How Do You Calculate f(8) Using the Fundamental Theorem of Calculus?

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In summary: There are other ways to do this, but this is simple.) So I would have typed this:$$\int_{3}^{8} f(x) dx$$And I get this:$$\int_{3}^{8} f(x) dx$$If you want to experiment with any of this, you can use the "Reply with quote" button to see how things are done.If you want to learn how to type LaTeX, go to the LaTeX section here on PF and read the sticky threads at the top of the page. There is also a link at the bottom of most pages to help with LaTeX.Anyway, with that explanation out of the way ...One way to think of an integral is as the area
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B18
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Homework Statement



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

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.
 
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  • #2
B18 said:

Homework Statement



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

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

Isn't [itex]\displaystyle \int f'(x)\,dx=f(x)+C\ ?[/itex]
 
  • #3
Yes i apologize, the question has "f is continuous" not F.
 
  • #4
B18 said:
Yes i apologize, the question has "f is continuous" not F.
In addition to that ...

My rhetorical question was meant to be a hint.

Try it.
 
  • #5
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
B18 said:
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.
Well, f(8) is 26, but I don't believe you can determine C from the information given.

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

That's about all you need.
 
  • #7
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
B18 said:
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?
The answer looks good.

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
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
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
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
Having trouble finding how to place the bounds on the integral correctly. Sorry :(
 
  • #13
B18 said:
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)

Yes.
 
  • #14
B18 said:
Having trouble finding how to place the bounds on the integral correctly. Sorry :(

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

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

FAQ: How Do You Calculate f(8) Using the Fundamental Theorem of Calculus?

What is the fundamental theorem of calculus (FTC)?

The fundamental theorem of calculus states that differentiation and integration are inverse operations of each other. This means that if we integrate a function and then differentiate the resulting integral, we will get back the original function.

How can FTC be used to find an F(x) value?

FTC can be used to find an F(x) value by evaluating the integral of a function at a specific x-value. This can be done by first finding the antiderivative of the function and then plugging in the x-value into the antiderivative to get the F(x) value.

What is the difference between the first and second parts of FTC?

The first part of FTC, also known as the evaluation theorem, states that we can find the value of an integral by evaluating the antiderivative of the function at the endpoints of the interval. The second part of FTC, also known as the differentiation theorem, states that we can use the derivative of a function to find the value of its integral.

Can FTC be used for all types of functions?

Yes, FTC can be used for all types of continuous functions. However, if a function is not continuous or has a discontinuity at the endpoint of the interval, then FTC cannot be used to find the value of its integral.

Is there a specific method for using FTC to find an F(x) value?

Yes, to use FTC to find an F(x) value, we need to follow these steps: 1) Find the antiderivative of the function, 2) Plug in the x-value into the antiderivative, 3) Evaluate the integral at the upper and lower limits of the interval, and 4) Subtract the lower limit value from the upper limit value to get the F(x) value.

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