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Problem 58 Hard Difficulty

The function $ f(x) = \sin (x + \sin 2x), 0 \le x \le \pi, $ arises in applications to frequency modulation (FM) synthesis.
(a) Use a graph of $ f $ produced by a calculator lo make a rough sketch of the graph of $ f'. $
(b) Calculate $ f'(x) $ and use this expression, with a calculator, to graph $ f'. $ Compare with your sketch in part (a).

Answer

a. see work for graph
b. $f^{\prime}(x)=\cos (x+\sin 2 x)[1+2 \cos (2 x)]$

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Video Transcript

for this problem we have f of X equals the sign of X plus the sign of two X And the first thing we want to do is graph it on a calculator. So we grab a calculator, we go toe y equals and we type it in. We want the domain to be zero to pi so we goto window and weaken type in zero to pi for the X range And then I'm just going to use negative 626 for why? And we'll see how that goes. Only press graph. Okay, so what we want to do is use this graph to make a rough sketch of the derivative. All right, so here's the graph that we just found on the calculator, and now we're going to use it to make a sketch of F Prime of X. So notice that right here on the graph, we have an increasing function, but it's getting less and less steep, and the slope is getting closer and closer to zero. So that means that the derivative would be positive, but it would be getting closer to zero. And then throughout the middle section we see that it's relatively flat, so the slopes are very close to zero. But it looks like it is slightly going up and down, so we might have parts that are kind of sometimes below zero and sometimes above zero and sometimes below zero in, sometimes above zero. And then we see in the last section that the graph is decreasing. And so that means we would have a negative derivative. So roughly like this. Now what we want to do is actually calculate the derivative. So we're going to use the chain rule. And so we have The derivative of Sign is co sign so co sign of the inside multiplied by the derivative of the inside. So the derivative of X would be one on the derivative of Sign two X. We would use the chain rule on that as well. And we would get coastline of two x times, the derivative of its inside, and that would give us an extra tube. So who put that into the calculator and graph at Let's see what we get. So here we see it all typed in co sign of X plus sign of two X times one plus two times co sign of two X. Okay, so let's look at the graph and here's what we see and we're going to compare it to the sketch. And here we have a side by side comparison, and you can see that the sketch by hand does look very similar to the sketch from the calculator or the graph from the calculator.