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Question

Decide if the function is an antiderivative of $f(x)=2 e^{2 x}$
$$F(x)=e^{2 x}+\int_{0}^{1} e^{2 t} d t$$

Noah Duncan · Accepted Answer

Okay, So in this problem, we need to determine whether this function big f of X, which is either the two x plus the definite innit? A girl from +01 of each of the two t d t Is that function an anti derivative of little F of X, which is to eat of the two x and answer. This question is going to be yes. If the derivative of big after backs big F crime of axe is equal to a little f of X because that&#x27;s what it means for big F of X to be an anti derivative. We&#x27;re basically just undoing big ffx here with a derivative so that it just becomes little f again. And so all we need to do for this promise we just need to take the derivative of big F Prime Effexor, the derivative big F of X, and see if that gives us little after Becks back. And so the derivative of this termite look complicated. Um, but really, let&#x27;s just go to turn my terms. The derivative of each of the two X is gonna be too, you know, the two X because we&#x27;ve got this constant in the, um exploded here, so we actually need to bring that down is part of a chain roll. So the derivative the first turn it&#x27;s going to eat of the two X and then the second term is actually just a number definite ending roles are going to give us numbers. Um, they are not functions. So this is really just a constant. And the derivative of constant, we know is actually zero. So the derivative big F private exes actually just get me to eat of the two X and this is this is equal to little FX. So the answer to this question is, yes, that big FX is an anti derivative of little ffx because big F prime of X.

Noah Duncan · Answer

Okay, So in this question, we&#x27;re trying to answer whether this function big f of X, which is even the two X, is an anti derivative of little F of X, which is to eat of the two X And the answer. This question is going to be yes, if the derivative of big F of X big F prime of acts is equal to a little f of X, because that&#x27;s what it means to be for a function to be an anti derivative of the other one. So to check to see if this problem is yes or no, all we need to do is take the derivative of big FX and said equal to a little effort, Becks. So the derivative of big FX, just the derivative e to the two X, um which is going to through the chain rule B E to the to excuse the derivative. You know that something is eating something. Then we need to take the derivative of thing on exponents, which is the derivative of two X, which is to say we put the two in the front. So the derivative of big FX is to eat Of the two X, which is indeed equal to little FX. So the answer to our question is yes, Big F of X.

Noah Duncan · Answer

So in this problem, we needed to determine whether big F of X, which is to eat of the two X, is an anti derivative of little FX, which is also to eat of the two X and the answer, this question is going to be yes, on Lee if f prime of X Big F Prime of X is equal to a little aftereffects because that&#x27;s what it means for a big f of X to be a little Anant I derivative of little FX. So we need to do here is we need to take the derivative big effort, Becks, and see if it gives us the same thing. Is little effort, Becks. And so the derivative of to eat of the two X is gonna be too times the derivative of the two X, which is again to eat of the two x. So multiplying the twos together, we&#x27;re gonna get for each of the two X here, which is actually not equal to little FX. So the answer here is no. That big F of X is not an anti derivative of little

Noah Duncan · Answer

Okay, So in this question, we need to figure out whether this function big f of X, which is equal to X times eat of the two X is an anti derivative of little affects, which is to either to x and answer. This question is, yes. If the derivative of big FX is equal to a little FX because that&#x27;s what it means for big F of X to be an anti derivative of little FX. The anti derivative is where you can take the derivative of that function, and it&#x27;ll give you back the original function. So here we just need to take the derivative of big FX and see if it gives us a little after backs back. And we see here that we&#x27;ve got the derivative F vex Big F of X is gonna be the derivative between these two functions here. We actually have to do the product rubble. Um, so to do that, let&#x27;s say that axe is the first function, even even the two exes a second function. So we&#x27;ll do the first function times the derivative of the second, which is to each of the two x the derivative, either the two exes to eat of the two x plus the second you the two x times, the derivative of the first, which is a derivative of X, which is just one. And so this function is going to be, ah, two like this two X you know, the two X plus, even the two X And so clearly this does not equal little after Becks. So the answer here is no.

Noah Duncan · Answer

Okay, So in this question, we&#x27;re gonna try to determine whether big F of X, which is five feet the two X is an anti derivative of little FX, which is to eat of the two X and the answer. This question is going to be yes. If big F of X, the derivative of big FX, is equal to a little FX because that&#x27;s what it means for big FX. To be an anti derivative of little effort Vex, I mean, so we can take the derivative off Big F of X, and it&#x27;ll give us a little effort, Vex back. So to solve this problem, we just need to take the derivative of big F of X and see if it equals little F of X. So the derivative up here of big FX is just going to be We&#x27;re gonna leave the five in front, take the derivative of eat of the two X, which is going to be eat of the two X because the derivative eat of the something is eating something. But we need to take the derivative of the thing in the exponents. Well, so that&#x27;s gonna be, um, times, too. So the derivative. Two exes too. And so we&#x27;ve got this five in this, too, that we need to combine. So this is gonna give us 10. Eat the two X, and this does not equal little affects. So the answer to our question here, then, is no Big f of X is not an anti derivative little effort backs because big F prime of X does not.

Noah Duncan · Answer

Okay, So when this question we&#x27;re gonna try to answer whether big F of X, which is even the two X plus five, is an anti derivative of little F of X, which is two e two, the two X. So the answer this question is going to be yes. If the derivative of big FX F Prime of X is equal to FX and it&#x27;s gonna be no otherwise So let&#x27;s just check whether big a big F prime of X is equal to little FX, and that will give us our answer. So a big F prime of X is going to be equal to the derivative of this first term, which is going to be, uh, eat of the two X. Um, but then the chain rule is gonna mean that we need to take the derivative of the export here, So that&#x27;s gonna be two tacked on to that. So the derivative of each of the two X is to eat of the two X and then we&#x27;re gonna do the derivative of five here in the derivative of a constant zero. So we&#x27;re just gonna be left with this term and f prime of X Big F prime of X is going to be equal to each of the two X, which is indeed equal little FX. So the answer to our question is, yes, that big.

Problem

Decide whether the expression is a number or a fa…

Problem 6 Easy Difficulty

Decide if the function is an antiderivative of $f(x)=2 e^{2 x}$
$$F(x)=e^{2 x}+\int_{0}^{1} e^{2 t} d t$$

Answer

$F(x)$ is an antiderivate of $f(x)$

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Deborah Hughes-Hallett, Patti Frazer Lock, Andrew M. Gleason

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$F(x)$ is an antiderivate of $f(x)$

Applied Calculus

Catherine R.

Anna Marie V.

Caleb E.

Joseph L.

Integration Review - Intro

Definite vs Indefinite

Deborah Hughes-Hallett, Patti Frazer Lock, Andrew M. GleasonApplied Calculus

Catherine R.

Anna Marie V.

Caleb E.

Joseph L.

Deborah Hughes-Hallett, Patti Frazer Lock, Andrew M. Gleason

Applied Calculus