mth 158
Your 'question form' report has been received. Scroll down through the document to see any comments I might have inserted, and my final comment at the end.
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In chapter 3. Problems like the average rate of change. There is an example in the book that I do not understand. Below is the question.
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Find the average rate of change of f(x) = 3x^2
From 1 to 3
The average rate of change of f(x) = 3x^2 from 1 to 3 is
f(3) - f(1) / 3 - 1 the book starts out with this as the first step. But, I don't understand. I know the numberator is the change in y so where do they get 3 and 1 for the y values? The denominator is the change in x and I see that 3 - 1 would be logical here.
= 27 - 3 / 3 - 1 this is the second step. Where did the 27 come from? Is that the f(x) = 3x^2? And if it is, that doesn't make much sense to me either. It appears to me that it would be like this:
f(3x^2) - f(3x^2) / 3 - 1 but that really doesn't make sense either.
You appear to be confused on the f(x) notation.
The definition of this function is
f(x) = 3 x^2
means, for example, that when x = 1 we substitute 1 for x; in this case f(x) = 3 x^2 becomes
f(1) = 3 * 1^2.
Applying the definition to x = 3 we substitute 3 for x, obtaining
f(3) = 3 * 3^2.
Thus the average rate of change is
( f(3) - f(1) ) / (3 - 1) = ( 3 * 3^2 - 3 * 1^2) / (3 - 1) = (27 - 3) / (3 - 1) = 24/2 = 12.
I am a little confused on this. Where does the x^2 come into any of this?
I think the 27 - 3 is where the 3 is raised to the 2nd power and then multiplied by 3 (which is the 3x^2) I'm not sure, but that makes more sense. If that's true, then 3 would be the value in 27 - 3.
But then that leaves me with a question of how can the change in x be the same numbers as the y value?
I am guessing at this point in the question that I have confused you totally as to what I'm saying, and I don't know how to ask it any other way. Maybe if you just explained how the values were put into the equation, I might get it.
See if my explanation helps. You're welcome to follow up with additional questions.