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Numerade Educator



Problem 60 Hard Difficulty

(a) Prove that the equation has at least one real root.
(b) Use your graphing device to find the root correct to three decimal places.

$ \arctan x = 1 - x $


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Video Transcript

this problem Number 60 of the Stuart crack the safe edition Section 2.5 a party proved that the equation has at least one real root in part B. Use your graphing device to find the route correct to three decimal places. The equation, given his arc Tangent of X, is equal to one minus X on our first step will be to rearrange this equation so that all the terms on the left side mark 10 X plus X minus one course zero or the left side we consider is the function F and any roots are the values of X that make this function equal to zero for party. We're just going to prove that these exists in part, we will be able to find what the root is using our crafting calculators. The the approach to proving that this function has at least one route is to understand the behaviour of this function. For a certain interval, we're going to choose, Do you, you know, largest interval possible and understand what this function looks like when we approach infinity and also negative infinity. So for the first limit as X approaches infinity, the function approaches infinity because this function is our tangent, which only has a limited range range and a maximum value of power to plus infinity X approaches infinity minus one. So this whole function will be dominated by this X term and it will be infinity. Likewise, as we approach negative infinity, the function will be dominated by the X term as well, and this function approaches negative infinity. So the the reason that we can say this function has at least one route is because through the intermediate value theorem, because this function is a continuous function. Since the combination of trigonometry functions, linear functions and constant values, it is a continuous function, and for this given interval from infinity to infinity, it must take on every single value of what the function is evaluated at. So it's negative. Infinity and negative infinity. It's positive affinity, a positive infinity. Since it's continuous, we know that they must take on every single value between these two numbers, including zero, so there must be at least one real root. According to the Intermediate Value Theorem, we will graph in part be using your graphing calculator. You can either plot this function equal to this function and see where they intersect. Or just take this function that we called f plot it and see where it crosses the X axis because that's where X or why why will be equal to zero? And we can figure out exactly what the value is to three decimal places. We see that the function crosses the X axis at 0.5 to 3, or approximately 0.52 Oh, is what we will consider the solution to this problem. This is a corrected through decimal places where the route occurs, where the function will be equal to zero, and we can definitely double check that I plug them into the equation, and we should get a value very close to zero.