Problem 18 Easy Difficulty

Make a careful sketch of the graph of $ f $ and below it sketch the graph of $ f' $ in the same manner as in Exercises 4-11. Can you guess a formula for $ f'(x) $ from its graph?

$ f(x) = \ln x $

Answer

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Video Transcript

this problem. Number eighteen of the Stuart Calculus eighth edition, Section two point eight Make it careful. Sketch. With the graph of F and blow, it sketched the graph of Prime in the same manners and exercises for through eleven. Can you guess a formula for F prime of X from its craft and the function F is from X equals seven max so we can plot Ellen Rex as we're familiar with dysfunction. It has the shape, whereas expert's negative infinity or as experts zero from the left from the right, the function of purchase negative. Infinity as experts is infinity. The function approaches infinity, and another interesting point is that as ah at X equals one in the function, Ellen of X equals zero two. We can use this information to determine with the graph of Prime should look like and when we do that by using on the slope of the tangent lines to every at every point of dysfunction. So one of the easier ones scene is that let's say this is a slip of the tension line. At this point, as we get closer and closer to Texas is impunity. This function, uh, appears to level off the slopes decrease and approach zero. So we can show that here that eventually the slopes, our may decrease towards zero. And what we see here going towards ex zero from the right, see that the spokes become more and more steep. One more negative are more, more positive. And therefore, as X apprentice here from the rain, disfunction will be going towards positive infinity. And if we were to exactly figure out what's hope is here at X equals zero are at X equals one. We ended up determining that soap is indeed one here. X equals one. So at X equals one plus look is one. Thats the point one one and that's a character's a point here that function will need to go through. So this is the general shape of what crime should look like. And if we were to think kiss dysfunction is similar to the function one over X, meaning that we expect the derivative of Alan X to be similar to this Formula one a rex. And it is indeed true. The derivative of Alan X is one of wrecks on, so we were able to determine the shape of the derivative by using any slopes of the tension lines at each of the point of dysfunction, F equals Eleanor IX.