Find & Simplify Difference Quotient of f(x)=sq root x

In summary, the question is asking for the difference quotient of a function that takes a x+h input and produces a sq root(x+h) output. The difference quotient is found by multiplying numerator and denominator by sq root(x+h)+ sqrt(x).
  • #1
kuahji
394
2
The question is "find and simplify the difference quotient."

Given function

f(x)=sq root of x

So what I did is insert (x+h) under the radical & got

sq root of (x+h), then I subtracted the sq root of x (original function)

My answer was sq root [(x+h) - sq root (x)] / h

The answer in the back of the book is 1 / [sq root (x+h) + sq root (x)]

I'm not understanding where I'm going wrong, the other problems didn't give me a problem, just this one.
 
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  • #2
multiply numerator and denominator by [sqrt(x+h) + sqrt(x)] ...

[sqrt(x+h) - sqrt(x)]/h * [sqrt(x+h) + sqrt(x)]/[sqrt(x+h) + sqrt(x)] =

[(x+h) - x]/[h[sqrt(x+h) + sqrt(x)]] =

h/[h[sqrt(x+h) + sqrt(x)]] =

1/[sqrt(x+h) + sqrt(x)]
 
  • #3
Ok, thanks!

So is there a reason why its listed in that form instead the other? I think that is what I'm not understanding.
 
  • #4
you'll find out why when you have to take the limit of the difference quotient as h -> 0 ... it's a calculus concept.
 
  • #5
kuahji said:
The question is "find and simplify the difference quotient."

Given function

f(x)=sq root of x

So what I did is insert (x+h) under the radical & got

sq root of (x+h), then I subtracted the sq root of x (original function)

My answer was sq root [(x+h) - sq root (x)] / h
Not sure if this is a typo of a major misunderstanding. What you said you did is correct but you should have [sq root(x+h)- sq root(x)]/h. Do you see the difference? That is:
[tex]\frac{\sqrt{x+h}- \sqrt{x}}{h}[/tex]
where yours is
[tex]\frac{sqrt{x+h- sqrt{x}}}{h}[/tex]

The answer in the back of the book is 1 / [sq root (x+h) + sq root (x)]

I'm not understanding where I'm going wrong, the other problems didn't give me a problem, just this one.
As was said before, rationalize the numerator: multiply numerator and denominator by sq root(x+h)+ sqrt(x).
[tex]\frac{\sqrt{x+h}-sqrt{x}}{h}\frac{\sqrt{x+h}+\sqrt{x}}{\sqrt{x+y}+\sqrt{x}}[/tex]
 

1. How do I find the difference quotient of f(x)=√x?

To find the difference quotient of a function, you will need to use the formula:
(f(x+h)-f(x))/h
In this case, you will plug in √(x+h) for f(x+h) and √x for f(x). Then, simplify the equation to get the difference quotient.

2. Is there a simplified form of the difference quotient for f(x)=√x?

Yes, the simplified form of the difference quotient for √x is 1/2√(x+h)+√x)/h. This can be derived by simplifying the initial formula and using the property of radicals.

3. What does the difference quotient represent for a function?

The difference quotient represents the slope of a secant line that passes through two points on a curve. In this case, it represents the average rate of change of the square root function between two points.

4. Can the difference quotient be used to find the derivative of a function?

Yes, the difference quotient is one of the first principles used to find the derivative of a function. It is used to approximate the slope of a curve at a specific point, which is the definition of a derivative.

5. Are there any restrictions when using the difference quotient for f(x)=√x?

Yes, there are some restrictions when using the difference quotient for √x. The value of x+h must be greater than or equal to 0, since the square root function is undefined for negative numbers. Additionally, h cannot equal 0, as this would result in a division by 0 error.

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