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Problem 58 Medium Difficulty

Draw a graph of $ f $ and use it to make a rough sketch of the antiderivative that passes through the origin

$ f(x) = \sqrt{x^4 - 2x^2 + 2} - 2 $, $ \quad -3 \leqslant x \leqslant 3 $

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Video Transcript

So we're gonna be drawing the graph F. And then we're going to be looking to find its anti derivative. So the graph and F. Of this case is the function square root. Sure equals the square root of extra forest. Nice. Two X squared. That's true. I asked you this is our graph right here and we want to show the anti derivative of the function. So we see that here uh F is positive from negative infinity to about 0.165 Um So since that's positive, are anti derivative will be increasing. And then when it reaches that point um it's going to be from negative 1.65 to 1.65 It's deco it's negative. So our function will be decreasing and then it's gonna reach a point where it increases. So we'll look at this here. This will be something like um X. Cubed plus two X. Mm. Uh huh. So it'll look something like this but centered at the origin. So if this was shifted over more centered at the origin, we would get something like this. Um This is, but if we had something more like that shifted the origin, that would be the behavior we're looking at the increase, which is the point of decreases increases again.