Simple Precalculus: Average rate of change

AI Thread Summary
To find the average rate of change of g(x) = 1/x from x = 1 to x = a, the formula used is (g(a) - g(1)) / (a - 1). The calculations yield g(1) = 1 and g(a) = 1/a, leading to the expression ((1/a) - 1) / (a - 1). The correct simplification reveals the average rate of change as -(1/a). The discussion highlights the importance of careful algebraic manipulation and recognizing basic rules to avoid errors in problem-solving.
DarrenM
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Homework Statement


Find the average rate of change:
g(x)=1/x; x = 1, x = a


Homework Equations


(f(b)-f(a))/(b-a) = Average rate of change


The Attempt at a Solution


I've spent far too much time on this problem, but I know I'm making a stupid error and I just can't pin it down. So far, the steps I'm pretty confident about:

g(1) = 1/1 = 1
g(a) = 1/a

Average rate of change = ((1/a)-1)/(a-1)

According to the book the answer is -(1/a)... but I'm just not seeing it.
 
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1/a-1 = (1-a)/a :wink:
 
Yea, got it now. It was just as ridiculous an oversight as I had originally thought. I kept getting to the point in the problem where I had (1-a)/(a(a-1)) and going, "Oh, I can't divide those binomials!" Factor out a -1 from the numerator, cancel out the resulting binomial with the denominator and there's the answer... staring me right in the face. I'm rather embarrassed that I had so much trouble with such a simple problem, and that I forgot such a basic rule. Thanks for the help.
 

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