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Problem 46 Hard Difficulty

Compare the functions $ f(x) = x^{0.1} $ and $ g(x) = \ln x $ by graphing both $ f $ and $ g $ in several viewing rectangles. When does the graph of $ f $ finally surpass the graph of $ g $?

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all right. Our goal in this problem is to graph both of these functions Y equals X to the 0.1 power and Michael's Natural log of X, and to change the window dimension several times and on to figure out if f of X or win f of X finally surpasses G of X. So perhaps we start with a standard window negative 10 to 10. That's always a good place to start, and we can see that the red one, which is G of X, the natural log. It doesn't take long for it to surpass the blue one. And for a long time it's higher than the blue ones. So G is higher than F. We want to know Windows F go back higher than G, so we're gonna change the window and let's go further out. Suppose we go out to 100 on the X axis and let's take a look. Well, G is still higher than F. Let's go out to 1000. We don't really need to go down so low on the Y axis. We could change that if we want to. O K G is still higher than F, and it's going to be quite a while until F surpasses G. So we need to put some much bigger numbers in for X max. We're going to have to keep fiddling with it and keep fiddling with it. I did tend to the 10th Power and I changed the Y Max as well. 2 50 I believe. And let's see what that gives us. Well, G is still higher than F. The loggerhead McGrath is still higher than the polynomial or the power function. At this point, let's go even bigger. Let's let X Max equal 10 to the 15th who they're getting closer now. F is getting closer to G, so it looks fairly promising. Let's go ahead and go with 10 to the 16th Power and maybe I'll change my Why Max again, change it to 70. Okay, we do see finally, that f is surpassing G. So the power function eventually way out there does go higher. Then the lager and MK function and let's figure out where that is. So we compress second trace to take us into the calculate menu. Then we go to number five, which is intersect cursor on the first curve. Yes, press enter person on the second curve. Yes, press enter, Move the cursor closer to the point of intersection and press enter. And we see at the bottom of the screen that it says the point of Intersection is at X equals approximately 3.13 times 10 to the 15th Power. So that's when f finally surpasses G. That is pretty far out.

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