The graph of a function is shown in the figure. Make a rough sketch of an antiderivative $ F $, given that $ F(0) = 1 $.
$-\ln x+x \ln 2-\ln 2$
So given the graph of the function the picture, we want to give a rough estimate um for the sketch of the anti derivative F. And we know that F of zero is going to be equal to one. So the reason why we have those initial conditions and why they're extremely important is because we know that when we take F prime of X. A. So let's say F was equal to some value, you know, C. X plus the right. Well, we know that when we take F. Crime that that the value is going to go away. So when we take the integral of the derivative, we're going to get F back but we need to compensate for that. See that we might have lost. So as a result of that, we use the initial condition, the anti derivative that we're going to end up getting at least it's similar to that. It's going to be F equals the natural log of acts. For the negative natural log of acts plus X. Natural life of two minus, and that's where the initial condition comes in minus the natural log of two. So that's right here, it's going to be the final graphical representation that we see.