval \[\left[ -4,4 \right]\]?

Answer 1

Hint: From the question given, we had been given that, \[f\left( x \right)=3{{x}^{3}}-2x\]. And, we have been asked to find the average or mean slope of the function on the interval \[\left[ -4,4 \right]\]. To find the average or mean slope of the function in the given interval, first, we have to evaluate the given function in the provided intervals. Then, we have to divide by the interval.

Complete step by step answer:
Now considering from the question we need to find the average or mean slope of the function $ f\left( x \right)=3{{x}^{3}}-2x $ in the interval $ \left[ -4,4 \right] $ .
First of all, we have to evaluate the given function in the provided intervals.
As the process mentioned, first let us evaluate \[f\left( 4 \right)\].
We have been already given in the question,
\[f\left( x \right)=3{{x}^{3}}-2x\]
\[\Rightarrow f\left( 4 \right)=3{{\left( 4 \right)}^{3}}-2\left( 4 \right)\]
On furthermore simplifying we get,
\[f\left( 4 \right)=192-8\]
\[\Rightarrow f\left( 4 \right)=184\]
Now, as the process mentioned, let us evaluate \[f\left( -4 \right)\].
We have been already given in the question,
\[f\left( x \right)=3{{x}^{3}}-2x\]
\[\Rightarrow f\left( -4 \right)=3{{\left( -4 \right)}^{3}}-2\left( -4 \right)\]
On furthermore simplifying, we get,
\[f\left( -4 \right)=-192-\left( -8 \right)\]
\[\Rightarrow f\left( -4 \right)=-192+8\]
\[\Rightarrow f\left( -4 \right)=-184\]
Now, as we have been already discussed above, to find the average or mean slope of the function, we have to evaluate the function in the interval and then divide by the interval,
So, to find mean slope, we have to do \[\dfrac{f\left( 4 \right)-f\left( -4 \right)}{4-\left( -4 \right)}\]
Let us evaluate this to find the average or mean slope.
\[\dfrac{f\left( 4 \right)-f\left( -4 \right)}{4-\left( -4 \right)}=\dfrac{184-\left( -184 \right)}{8}\]
\[\Rightarrow \dfrac{f\left( 4 \right)-f\left( -4 \right)}{4-\left( -4 \right)}=\dfrac{184+184}{8}\]
\[\Rightarrow \dfrac{f\left( 4 \right)-f\left( -4 \right)}{4-\left( -4 \right)}=\dfrac{368}{8}\]
\[\Rightarrow \dfrac{f\left( 4 \right)-f\left( -4 \right)}{4-\left( -4 \right)}=46\]
Hence, we got the value of the average or mean slope of the function.

Note:
 We should be well aware of the concepts like slope and average slope to get this type of problem solved. We should be very careful while doing the calculation. We should also be well known about evaluating functions in the given intervals. We should make good use of the interval provided in the given question. Generally, sometimes many of us don't know this way and we go on like calculating values for all the integers $ x $ values of $ f\left( x \right) $ and then finding the mean of them is such a lengthy process and not efficient.