Find an F(x) Value Using FTC

1. Dec 5, 2012

B18

1. The problem statement, all variables and given/known data

If f is continuous, f(3)=10, and ∫ from (3,8)f'(x)dx=16, find f(8)

3. The attempt at a solution
I attempted to do this by evaluating 16x] from 3 to 8 using FTC. I got 80. And in no way shape or form did i get any closer to the answer from there.

Last edited: Dec 5, 2012
2. Dec 5, 2012

SammyS

Staff Emeritus
I suspect that you mean that f is continuous. There is no F mentioned anywhere else.

Isn't $\displaystyle \int f'(x)\,dx=f(x)+C\ ?$

3. Dec 5, 2012

B18

Yes i apologize, the question has "f is continuous" not F.

4. Dec 5, 2012

SammyS

Staff Emeritus

My rhetorical question was meant to be a hint.

Try it.

5. Dec 5, 2012

B18

Would this be a correct way of finding the answer?
f(3)=10 so f(8)=10+C
then ∫(3,8) f'(x)dx=16
f(x)+c=16
c=16 so..
f(8)=10+16=26.

6. Dec 5, 2012

SammyS

Staff Emeritus
Well, f(8) is 26, but I don't believe you can determine C from the information given.

According to the FTC: $\displaystyle \ \int_{3}^{8}f'(x)\,dx=f(8)-f(3)\ .$

7. Dec 5, 2012

B18

i did ∫(3,8)=f(8)-f(3)=16
f(8)=16+f(3)
f(8)=16+10=26
correct way of completing this problem?

8. Dec 5, 2012

Staff: Mentor

Your notation could use some work though.

$$\int_3^8 f'(x) dx = 16$$
Replacing the integral above, for the reason that SammyS gave, we have
f(8) - f(3) = 16
=> f(8) - 10 = 16 (It's given that f(3) = 10.)
=> f(8) = 26

9. Dec 5, 2012

B18

Thanks guys. I am going to sketch a picture or two so i fully understand FTC and how this problem was solved. Thanks again.

10. Dec 5, 2012

Staff: Mentor

I'm not sure a picture would be helpful, but don't let that stop you. The idea is pretty simple.

The 2nd part of the FTC is usually presented like this (with some of the fine print about continuity omitted)If F is an antiderivative of f, then
$$\int_a^b f(x)dx = F(b) - F(a)$$

In your problem, f is an antiderivative of f', so
$$\int_3^8 f'(x)dx = f(8) - f(3)$$

Antidifferentiation and differentiation are pretty much inverse operations, so if you antidifferentiate something that is already a derivative, you get back the original function.

11. Dec 5, 2012

B18

Your right a picture is not extremely beneficial. So in the definition of the FTC you provided the f(x)dx... the f(x) would be considered (a) the function and F(a) and F(b) are antiderivatives/integrals that are being evaluated at 3, 8 respectively. So to make sure i completely understand this for our test tomorrow, if we had ∫(3 to 8) f''(x) is it safe to assume that equals f'(8)-f'(3)

12. Dec 5, 2012

B18

Having trouble finding how to place the bounds on the integral correctly. Sorry :(

13. Dec 5, 2012

Staff: Mentor

Yes.

14. Dec 5, 2012

Staff: Mentor

Sammy and I used LaTeX, which is essential for doing nice definite integrals. The script looks like this

[ tex]\int_{3}^{8} f(x) dx [/ tex]

To make this display without rendering as an integral, I had to insert a couple of extra spaces at the beginning of the tex and /tex tags. If you remove the spaces, it looks like this:
$$\int_{3}^{8} f(x) dx$$

Instead of [ tex ] and [ /tex ] tags, I usually use  at the beginning and end, again without the space.