Why do antiderivatives of even functions not follow the usual graphing rules?

[DivGradCurl]

The general idea I have in mind when it comes to analyzing a graph that has a derivative $$f$$ and its antiderivative $$F$$ ($$\mathrm{C}=0$$) is the following:

- When $$f$$ is positive, $$F$$ increases.
- When $$f$$ is negative, $$F$$ decreases.

However, I came across a couple of problems in my textbook that don't seem to work exactly that way. That's when either the derivative or the integral is an even function. Here is what I'm talking about:


$$\int \frac{x}{\sqrt{x^2 +1}} \: dx = \sqrt{x^2 +1} + \mathrm{C} \qquad (1)$$

$$\int \tan ^2 \theta \sec ^2 \theta \: d\theta = \frac{\tan ^3 \theta}{3} + \mathrm{C} \qquad (2)$$

The graphs are located at http://photos.yahoo.com/thiago_j

Note: the blue curves represent $$f$$, while the red ones represent $$F$$. The elements from Eq. (1) are depicted in "calculus-5-5---34" while those from Eq. (2) appear in "calculus-5-5---36".

Is this correct?

Do I need to modify the domain so that I only show the part of the plot that work as expected?

Thank you very much

 

[OlderDan]

What do you think is wrong with the curves for #1? f is the integrand represented by the blue curve and R is the antiderivative represented by the red curve, so the blue curve is the derivative of the red curve

 

[luther_paul]

I m trying to understand your problem, but i can't see our point. The graphs of the functions seems to be ok. on equation (1), $$f$$ is negative from $$(-\infty,0)$$ and $$F$$ is decreasing on that interval. And also, $$f$$ is positive on $$(0,+\infty)$$ and $$F$$ is increasing. Same is true for the second one. I think nothing is wrong with the graphs, or maybe i just didn't get your problem.

 

[whozum]

Thats exactly what's happening in this plot.. When the derivative is negative, the red graph is decreasing, when its positive, its decreasing..

 

[HallsofIvy]

Thats exactly what's happening in this plot.. When the derivative is negative, the red graph is decreasing, when its positive, its decreasing..

I think you meant to say "when it's positive, it's increasing"!

 

[DivGradCurl]

Well, I've just taken a second look at both graphs and it now makes sense. I was a little confused. Thanks for your input, guys.

 

[whozum]

I think you meant to say "when it's positive, it's increasing"!

Yup, I really should re-read my posts before submitting.