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(b) Illustrate part (a) by graphing the curve and the tangent line on the same screen.

a)$y=-\frac{3}{5} x+\frac{16}{5}$

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this poem. Number twenty nine of the Stuart Calculus eighth edition section two points of it part A. If fx's five x over the quantity one plus X squared find of prime of to and use it to find an equation that enjoying to the curve why equals five X right by the quantity one X plus X squared at the point two two for part A we're going to use Ah, this equation our definition number one here shown on the right for the value of equals two came, so this will be a limit as X approaches to of the function five x half of x function Fabrics over one plus X squared minus F d. So what is F evaluated at X equals two or equals two? We have five times two is ten divided by one plus two squared. That's five ten divided by five is two and that nominated We have X minus two. Our next appear would be to multiply the top on the bottom by the denominator of the fraction of the numerator. In order to get rid of that fraction, we're multiplying top and bottom by almost X squared. So in the next step we will have ah limit his expertise to of five X minus two times a quantity One plus x word are all directed by X minus two o We also multiplied by one plus X squared in the denominator explains too Time's quantity Want sex word the numerator can be rearranged and then we're going to be looking to factor If we rearrange the numerator, we could do it as a negative Ah get one on the outside two X word minus five X plus two And then we'LL see exactly the reason why we did this Pulling out the negative In our next step, we should get the fact that for most to X squared minus five exposed to is X rays too two X minus one All the right of my quantity X minus two more provided by the quantity one plus x squared. Okay, here we see that X minus two is president generator and the denominator s So that cancels out. And so our limit we'll just be negative quantity to excellence one right by the quantity one plus X squared as expertise to this quantity approach is negative. The quantity two temps to minus one, which is negative. Three and generator on the quantity. One plus two squared in the denominator, which is five. So after I move to equals this whole limit and this limit for equals two gives us a slope of native three trips. The equation of the ten Jalin. Can we find these in point Slope Form? Now we know the Slope M is negative. Three over five and were given a point x one y one. The point is to to so where women is two equals two native, three fifths times the quantity X minus two If we saw for why we work a negative three Feds X Here we have negative three times in two over five at six over five. And then we're adding two, which is ten or five. Uh, so it will be plus sixteen over five. And this is your final answer for our equation of the tension line part being illustrate. Party, right, graphing the curve and the tension line on the same screen. The curve is here. Affix is equal to five x over the quantity one plus x squared and the tangent line is why equals negative three foot six plus sixteen fifth. So here we planted a function of the vics and the equation of the henge align, and we see that the curve being read the tangent line touches his curve only once, exactly at the point two two.