Fundamental Theorem of Calculus Part IV

[Hckyplayer8]

Unlike the previous ones, I'm not positive what this one is asking. If one were two establish a flowchart as follows

function --> derivative -->second derivative

you can reverse the process by going

integrand --> antiderivative -->function (which would be the antiderivative of the antiderivative)

So pending that is all correct, then the derivative of an integrand is just a regular derivative and I should be using differentiation?

 

[Steven G]

What is the rule you were given to find the derivative of an integral?

You can always say \(\displaystyle \int tsint dt = F(t) + C\)
See if you can work with this (assuming that you did not learn a formula to compute the derivative of a definite integral).

 

[Romsek]

Another form of the fundamental theorem of calculus.

\(\displaystyle \dfrac{d}{dx} \left(\displaystyle \int \limits_{a(x)}^{b(x)}~f(t)~dt\right) = f(b(x))\dfrac{db}{dx} - f(a(x))\dfrac{da}{dx}\)

 

[pka]

Another form of the fundamental theorem of calculus.
\(\displaystyle \dfrac{d}{dx} \left(\displaystyle \int \limits_{a(x)}^{b(x)}~f(t)~dt\right) = f(b(x))\dfrac{db}{dx} - f(a(x))\dfrac{da}{dx}\)

One additional condition: both \(\displaystyle a(x)~\&~b(x)\) are differentiable functions.

 

[Hckyplayer8]

So just keep doing what I've been doing?

Find the antiderivative, and plug in and find the difference of the upper and lower bound?

 

[Dr.Peterson]

So just keep doing what I've been doing?

Find the antiderivative, and plug in and find the difference of the upper and lower bound?

No. Did you not see what post #3 tells you to do? The point is that you don't need to find the antiderivative. (You could do that -- but there are some functions whose antiderivative you wouldn't be able to find, and many for which that would be far more work.) Surely there are examples of this in your textbook.

What are a, b, and f in your problem?

 

[Hckyplayer8]

No. Did you not see what post #3 tells you to do? The point is that you don't need to find the antiderivative. (You could do that -- but there are some functions whose antiderivative you wouldn't be able to find, and many for which that would be far more work.) Surely there are examples of this in your textbook.

What are a, b, and f in your problem?

I read that wrong. Let me try that again. Interpreting that post to plain text...

The derivative of an integral is the function of the upper bound times the derivative of the upper bound minus the function of the lower bound times the derivative of the lower bound?

 

[pka]

Find the antiderivative, and plug in and find the difference of the upper and lower bound? NO!

EXAMPLE: \(\displaystyle {D_x}\left( {\int_{\cos (x)}^{\log ({x^3} + 1)} {({t^2} + 2)dt} } \right) = \left[ {{{\left( {\log ({x^3} + 1)} \right)}^2} + 2} \right]\left( {\frac{{3{x^2}}}{{{x^3} + 1}}} \right) - \left[ {{{\cos }^2}(x) + 2} \right]\left( { - \sin (x)} \right)\)

 

[Dr.Peterson]

I read that wrong. Let me try that again. Interpreting that post to plain text...

The derivative of an integral is the function of the upper bound times the derivative of the upper bound minus the function of the lower bound times the derivative of the lower bound?

Essentially, yes. Note the important thing: the derivative undoes the antiderivative, so that most of the result is just values of the original function.

Be sure you understand how pka's example demonstrates the formula; it's an excellent one.

 

[Hckyplayer8]

Thank you all for posting.