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Problem 27 Hard Difficulty

Compare the functions $ f(x) = x^10 $ and $ g(x) = e^x $ by graphing both $ f $ and $ g $ in several viewing rectangles. When does the graph of $ g $ finally surpass the graph of $ f $?

Answer

The graph of $g$ finally surpasses the graph of $f$ when $x \approx 35.8$

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Video Transcript

for this problem. We want to compare the graphs of F of X equals X to the 10th Power and G of X equals E to the X power. So we type both of those into the graphing calculator. We're going to look at them using several different viewing windows and figure out the point of intersection when the exponential graph finally surpasses the polynomial graph. So I'm going to start with just a standard with viewing window negative 10 to 10. So go to zoom six. So we see the blue one is X to the 10th and we see the red one is y equals e to the X. So we can see that in this area right here. The blue one is higher than the red one. So the exponential graph has not surpassed the polynomial yet. Now we can change our window dimensions. Perhaps we can go a little higher with her y values. No, we go up to 1000. Take a look at that. We can see that the polynomial graph is still quite a bit higher than the exponential graph. So let's make a few more changes. Maybe we go out to X equals 20 and perhaps we go all the way up to 10 to the eighth power. Well, the polynomial is still greater than the exponential graph, so let's go out a little further. Let's go out to X equals 50 and let's go all the way up to why equals 10 to the 16th power. Okay, now we've seen a change. So we see a point of intersection. We see the point where the exponential graph surpasses the polynomial graph. And let's find that point of intersection so we can go to the calculate menu. We can choose number five Intersect. Put the cursor on the first curve, press enter, put the cursor on the second curve, press enter and then move the cursor closer to the intersection. Point press. Enter and we get the intersection point, as is at about X, equals 35.8. And that's at a Y value of 3.4 times 10 to the 15th Power