{Use of Tech} Fixed points An important question about many fu...

Question

{Use of Tech} Fixed points An important question about many functions concerns the existence and location of fixed points. A fixed point of f is a value of x that satisfies the equation f(x) = x; it corresponds to a point at which the graph of f intersects the line y = x. Find all the fixed points of the following functions. Use preliminary analysis and graphing to determine good initial approximations.

f(x) = tan x/2 on (-π,π)

f(x) = tan x/2 on (-π,π)

Accepted Answer

Welcome back, everyone. The function G of X equals the tangent of X divided by 4, and open parentheses 22 closed parentheses is given. A fixed point of G is where G of X equals X. Using preliminary analysis and graphing, what is a good initial approximation for a fixed point of G? Here we have a graph of G of X and Y equals X, and for our answer choices, A says X equals 0 and is approximately equal to -5.57 and 5.57. B says it's approximately equal to -5.57 and 5.57. C says it's approximately negative 5.57 and 0, and the D says it's approximately 0 and 5.57. Now, here, if we go to our graph to find the fixed point of the function G of X equals the tangent of X divided by 4, we need to solve for the equation G of X equals X. Now there are many ways that we could solve it, but notice here that because we're just trying to find the value of X where G of X equals the tangent of X divided by 4, and we already have a plot of G of X and the line Y equals X on the same graph, we can visually identify the point where the two curves intersect because at that point, that means our equations will be equal. Now, in this case, by looking at our graph, notice that there are 3 points of intersection, OK? Now, for the first point of intersection, notice here that we can tell it's approximately 5.57, OK? So, here, it's approximate, sorry, negative 5.57. For our second intersection, we know that it intersects at the at the origin of the graph, which makes sense because when X equals 0, Y equals 0 in both cases. And finally, for our third point of intersection, we can tell that it's approximately 5.57. Therefore, X equals 0. Or rather, our intersection occurs at around the point where X equals 0, OK. And where X is approximately equal to -5.57 and 5.57. If we look back at our answer tras, that tells us A is the correct answer. Thanks a lot for watching everyone. I hope this video helped.

Key Concepts

Fixed Points

Graphing Functions

Preliminary Analysis

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