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Find the equation of the tangent line to the curve y = 1? + 1 at the point P(2,5). Find the equation of the tangent line to the curve y = at the point P(0, %):...

Question

Find the equation of the tangent line to the curve y = 1? + 1 at the point P(2,5). Find the equation of the tangent line to the curve y = at the point P(0, %):

Find the equation of the tangent line to the curve y = 1? + 1 at the point P(2,5). Find the equation of the tangent line to the curve y = at the point P(0, %):



Answers

Find an equation of the tangent line to the curve at the given point.

$ y = \sqrt{x} $,

$ (1, 1) $

Our goal here is to find the equation of the tangent line to the curve. Y equals two to the X at the points you're a one. The slope of the tangent line will be the derivative at that point, So let's find the derivative. The derivative of two to the X is two to the X Times natural log tube, so the derivative evaluated at zero will be two to the zero power types. Natural log to and to to the zero power is one. So we have one times natural log to which is natural log to So that is the slope of the tangent line. Now let's use the point slope form. Why minus y one equals m times X minus X one to find the equation of the line. So we substitute one in for why one? Since that's the Y coordinate of our point. We substitute natural log to infer the slope, and we substitute zero in for X one. Since that's the X coordinate of our point, and now we simplify. So we have y minus one equals natural log to Times X, and then we can add one to both sides. And the equation of the Tanja line is why equals the natural log of two times X plus one

Question 38 says find the equation for a tangent line. Two equals one minus x over one plus x. At x equals two. So are tangent line is gonna look like Y equals um x minus two. And then the y value at two is negative one third. So we need our to find em is going to be the derivative of why with respect to X evaluated at X equals two mm. Mhm. The derivative of that function equals, and we're gonna be using the question rule here, derivative of the top is negative one times the bottom minus the derivative of the bottom, which is just positive one times the top over the bottom function squared, Evaluating at x equals two gives us negative one plus two minus one minus two. All over one plus two squared. So we get negative three Plus 1/9 which is -2 nights. So we can now replace the M. With well we know the slope to be which is to over nine negative two event

Here on this problem. We want to find the equation of the Tanja line to this curve at a 0.11 And so first we need are derivative. And so using Archangel. Here we have the Y. The X is equal to two times X minus one to the first units using our general. And so now we evaluate this at the 0.11 Then so we plug in one for X was to be two times one minus one. The first, which gives us zero and so are Slope is zero. We have our soap. We also know the point it goes through is 11 It's a point is 110 So that tells the equation of our line is why equals one? That is the equation of our Tangela and sort of draftees on the same graph. You'll notice that our original curve is a parabola with the vertex of 11 and it opens upward on our tangent line was the line y equals one And that's just this horizontal line everywhere. The Y value is one and so in red here is y equals one, and in black is y equals X minus one squared plus one

He It's clear. So when you right here. So our first gonna differentiate using the quotient role. So we have tea. Why? Over t x is equal to one plus each The x the derivative of one plus X minus one plus eat x The derivative times one plus x This is all over one plus Eat the X square. This is equal to one plus eat the X times one minus Eat the x one plus X All over one plus Eat the X square. This becomes equal to one minus x Eat the X over one plus each the X square and to find the slope of the 10 gents, we're gonna plug in zero when we get 1/4 and the equation becomes why minus one have equals 1/4 times X minus zero and this becomes equal to x over four plus 1/2


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