Fundamental Theorem of Calculus

[ralfsk8]

So would the answer be:

x^2 + 144?

 

[Mark44]

For what? You should be writing an equation; i.e., something with = in it.

 

[ralfsk8]

I am completely lost :(

 

[Mark44]

You're given that $F(x) = \int_5^x \sqrt{t^2 + 144}~dt$

b) Find F'(5)

To do this part, first find F'(x), and then evaluate this derivative at x = 5. This is where the Fund. Thm. of Calculus comes into play.

 

[ralfsk8]

I'm comparing my answers with a similar problem in the book and there aren't any equations, only numerical answers.

 

[Mark44]

But your work should consist of equations, like this:

F'(x) = <some function of x that you work out>
F'(5) = <some number>

If you just throw something up in a post, I have no idea what you are doing.

 

[ralfsk8]

I understand the bit about the Fundamental Theorem of Calculus coming into play but I thought that you just substitute the bounds where the variables are. For example, the book gives us the example of:

Integral with lower bound of 1 and upper bound of x, (t^3)dt. The answer is given as merely x^3

 

[Mark44]

I understand the bit about the Fundamental Theorem of Calculus coming into play but I thought that you just substitute the bounds where the variables are. For example, the book gives us the example of:

Integral with lower bound of 1 and upper bound of x, (t^3)dt. The answer is given as merely x^3

OK, let me ask you: What does the answer you wrote represent?

 

[STEMucator]

IMHO, memorizing that formula is NOT a good idea, especially if that memorization comes at the expense of understanding.

Point taken I suppose haha, but grant me that I'm not going to write out and explain the whole proof of the theorem lol.

 

[Mark44]

Or, in other words, what question is x³ the answer to?

 

[ralfsk8]

Does it mean that the function is differentiable within those bounds?

 

[Mark44]

What function? Try to ask questions that are more precise.

 

[ralfsk8]

Okay I finally got the answer but only by comparing them to other online resources. I'm still not entirely sure on how to do the actual problem. I wouldn't mind discussing this further but if you guys need to go do other things, that's okay. Thanks for the help anyway.

 

[SammyS]

Okay I finally got the answer but only by comparing them to other online resources. I'm still not entirely sure on how to do the actual problem. I wouldn't mind discussing this further but if you guys need to go do other things, that's okay. Thanks for the help anyway.

ralfsk8,

Suppose that G(t) is an anti-derivative of $\displaystyle \sqrt{t^2+144}\ .$

We write that as $\displaystyle G(t)=\int\,\sqrt{t^2+144}\ dt\ .$

So if we have a definite integral such as $\displaystyle \int_{a}^{b}\,\sqrt{t^2+144}\ dt\,,$ we can evaluate that as G(b) - G(a), according to the Fundamental Theorem of Calculus. Correct?

Now, in the case of the problem in this thread, we have:

$\displaystyle F(x)=\int_{5}^{x}\,\sqrt{t^2+144}\ dt\,=G(x)-G(5)\ .$​

Therefore, $\displaystyle F'(x)=G\,'(x)\,,$ since G(5) is a constant.

But G(x) is the anti-derivative of $\displaystyle \sqrt{x^2+144}\,,$ so that $\displaystyle G\,'(x)=\sqrt{x^2+144}\,.$ Correct?

Therefore, $\displaystyle F'(x)=\sqrt{x^2+144}\,.$