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Problem 5 Medium Difficulty

Use a graph to find a number $ \delta $ such that
if $ \displaystyle \left| x - \frac{\pi}{4} \right| < \delta $ then $ \left| \tan x - 1 \right| < 0.2 $

Answer

$$\delta=0.0906$$

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Video Transcript

this problem. Number five of the Stuart Calculus eighth edition section two point for Use a graft to find the number Delta such that if the absolute value of X minus power for is Lesson Toyota in the absolute value of tangent, my X minus one is less than zero point two. So we will begin. Renee's a graph. So here we have a A graph. What to tent from tension function Looks like around this point of interest point for pyro for and won. You can use your graphing calculator or any graphing Tulloch's well to follow along. Just plot the tendon function around this area, and we will begin to analyze a graph. But first will rewrite this second inequality as and they get a zero point two is less than tangent of X minus one his list in zero point two, and if we had one towards sites, we get through one eight, lest in Tangent of X, it's less then one point two, and this is the range for tension that we're looking for. Remember the attention being our function. So we're looking between the values of tangent for one point two just right there and zero point eight, which is approximately right here. So we see that there's an X value associated with each of these. So there's a value lesson by Ra for associated with tension of X equals zero money and there's a value greater than paper for, um, for ten for X, um, that is associated with tension of X equals to one point two. So we want to find what the X values are. So one idea or one way that you can calculate these exes is to use your crafting calculator and to trace the function. So move your cursor along the function until your cursor lands at approximately why equals zero point eight. So if you were to do this, you'd be able to find the X value for which the tangent function of that X value is equal to zero point eight. So if you would do this for zero point eight, he wouldn't get zero point six seven five as your ex value. And if you were to do this with one point two, you would be able to get the value associated with that at zero point eight seven six approximately. And these air the values less than in greater than Piper for associated with this range that we're giving at the beginning. In order to compare it to this first inequality Organa subject camera four teach term and for the upper bound. If we subject power form, we should get approximately zero point zero nine. And if we subtract power for from zero point six seven five, we should get approximately negative zero point one one. So we will compare that with this first inequality, which can be re written. I think the little toe Liston X minus part before less than Delta, and we see that D conditions that must be met are the delta must be the delta must be less than or equal to zero point zero nine more or less than or equal to zero point one one incense. The only values that satisfy both our values that are less than equal to zero point their name. We can choose the value any value less than or equal to zero point your name, any positive value. And this satisfies our answer for this problem.