Tangent line to y = √x Does any tangent line ...

Question

Tangent line to y = √x Does any tangent line to the curve y = √x cross the x-axis at x = −1? If so, find an equation for the line and the point of tangency. If not, why not?

Accepted Answer

Hi everyone, let's take a look at this practice problem. This problem says a tangent line to the curve Y is equal to 2 X2d intersects the X-axis at X equal to 2. Determine the equation of the tangent line and its corresponding point of tangency to the given curve. Now, the first step to solve this problem is we need to determine the slope of the tangent line at any point on our curve Y is equal to 2 X2. So we're going to call the slope of the tangent line, M tangent. And this is going to be equal to, and here we call your formula for the slope of a tangent line at any point on a curve. And so this is going to be equal to the limit, as H approaches 0 of the quantity of F of X plus H. Minus F of X in quantity, divided by H. Now, in our case, F of X is going to be equal to 2 x 2d. So, we'll substitute in that formula for F of X. So this is going to be equal to the limit. As h approaches 0, of the quantity of 2 multiplied by the quantity of X plus H in quantity squad minus 2 x 2 in quantity divided by H. So multiplying out the binomial and the numerator, this is going to be equal to the limit. As H approaches 0, of the quantity of 2 multiplied by the quantity of X2 plus 2 HX plus H2 in quantity minus 2 X 2 in quantity divided by H. So simplifying our numerator, this is going to be equal to the limit as H approaches 0 of the quantity, and in the numerator for the first term, we'll have 2 x 2, but also have a minus 2x2d later on. So those two terms will cancel out. For the next term, we'll have 2 multiplied by 2 HX which will give me 4 HX. Then we'll have plus 2 multiplied by 82, which is 28 squad. And that numerator is going to still be divided by H. If I look at my numerator, I can factor out an H out of both terms in the numerator, and that will cancel out with the h in the denominator. So that means that this is going to be equal to the limit, as H approaches 0 of the quantity of 4 X plus 2 H. And evaluating this limit by direct substitution, this is going to be equal to 4 X plus 2 multiplied by 0, which is equal to 4 X. So that is the slope of our tangent line at any point on our curve Y. So now we need to determine the point of tangency. And here we don't have any information given about the point of tangency. So we're just going to set the X value for the point of tangency equal to A. So we'll set X equal to A for that point. So that means that our y value is going to be equal to 2A2d. So we're really looking for the point of A2A2d. And we can actually write down an equation for the tangent line that goes through this point using the point slope form. So we'll have Y minus our particular Y value, which is to a squad, and that's going to be equal to our slope, which is 4 X, but X is equal to A, so we'll have a slope of 4A, and that gets multiplied by the quantity of X minus A. So now we need to figure out the value of A. And we're given another piece of information in our problem, that when X is equal to 2, That means that Y is equal to 0, since that is going to be the X intercept. And so, we can substitute in these values for X and Y into our equation for our tangent line. So doing so, we'll have 0 minus 2A squared is equal to 4 A. Multiplied by the quantity of 2 minus A. So we can simplify this expression. So doing so, we'll have minus 2A squad, is equal to 8A. Minus 4A squad. So to solve this equation for A, we're gonna add 4A2d to both sides, and so we'll get 2 A squared is equal to 8A. And so here we have to be careful, because if we were to divide by A, we could possibly be dividing by 0. But if we look at what A equal to 0 represents, that means that the point of tangency is going to be 0.0, which would be the origin, and that means that the tangent line cannot intersect the origin as well as the point X equal to 2.0. So, since it can't intersect both points at the same time, that means that A equal to 0 is not a solution to this equation. So we can divide both sides by A, since A cannot be equal to 0. And in doing, and we can also divide both sides by 2. In doing so, we'll get a as equal to 4. So that means that our point of tangency is going to be 4.2 multiplied by 4 squad, which is 32. So that is the point of tendency that we were looking for. Now we need to actually figure out our equation for our tangent line. Using the value of A. So if we look at our equation for a tangent line, we will have Y minus 32 is equal to 4 multiplied by 4, which is 16, multiplied by the quantity of X minus 4 in quantity. So, distributing the 16 on the right hand side will have Y minus 32 is equal to 16 X minus 64. And then add 32 to both sides, we will have Y is equal to 16 X minus 32. And so that is the equation for the tangent line that we were looking for. So I hope that this explanation has been useful, and I'll see you in the next video.

Tangent line to y = √x Does any tangent line to the curve y = √x cross the x-axis at x = −1? If so, find an equation for the line and the point of tangency. If not, why not?

Key Concepts

Tangent Line

Derivative

Intersection with the x-axis

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