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Problem 26 Medium Difficulty

Compare the functions $ f(x) = x^5 $ and $ g(x) = 5^x $ by graphing both functions in several viewing rectangles. Find all points of intersection of the graphs correct to one decimal place. Which function grows more rapidly when $ x $ is large?

Answer

$g(x)$ grows more rapidily when $x$ is large

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Video Transcript

in this problem, we're going to use a graphing calculator to compare the functions f of X equals X to the fifth Power and G of X equals five to the X power. So we type those into our y equals menu, and what we want to do is look at them using several different viewing wreck rectangles, and then we're gonna find the points of intersection. And so, for starters, for a good viewing with rectangle, I'm going to start with Zoom decimal, which is choice number four, and we can see that the red one is the five to the X power and the blue one is the X to the fifth power. Okay, so the five to the X power appears to be greater than the X to the fifth power at this point in time. Okay. And that viewing window goes from negative 6.6 to 6.6 on the X axis and from negative 4.1 to 4.1 on the Y axis. So now let's change the numbers. Now I'm going to go with Zoom Standard number six, and that takes me from negative 10 to 10 on the X and y axes, and it appears as if we might be getting closer to an intersection point. Now I'm going to manually input some different window dimensions. And I don't think we need all these negative values over here. So we're gonna concentrate on going further in the positive direction and going further up in the positive Y direction. So what if we try negative too t eight to, let's say, on the X axis. And what if we try? Um, negative 10 to 10,000 on the Y axis? Let's go way up there and let's make our scale go by one hundreds. The scale isn't that important. It just is showing you how frequently you have a tick mark on your axis. All right, well, we see here that the the blue graph, which is y equals X to the fifth, is overcome by the red graph somewhere around here. So that tells us that the exponential function does grow more rapidly once we get to this point, whatever that is. So let's go ahead and find that intersection point so we can go to the Calculate menu, which is second trace and then choose number five intersect put the cursor on the first curve. Press enter, but the cursor on the second curve. Press enter, and then you can move your cursor over closer to the intersection. Point and press enter, and we can see that that intersection point has X coordinate. Five. So that tells us that they intersect at the 0.5 comma 3125 and when X is large, the higher graph the red one. The exponential graph is growing more rapidly, So f of X equals five to the X, or its G of X equals five to the X grows more rapidly when X is large.