Need help with a Domain

B

buyerat

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Sep 20, 2009
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Ok this is the original problem
Use the definition of the derivative (don't be tempted to take shortcuts!) to find the derivative of the function
f(x) = sqrt{8 + 5 x}.
Then state the domain of the function and the domain of the derivative.

I ended up getting the derivative correct : 5/(sqrt(8 + 5x ) + sqrt(8 + 5x))

I am now having trouble with the domain. I am pretty sure all I have to worry about is making sure the final number under the radical does not reach 0 or negative, but I cant seem to get the right answer. After doing some work I found that (-8/5, Infinity) is the domain of the derivative. Unfortunately I figured this would be the domain of the function and not necessarily the derivative so that only made me a little more confused. Any help would be much appreciated. Thank you
 
The definiton of a derivative is \(\displaystyle \lim_{h\to 0}\frac{f(x+h)-f(x)}{h}\)

Therefore, we have \(\displaystyle \lim_{h\to 0}\frac{\sqrt{8+5(x+h)}-\sqrt{8+5x}}{h}\)

Now, multiply top and bottom by the conjugate, \(\displaystyle \sqrt{8+5(x+h)}+\sqrt{8+5x}\), and it will simplify nicely down to the derivative of \(\displaystyle \sqrt{8+5x}\)

When dealing with domains and radicals, we can not have negatives inside the square root, nor division by 0.
 
I got the (-8/5, infinity) for the domain. But I am doing it over the internet and plugged in the answer and it didnt take it. It did take it for the domain of the derivative, but not the function like I thought it would.
 
Be careful. Note that -8/5 is in the domain because it results in 0. Is that a bracket or a parenthesis?.
 
I'm not sure what you're asking or what I need to do. I put (-8/5, infinity) into the online website for the problem and got that as a domain for the DERIVATIVE, but I thought originally that was the answer for the domain of the FUNCTION. So I just plugged it into the derivative domain to see and it was correct there and I have no idea why and now I have no idea what the domain is of the function.
 
You merely have a notation error. Look close. You have \(\displaystyle \left(\frac{-8}{5}, \infty\right)\).

Look at that left parenthesis. Shouldn't that be a bracket because -8/5 is in the domain?.
 
Oh, but a 0 under the radical is ok? Just no negatives right. I think I get it now. Thanks very much for all your help everyone.
 
Yes, a 0 is OK. \(\displaystyle \sqrt{0}=0\). That's right,no negatives in the radical.

Therefore, \(\displaystyle \left[\frac{-8}{5},\infty\right)\) for f(x).
 
Sorry it's been about 10 years since I've taken a math class. Of course it's obvious once I hear it again, but thanks for the zing.
 
I made an A on the first test /shrug. I have another degree and a masters. Someone can be smart even though they have barely looked at math in 10 years. All the small stuff it takes me about 1 breath to hear it and relearn something I 'know' but have not seen in many years. Helps to just hear it again to jog memory. I admit that one was obvious :D
 
buyerat said:
… I have another degree and a masters … it takes me about 1 breath to hear [something forgotten] and relearn [it] …
Click to expand...


Ah, well, with your two degress and your MS, you've clearly studied calculus before. Combined with your quick-recall abilities, the repeat courses should all be smooth sailing.

See ya here again, when your noggin needs jogg'n 8-)

 
Thank everyone! It has been a long time since I've worked on my domain notation skills~.
 
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