- #1

scorpa

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Right now I'm taking derivatives in relation to curve sketching, and I just wanted to make sure I am doing these right.

The first question is to determine using the first derivative where the graph of y =

__x__is rising.

(x^2)+1

This is what I have done so far:

I tried to find the derivative using the quotient rule...

__((x^2)+1)(d/dx)(x)-(x)(d/dx)((x^2)+1)__

((x^2)+1)^2

__((x^2)+1)(1)-(x)(2x)__

((x^2)+1)^2

__(-x^2)+1__

((x^2)+1)^2

That is the value I found for the first derivative, although I am unsure whether I have done it right or not. Then to find where the curve was rising I said that since the derivative must be greater than 0, the value I found for the derivative must be greater than zero. Then I tried solving that and I ended up with -x^2 + 1 > x^4 + 2x^2 + 2 and although I realize that that is not quite finished yet, it just doesn't seem right to me. I am obviously going wrong somewhere, but where I do not know.

The second question asks you to determine where f(x) = x^2(1-x) is concave upward using the second derivative. This is what I did:

(x^2)(d/dx)(1-x) + (1-x)(d/dx)(x^2) =

(x^2)(-1) + (1-x)(2x) =

(-3x^2) + 2x This is the first derivative I found using the product rule and the steps shown above.

Then I found the second derivative to be -6x + 2

Have I found the derivatives correctly for these questions? It just seems like something is very wrong, mostly with the first question. Any help you guys can give me I would really appreciate. Thanks in advance.