Using a given integral to determine concavity

[finalproject]

Hi all, I'm working on a project for my summer math class and one of the questions that has been stumping me is the following:

"If A(x) = the definite integral from 0 to x of (R(t) dt), answer the following questions about A(x):
-Where is A(x) concave up / down, and explain using the given graph of R(t) why there are no local or minimum values on the graph A(x)."

I'm having difficulty even conceptualizing how to do this - I know that I need to find the second derivative to see the concavity of the function, but I can't figure out how to find it. Also, I know I haven't posted a picture of the graph, but if you could help me understand what features of the function would indicate the lack of local min's / max's that would be appreciated. If that's not possible, I can try and snap a picture to upload.

Thanks in advance for any help - I really appreciate it!

 

[DrPhil]

Hi all, I'm working on a project for my summer math class and one of the questions that has been stumping me is the following:

"If A(x) = the definite integral from 0 to x of (R(t) dt), answer the following questions about A(x):
-Where is A(x) concave up / down, and explain using the given graph of R(t) why there are no local or minimum values on the graph A(x)."

I'm having difficulty even conceptualizing how to do this - I know that I need to find the second derivative to see the concavity of the function, but I can't figure out how to find it. Also, I know I haven't posted a picture of the graph, but if you could help me understand what features of the function would indicate the lack of local min's / max's that would be appreciated. If that's not possible, I can try and snap a picture to upload.

Thanks in advance for any help - I really appreciate it!

If R(t) is >0 everywhere, then A(x) is increasing everywhere.

\(\displaystyle \displaystyle A(x) = \int_0^x R(t)\ dt \)

To find the derivative of A(x), apply the Fundamental Theorem of Calculus. Then take the second derivative in the usual way: the 2nd derivative of A(x) is . . . related to R(x)?

 

[finalproject]

If R(t) is >0 everywhere, then A(x) is increasing everywhere.

\(\displaystyle \displaystyle A(x) = \int_0^x R(t)\ dt \)

To find the derivative of A(x), apply the Fundamental Theorem of Calculus. Then take the second derivative in the usual way: the 2nd derivative of A(x) is . . . related to R(x)?

So applying the FTC, I would have A(x) = F(x) - F(0) where F(x) is the antiderivative of R(t) dt?

 

[pka]

So applying the FTC, I would have A(x) = F(x) - F(0) where F(x) is the antiderivative of R(t) dt?

No. You know at once \(\displaystyle A'(x) = R(x)\). If you know that \(\displaystyle R(x)\) is differentiable then \(\displaystyle A''(x) = R'(x)\)

 

[finalproject]

No. You know at once \(\displaystyle A'(x) = R(x)\). If you know that \(\displaystyle R(x)\) is differentiable then \(\displaystyle A''(x) = R'(x)\)

So then I would look at the graph of R(t), determine where the slope is positive, and for that interval A(x) would be concave up? And the opposite for concave down?

 

[pka]

So then I would look at the graph of R(t), determine where the slope is positive, and for that interval A(x) would be concave up? And the opposite for concave down?

You can do that. You did not report that you had the graph of \(\displaystyle R(x)\).
If you are given that graph, then that is exactly how to proceed.

 

[finalproject]

You can do that. You did not report that you had the graph of \(\displaystyle R(x)\).
If you are given that graph, then that is exactly how to proceed.

Sorry, I suppose I didn't make that clear. Thanks for all the help though - I really appreciate it!

I'll be sure to come back here if I have any more questions.