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Mth 271
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.
** Question Form_labelMessages **
Question 31 chapter 1 section4
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Evaluate the difference quotient and simplify the result.
31 g(x)= sq. rt. of x+1
g(x+ delta x) - g(x)/ delta x
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You aren't consistent in using parentheses where they are needed. So I can't tell whether the function is sqrt(x+1) or sqrt(x) + 1.
If it is the former then the 1 has to stay within the parentheses. If the latter your expression is OK.
Assuming that your expression means sqr(x) + 1, then
sq. rt. of (x+delta x) +1 - sq. rt. of x+1
should be written
sq. rt. of (x+delta x) +1 - (sq. rt. of x+1)
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Using the solution on Calc Chat
I understand that I am subbing sq. rt. of x+1 into the numerator to get:
What I do not understand is
1. Why did I not sub. sq. root of x+1 into the denominator also?
In the first step the denominator is delta x. It seems that it should be delta sq. root of x.
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The difference quotient represents an average rate of change of f(x) with respect to x, which by definition is (change in f(x)) / (change in x).
The numerator of the difference quotient is the change in the value of the function, the denominator is the change in the value of x.
The numerator therefore represents the change that results in the value of the function when x changes by `dx, and the denominator is just `dx, which is the change in x.
So the denominator will always be `dx.
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2.Why was both the numerator and denominator multiplied by sq. rt. of x+ delta x +1 + sq. rt. of x +1?
I thought an expression was simplified if there were no radicals in the denominator. This takes the radicals out of the numerator and puts them in the denominator.
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I need more notes on this. I am not following the one example in the book at all, and there is no explanation of the steps on Calc Chat.
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#$&*
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The goal isn't to simplify the expression, it is to find the limiting value of the expression as `dx approaches zero. The multiplication of the denominator accomplishes this.
If the parentheses are as I'm assuming, the numerator
sq. rt. of (x+delta x) +1 - (sq. rt. of x+1)
becomes
sq. rt. of (x+`dx) +1 - sq. rt. of x-1
which is equal to
sqrt(x+`dx) - sqrt(x).
Multiplying numerator and denominator by sqrt(x+`dx) + sqrt(x) gives you
(x + `dx - x) / (`dx * (sqrt(x+`dx) + sqrt(x) ) )
which is just equal to
`dx / (`dx * (sqrt(x+`dx) + sqrt(x) ) ),
which in turn simplifies to
1 / (sqrt(x+`dx) + sqrt(x) ).
As `dx approaches zero, this approaches
1 / (sqrt(x) + sqrt(x) ) = 1 / (2 sqrt(x) ).
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This is a challenging problem and you're doing very well by asking good questions about it.
Do get in the habit of being very careful with your use of parentheses to indicate grouping.
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