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Problem 38 Hard Difficulty

Use the formula in Exercise 36 to find the curvature.
$$x=a \cos \omega t, \quad y=b \sin \omega t$$

Answer

$$
\kappa(t)=\frac{|a b|}{\left(a^{2} \sin ^{2} \omega t+b^{2} \cos ^{2} \omega t\right)^{\frac{3}{2}}}
$$

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Video Transcript

in this problem, we want to use the formula found and exercise 36 which is on the top right corner to find the curvature of the curve represented by functions. X equals eight times who sign of Omega T and why equals beer times Sign of Omega T. And for this problem, we're just going to assume that A, B and Omega are constants. So let's take the first derivative and the second derivative of our X component function. So the first derivative of X, with respect to T is going to be see negative A times Omega Times sign of omega T using archangel there Second derivative then is going to be negative. A times Omega squared It was co sign of a mega t well in taking the first and second derivatives of why for why we get be Times, Omega Times Co sign of Omega T and for why Double prime. We get negative B times, omega squared times the sign of omega T. All right, let's apply our formula. So we have capital of tea. It's going to be equal to, and our numerator have first derivative of X time. Second derivative of why minus first derivative of why times second derivative of X and our denominator. We have the first derivative of X squared, plus the first derivative. Why squared on that denominator is raised to the three house power. All right, let's make this a little bit cleaner. So inter numerator within the absolute value signs we're going to have hey times be times a mega cubed time sine squared of a mega t that we're gonna have plus a B a mega cubed again Times co sine squared of dignity In our denominator we're going to have a squared omega squared sine squared of omega T plus B squared omega squared ho sine squared a mega t that denominators raised to the three house power So in the numerator we see that we can factor out uh, a b omega cubed. And then we're left with sine squared plus co sine squared which ends up just equaling one. So inter numerator we have a times B times of mega cubed, an inner denominator. What I'm gonna do is I'm going to factor the Omega squared out of each term and end up with is Omega squared. Race to the three halves power times a squared science weird of Omega T plus B squared co sine squared a mega T raised to the three house power. Now remember, eso weaken bright that term. I'm gonna do this off to the side. Um, we can write the term Omega Square raised to the three halves power as the square root of omega squared quantity cubed. So then what we get here is the absolute value of Omega Hume. So what I'm gonna right here in my numerator high rate, the absolute value of a B Omega Cube as absolute value of a B times absolute value of like a cubed. And my denominator will have absolute value of omega cubed times a a squared science weird omega T plus B squared neuroscience weird mega t raised to the three halfs power. And we know that if we take the absolute value of omega cubed war, we take absolute value of omega and then cube that result there's air going to be equivalent, so these two terms are gonna cancel out. So what we're left with is absolute value of a B over a squared times sine squared of omega T plus B squared times co sine squared Omega T all raised to the three house power. And that represents the curvature, uh, the curve represented by the functions X and Y.

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