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Let $\ell$ be the tangent line to the parabola $y=x^{2}$ at the point$(1,1)$ . The angle of inclination of $\ell$ is the angle $\phi$ that $\ell$makes with the positive direction of the $x$ -axis. Calculate $\phi$correct to the nearest degree.

$63^{\circ}$

00:53

Clarissa N.

Calculus 1 / AB

Chapter 2

Limits and Derivatives

Section 8

The Derivative as a Function

Limits

Derivatives

Campbell University

Harvey Mudd College

Baylor University

University of Nottingham

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This is a little bit of a tricky problem. We're told that we have this function that is a quadratic and we have this parabola and we're told that we have this angle of inclination which we're going to call five. And now our goal is to find that angle phi. Well, the first thing that we need to dio is find the derivative of our function. And in this case, I'm going to use the limit definition just for the sake of practice. So we gain our knowledge of limits, become more comfortable with this. So we know that F private X equals a limit as age approach zero of f of X plus h minus ffx all over h. And then we know that our function is X squared. So we're going to use that knowledge and plug it into our limit. We're going to get the lemon is age approaches zero of X plus h squared minus X squared all over h. And now I can simplify this a little bit knowing that I have a quadratic function. I'm going to get the limit. Is age approaches zero of X plus H plus x Times X plus H minus X all over h so we can simplify this Thio, Let our ages cancel. We're going to get the limit is h approaches zero of two x plus h times h all over H Now we can see that these ages air going to cancel and that makes our limit much easier to solve. So we're going to get the limit is h approaches zero of two x plus h And then we can just plug in 04 h and we'll get that are derivative f Prime of X is two X Now you, of course, could have done this using the power rule just not using the limit definition and you would have gotten the same thing. So then we're told the slope of the tangent We know the slope of the tangent Why eagles ffx is at X equals a were given this 0.11 So let's find f prime of one f Prime of one would be equal to two times one, which equals two. Now we have to incorporate this knowledge that the fact that we have this angle so we have this inclination this angle So the inclination of the tangent is this'll angle phi and we don't know that yet. So the slope that we're trying to find is the tangent of fi. Well, we know that tangent. If I equals two, that's what we just calculated at prime of one. But we want to know five. So how would I rearrange that? Well, if I would be equal to the arc tangent of two or another way to say that is the inverse tangent of to So then you can just plug that into a calculator and we'll find that that angle Phi equals 63 degrees. So I hope that this problem helped you understand. Not on Lee had to find the derivative using the limit depth definition, but also how we can incorporate our knowledge of tangents to find this angle of inclination in our parabola.

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