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6 pts) Another way to express the curvature of & plane curve y = f(z) is given by the formulaIf" ()|n(x) [1 +(f'(2))21372 Use that equation to prove t...

Question

6 pts) Another way to express the curvature of & plane curve y = f(z) is given by the formulaIf" ()|n(x) [1 +(f'(2))21372 Use that equation to prove that the curvature at an inflection point of f is 0.4. (12 pts) Let u(t) = fi(t)i + fz(t)j + fs(t)k and v(t) = 9(t)i + g2(t)j + gs(t)k; and let f(t) be a real valued continuous funetion Use the definition of the derivative of a vector function (s0__a limit) to prove the following differentiation rulcs:#[u(t) + v(t)] u' (t) +v(t)

6 pts) Another way to express the curvature of & plane curve y = f(z) is given by the formula If" ()| n(x) [1 +(f'(2))21372 Use that equation to prove that the curvature at an inflection point of f is 0. 4. (12 pts) Let u(t) = fi(t)i + fz(t)j + fs(t)k and v(t) = 9(t)i + g2(t)j + gs(t)k; and let f(t) be a real valued continuous funetion Use the definition of the derivative of a vector function (s0__a limit) to prove the following differentiation rulcs: #[u(t) + v(t)] u' (t) +v(t) #[f(t)u(t)] f'(t)u(t) + f(t)u'(t)



Answers

Use Theorem 10 to show that the curvature of a plane para-
metric curve $x = f ( t ) , y = g ( t )$ is

$$\quad \kappa = \frac { | \dot { x } y - y \overline { x } | } { \left[ \dot { x } ^ { 2 } + \dot { y } ^ { 2 } \right] ^ { 3 / 2 } }$$

where the dots indicate derivatives with respect to $t$

So for this problem, since we have, um, different notation how we're going to write this is we know that art of tea is x Y zero our crime of key. Since we can't use the dot notation with Dez most, we will just use the prime symbol. So of X crime. Why prime your, um and our little prime is going to end up giving us next a low crime quite of a prime and zero again. So when we compute the cross product, what will end up getting? Um, it's very simple because we have those zeros, um, in the 2nd and 3rd place. So all that will end up having as a result is X crime. Why? Double prime minus y crime X double crimes. That's the cross product of R prime and our double prime. So the way we can write this now is theme agnan itude of this, whatever that is going to be, we will signify with this right here and then that's going to be divided by whatever the magnitude of our prime is cubed. So if we look more closely, we know that it's going to be the square root of Okay, ex crime squared. Plus why Prime Squared all raised to the third. Another way we could write. That would be to just take this right here and raise it to the three halves instead of using the square root symbol. So that's exactly what we dio. And this is how we end up getting that final formula that we wanted to find.

In this problem, we have a Parametric er that is represented by the functions X equals f of tea and why equals g of t and this is a plane curve and we want to use theorem 10 which is over on the right to show that the curvature can be written as follows The absolute value of first derivative of X times, second derivative of why minus first derivative of why time Second derivative of X over first derivative of X squared plus first derivative of why squared. And that denominator is going to be raised to the three halves power. Now, just a note about notation. Typically, we don't often right derivatives with the dot super top of the of the function variable. But we're going to hear just the number of dots correspond Teoh the number of times we're taking the derivative of that function. So the first thing that I'm gonna do to tackle this problem is right, our plane curve as a vector function. So we're gonna write this as our of tea and for a plane curve. We know that the X uh component function is going to be FFT. Why component function is going to be GFT and because it's in, it's a curve that's in a plane R z coordinate here is just gonna be zero rz a component function. So in order to be able to apply fearing 10 we know that we need to have the first and second derivatives of the SPECTRE function are so particular first derivative. But I'm going to end up with is f prime of tea G prime of tea and zero I take the derivative again are a double prime is going to be equal to F double prime of tea G double private T and zero again And we want to take the cross product of the first and second derivatives. We're not gonna take the magnitude yet and remember that you can take the cross product by looking at the determinant of a three by three matrix. Go ahead and review that if you need to, but what we're going to end up with, I'm gonna write the final form here is zero zero and then our third component function is where we're going to have f prime of tea. Times G double prime of tea, minus F double prime of tee times G Prime of tea. Like I said, make sure that you review how Teoh find the cross product between two vectors if you need to. So then we're going to take the absolute value or sorry, the magnitude of that cross product. So our prime of t cross with our double prime if tea with the magnitude of that resulting vector. So we'll take the square root, our first component function squared and our second component functions squared. Just add up to be zero, so we don't need to write that. But we do. I need to write or third component function quantity squared under our square root sign. And remember that the square root of a squared value is the absolute value of the argument there. So we're looking at the absolute value F prime of tee times G double prime of T minus F double prime of tea, G prime of tea. So there's the numerator based off a theorem tent. Now what we want to do is also take the magnitude of our prime of tea, which we're gonna need for denominator. So our prime is right here. So if we take the magnitude of that we're gonna end up with our first component functions squared, plus our second component function squared. And then third component function squares just zero. So now we have enough to use theorem 10 and see. I'm gonna write it up here and plug things in to our formula here. So in our numerator, we're going to have absolute value. F private T G double prime of tea, minus F double prime of tee times G prime of tea and then in urgent nominator, we're going to have the square root uh, f prime of T squared plus g prime of T Square. And then that entire denominator is going to be raised to the third power. So what we want to do now is right. This in terms of the first and second derivatives of X and y so in our numerator f prime of tea corresponds to the first derivative of X. So I'm gonna write. That is X with the daughter over. Top G double prime of tea corresponds to second derivative of why rece attract f double prime of T, which is the second derivative of X times. First derivative of why actually gonna write that in the opposite order. So that way it corresponds to the relationship. We're trying to get to correspondence toe this name aerator right there. And then in our denominator, what we're gonna have is f prime of T is the first derivative of X Matt Quantity squared. We're adding first derivative of why squared and that's all under the square root sign and the race of the third power which we can also a right as raised to the three house power. And what we see is that this is indeed equal. Teoh, capital of T weaken right cap of tea in this notation.

Now we're asked to find the curvature for a smooth curve. That's given by some function of some function F. In the I direction plus some function G in the J direction. You can find a vector that is tangent to that curve by just taking the derivative, we can find the magnitude of that vector is just plus or minus square root of F. Prime square plus G. Prime square. And then our unit tangent vector is our tangent vector divided by its magnitude. We can then take the derivative of this guy to find a vector in the normal direction and we just just ugly. But if we crank through it we get one over the absolute value of gamma phew, fans this whole quantity and I won't write it out but we get this in the I direction and this in the J direction. Then we can find the curvature. Bye, taking the magnitude of this vector, dividing it by the magnitude of this vector. And after a whole lot of simplifications and things when you get down that that is the curvature is then the absolute value of G prime, F double prime minus F. Prime G double prime divided by absolute value of gamma key. Now we're asked to use this formula for a couple of cases. And so the first cases when f is T and G is the natural log of the sine of T. So f prime is one G prime is the co tangent of tea. That's double prime is zero G double prime is minus minus the consequent of tea. And if we plug into our formula and do some simplification we get that the curvature is sine of T. And in the second case we're given that G is the natural log of the hyperbolic co sign of T. And F is the inverse tangent of the hyperbolic sine of T. Taking derivatives. This becomes F. Prime becomes this is the hyperbolic secret of T. And then F double prime is minus the hyperbolic tangent times the hyperbolic second time's the hyperbolic tangent and then T. Um G prime is the hyperbolic tangent of T. G. Double prime is the hyperbolic second squared of P. And we can then plug all those into our formula and do a lot of simplification. And we get that curvature is the absolute value of secret hyperbolic secret of tea.

Now we're asked, defying the curvature for a smooth curve that's given by some function, some function F in the eye direction, plus some function G in the J direction. We can find a factor that is Tanja to that curve. By just taking derivative, you can find the magnitude of that factor is just pushing my square mood of prime squared plus two prime square and then you're a catch in Vector is our pageant director divided by its management. You can and take the derivative of this guy to find a vector in the normal direction and just just ugly. But if Frank through it, we get one over the absolute value of gamma times, asshole quantity, and I won't write it out. But we get this in the eye direction. This is the J direction. Then we can find the curvature. Bye, taking the magnitude of this rector, providing it by the magnitude of this factor and after a whole lot of simple occasions and things get down. That, that is curvature is than the absolute value of G prime F double prime minus F prime double crime provided by absolutely of gamma Q. Now I asked to use this formula for a couple of cases. And so the first cases when f is t Angie is the natural log of the sign of So f prime is one g prime is the co tension of tea that double primary zero double prime is minus minus tico secret of. And if we plug into our formula and do some simplification, forget that the curvature is sign of teeth and in the second case, were given that she is the natural log of the hyperbolic co sign of tea and f is the inverse tangent of the hyperbolic sign of teeth taking derivatives. This becomes f prime becomes this is the hyperbolic seeking of tea and then f double crime is minus the hyperbolic tangent times the hype hyperbolic seeking times, a hyperbolic qianjin and then, um g prime is the hyperbolic tangent of TNT. Double crime is the hyperbolic seeking squared, and we can then plug all those into our formula and do a a lot of simplification and we get that curvature is the absolute value of seeking hyperbolic seeking


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