## anonymous 5 years ago If f is the antiderivative of x^2/(1+x^5) such that f(1)=0, then what would f(4) be?

1. amistre64

anti derive it :)

2. amistre64

it gives you the equation to antiderive and the initial condition of (1,0).... the trick is figuring out a way to get it to antiderive

3. anonymous

The limits of integrals are from 1 to 4. I tried approximating it but f'(1) is not 0.

4. amistre64

i dont think this its asking for the interval of integration when it says f(1) and f(4); its asking you to find the equation that this was gotten from, pinning it down to the point (1,0) and getting the answer for (4,y)

5. amistre64

if im wrong, let me know :)

6. anonymous

Will it be right to say? $f = \int\limits_{a}^{b}x^2/(1+ x^5) dx$ $f' = x^2/(1+x^5)$

7. amistre64

that looks proper, except for the a and b part... it gave no interval to find an "area" for right?

8. anonymous

isn't the interval [1, 4]?

9. amistre64

nope, there is no interval. they want to know a specific value of f at x=4. they give you f(1) = 0 so that you have something to anchor this f(x) with. otherwise it just floats up and down the y axis like a roaming gnome.

10. amistre64

tell me, can a derivative have more than 1 antiderivative?

11. anonymous

Yes, I think.

12. amistre64

lets verify that :) whats the derivative of these 2 equations: y = 3x^2 +6x +10 y = 3x^2 +6x -3

13. anonymous

y'=6x+6

14. amistre64

when you suit it back up it begins to float up and down the y axis right?

15. anonymous

I see how integrating a derivative can produce a family of functions with a different vertical translations.

16. amistre64

good, then when we find a suitable antiderivative, we add a constant to it, a generic "+C" as a place holder; apply the "initial condition" that f(1)=0 and sove for "+C" then we have a valid function with which to determine the value of f(4)

17. amistre64

the real issue becomes, what is the integral of that function :) I have not seen an easy way to get it....

18. anonymous

Since my teacher rushed through approximation methods today, I guess Ill have to use that. Trapezoidal approximation maybe?

19. amistre64

thats still looking to find the area under the curve, but that is not the question you posted above. do you have the question right?

20. anonymous

Im pretty sure that was the question. But is the area under x^2/(1+x^5) f?

21. amistre64

No, the area underneath x^2/(1+x^5) is not f. "f" is the function that will originally be derived down to: x^2/(1+x^5)

22. amistre64

its like you found someones wallet and are trying to find the owner by the limited clues available to you.

23. amistre64

we know that it is some type of cubic rational function; that it has a critical point at x=0.... and if we take another derivative we might be able to see some other clues to it, but figureing out the actual function it came from will be tricky nonetheless

24. anonymous

I got an email from my teacher. She said to estimate it and gave three choices: 0.016, 0.376, 0.629.

25. amistre64

then you try that trap rule and see if that gets you an answer near one of these, if so, then go for it :) but i think I am right about it not being an "area" question.... but Ive been known to be wrong :)

26. anonymous

Haha... I'm so clueless in calc.

27. anonymous

Thanks anyway.

28. amistre64

wish I coulda been more help :) good luck!!