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Problem 10 Easy Difficulty

Trace or copy the graph of the given function $ f $. (Assume that the axes have equal scales.) Then use the method of Example 1 to sketch the graph of $ f' $ below it.

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Video Transcript

This is final number ten of the sewer calculus. Eighth edition, Section two point eight. A tracer. Copy the graph of the given function f soon that the axes of equal scales then use the method of example one to sketch the graph of F prime blew it. So we have copied here. The graph of information have given with this problem, and it has this shape shown here in blue. And we're going to Grafton, conservative of this graph of prime in red. Um, we want to identify certain parts of grass, the graph of F that are special. And then you determine what the dirt of his Addie chills points and then come to a conclusion about what the derivative function looks like. So here we have identified a few points that we want to line up for the grand from the dirt of death. Uh, let's graphs, um, easy points. First, there's a horizontal tangent line here, Uh, at this point at this first special point which we want to draw here is zero slope on the after a photograph and this also f prime graph. And this also occurs over here. There's another minimum. This is it. This is an inflection point up here and then this is a minimum down here, and those both correspond with the slip of zero. So our prime zero those points. Then we ask ourselves what's happening in between each of these sections to the left of this first zero. We have, ah, function that has very negative soaps. They get less than negative, less negative as they approach zero so that what that means is that the director function is below that X axis as it approaches zero. So it's more negative than it becomes less neither towards this point. Ah yes, and I look like this next between this first x x intercept on the F prime, McGrath and X equals zero. We want to ask what's going on. This is an inflection point, which means that the function oh are the slopes don't change from negative to positive, in fact, that they remain negative. He goes from zero to something that a slope, some of the tension line that's negative to increasingly more negative. A maximum maximum negative derivative at X equals zero. So we will show this as still some negative soaps reaching a minimum here at X equals zero to the slope is steepest here at X equals zero. But then the slopes become less negative as it approaches the next special point. So what this means is that the soap comes back up and then reaches the a point of zero slope here at the second special point and then afterward. This so goes from zero and increases positively towards very positive slope for the tension lines, meaning that a religious keep increasing afterward s o. This graph of crime is consistent with all the slopes of the tension lines of the graph of F, meaning that this is an accurate sketch for the graph of F prime.