# Fundamental therom problem

1. Sep 1, 2004

### Alem2000

My professer told us to think about this problem. I have the answer in my solutions manual but I want to know how I would go through the thinking process to solve it...(and others like it) The question says

"Find a function f and a number a such that

$$6+\int_{a}^{x}\frac{f(t)}{t^2}dt=2\sqrt{x}$$
for all x greater than zero"

Last edited by a moderator: Sep 1, 2004
2. Sep 1, 2004

### Tide

Differentiate the equation!

3. Sep 1, 2004

### Alem2000

There was no point in that reply.. :grumpy:
Anyone out there that can really help?

Last edited: Sep 1, 2004
4. Sep 2, 2004

### Tide

Begging your pardon but I told you exactly what you need to do!

5. Sep 2, 2004

### Muzza

WTF? Tide's post was immensely helpful and practically gives you the entire solution.

6. Sep 2, 2004

### Gokul43201

Staff Emeritus
I second that. Tide's given you a correct way to solve the problem.

7. Sep 2, 2004

### HallsofIvy

Since you did not grasp what Tide said, here's another way of looking at it:
$$\int_{a}^{x}\frac{f(t)}{t^2}dt=2\sqrt{x}- 6$$

Do you notice that the right hand side is a constant?

8. Sep 2, 2004

### Muzza

The right hand side is not a constant...

9. Sep 2, 2004

### HallsofIvy

OMIGOD! I stared at that repeatedly and saw $2\sqrt{2}$.

10. Sep 2, 2004

### HallsofIvy

Alem2000: as Tide said, differentiate both sides:
The derivative of $$\int_{a}^{x}\frac{f(t)}{t^2}dt$$ is $$\frac{f(x)}{x^2}$$ (that's the "fundamental theorem your title referred to) and the derivative of $$2\sqrt{2}= 2(x^{1/2})$$ is $$x^{-1/2}$$.

Set them equal and solve for x.

11. Sep 2, 2004

### Alem2000

:rofl: :rofl: :rofl: OOOOOO! I think I made that way more complicated then it was. Thanks alot Tide...sorry about the frustration

12. Sep 2, 2004

### HallsofIvy

I did it again! I wrote $$2\sqrt{2}$$ when I meant $$2\sqrt{x}$$!

13. Sep 2, 2004

### JonF

And now you are throwing factorial signs about, tisk tisk ;)