What are the extreme values of the function SQR[9 - x^2]?

[force]

What is the difference between a differential and a derivative ?

(SQR[9 - x^2])^2 = 9 - x^2 ok, so what is SQR[9 - x^2] = ?

 

[andrewchang]

for the first question, you can refer to this pdf: http://are.berkeley.edu/courses/ARE211/currentYear/lecture_notes/mathCalculus1-05.pdf

(\sqrt {9-x^2})^2= 9-x^2 and you're asking what \sqrt {9-x^2} equals..?
i don't understand what you are looking for.

 

[VietDao29]

Since you have: \sqrt{9 - x ^ 2} in your equation. Unless you are in the Complex (i.e C), x \ \in \ [-3, 3].
The equation is:
(\sqrt{9 - x ^ 2}) ^ 2 = 9 - x ^ 2
\Leftrightarrow 9 - x ^ 2 = 9 - x ^ 2, which is always true.
But watch out for the condition that x \ \in \ [-3, 3]. So what is the range of \sqrt{9 - x ^ 2}, what's the maximum, and minimum value for this expression?
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By the way, it would certainly be nicer, if you include your thanks in the post (or write something more, instead of writing only 2 lines of the problem you are seeking help for), or otherwise, it'll look like you are demanding others to solve the problem for you, or to guide you (and in this case, it does seem so... ).

 

[force]

well could you factor or simplify this expression SQR[9 - x^2] any further ?

consider y=f(x)= x^3 +3x
dy/dx=3x^2+3 is its derivative with respect to x, then what is its differential dy=f(x) dx ?
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ofcourse I appreciate assistance from you folks, thank you

 

[VietDao29]

well could you factor or simplify this expression SQR[9 - x^2] any further ?

Well, you could factor it to \sqrt{9 - x ^ 2} = \sqrt{(3 - x)(3 + x)}
However, that does not help much to find its maximum and minimum value.
Of course it's true that: \sqrt{9 - x ^ 2} \geq 0, so what's its minimum value?
\sqrt{9 - x ^ 2} \leq \sqrt{9} = 3, because x ^ 2 \geq 0 , \ \forall x \in \mathbb{R}, so what's its maximum value?
Of course \sqrt{9 - x ^ 2} is continuous on the interval [-3; 3], so what are the "possible values" of \sqrt{9 - x ^ 2}?
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In general, you could find the maximum or minimum value of:
ax² + bx + c by complete the square; or by differentiate it, then set its derivative to 0, and solve for x, test if it's a maximum or a minimum, then plug x back to ax² + bx + c.

consider y=f(x)= x^3 +3x
dy/dx=3x^2+3 is its derivative with respect to x, then what is its differential dy=f(x) dx ?

The bolded part is wrong, it should read: dy = f'(x) dx.
So: dy = 3(x² + 1) dx

 

[roger]

x^2 can't be bigger than or equal to 0 as you already stated xforall x \in \mathbb{R}

 

[VietDao29]

x^2 can't be bigger than or equal to 0 as you already stated xforall x \in \mathbb{R}

Err, I don't get this. Can you clarify it please...
I don't understand. Did I make any mistakes?

 

[roger]

you stated that the domain / range was the reals so x^2 can't be bigger than 9

 

[VietDao29]

you stated that the domain / range was the reals so x^2 can't be bigger than 9

I didn't state that the domain, or the codomain, or even the image is the real! I stated that the domain is [-3; 3] (post #3).
And what's on Earth does the fact x² cannot be greater than 9 have anything to do here?
I explain that:
\sqrt{9 - x ^ 2} \leq \sqrt{9} = 3 is true because: (x ^ 2 \geq 0 , \ \forall x \in \mathbb{R}, this statement alone is again true, and because it's true for all x in the reals, it must be also true for x on the interval [-3; 3], and thus the above statement \sqrt{9 - x ^ 2} \leq \sqrt{9} = 3 is entirely true).
And still, forgive me, but I don't understand what you said...

 

[roger]

That's alright. It was a misunderstanding. I don't understand the relevance of this though, to what the OP asked ?

 

[VietDao29]

That's alright. It was a misunderstanding. I don't understand the relevance of this though, to what the OP asked ?

Uhmm, as I interprete, the OP might be asking for the image of the function: \sqrt{9 - x ^ 2} if the following condition is met:
(\sqrt{9 - x ^ 2}) ^ 2 = 9 - x ^ 2. (that condition is always true if \sqrt{9 - x ^ 2} is defined).
So I ask him to find the minimum value and maximum value of \sqrt{9 - x ^ 2}.
Yes, it's true that the OP question's not bring very clear.