Question

The function f(x) = x3 describes the volume of a cube, f(x), in cubic inches, whose length, width, and height each measure x inches. If x is changing, find the average rate of change of the volume with respect to x as x changes from 1 inches to 1.1 inches.
A. 2.33 cubic inches per inch
B. -3.31 cubic inches per inch
C. 23.31 cubic inches per inch
D. 3.31 cubic inches per inch

Answer 1

Average rate of change over [1,1.1] \(=\dfrac{f(1.1)-f(1)}{1.1-1}\)

Answer 2

And f would be... 3, or am I crazy?

Answer 3

Well,
\(f(1.1)=1.1^3=1.331\)
\(f(1)=1^3=1\).

So, the average rate of change is \(\dfrac{1.331-1}{1.1-1}=\dfrac{.331}{.1}=.331\). If that's what you meant, then no, you're not crazy (probably).

Answer 4

You can also reason by looking at the graph. Over the interval \([1,1.1]\), \(f(x)=x^3\) is obviously increasing, right? So you can immediately eliminate the negative answer.

The other answers are simply too big. The interval \([1,1.1]\) is pretty small, and the values of \(f(1)\) and \(f(1.1)\) are pretty close to each other.

Answer 5

Nope, nope, absolutely crazy. For some reason I thought I needed to multiply by 3.
Makes absolute sense though. Thanks so much! And your name made me laugh.