Jon P. answered • 05/10/15
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This is an interesting question. I'm not actually sure how to solve it by working with the actual equation (arctan x = 0.9x). However, I can see the answer by graphing the two functions.
If you graph y = arctan x and y = 0.9x, then the places where the graphs intersect would be the places where the two functions are equal.
When you do this, you can see that the arctan function starts out at (0,0) and rises to the right (and falls to the left) with a slope of 1, but the slope gradually decreases and approaches 0 as the function approaches its horizontal asymptotes at y = π/2 and y = -π/2.
The graph y = 0.9x is a line that also starts at (0,0) and leaves the origin with a slope slightly less than 1 (0.9) along the whole line. That says that right after the arctan graph leaves the origin, it goes higher on the right and lower on the left than the graph of y = 0.9x, but then the line soon crosses the arctan graph.
So I see three places where the two functions are equal: (0,0) and the two places that the graphs cross -- one to the right of the origin, and one to the left.
Jon P.
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You could use the same logic as I gave without actually relying on a graph per se -- that is look at the mathematical behavior of the two functions and recognize that arctan starts out rising faster than 0.9x but then levels off, which implies that the two functions cross somewhere else besides (0,0). However, using the graph simplifies this significantly.
There's no other EXACT way to solve this (that is, solve the equation arctan x = 0.9x). But Robert's solution is definitely a good way to do it as an approximation -- as long as you're familiar with Taylor expansions, which is a technique that comes out of calculus. My guess was that you're not.
One hint was that the question asked for the NUMBER of solutions, not the actual solutions. That tells me that the writer of the question knows that there's no way to solve the equation exactly, and that he/she more likely intended you to use graphing and the general behavior of the functions to figure it out.
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05/10/15
Kayla C.
Yes, Ive tried graphing and was able to find the three intersections you mentioned! But my calculator seemed blurry so I was wondering if solving it manually would be more efficient. Would it be possible to solve it without a graph?
05/10/15