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Problem 10 Easy Difficulty
Suppose $ 4x^2 + 9y^2 = 36, $ where $ x $ and $ y $ are functions of $ t $.
(a) If $ dy/dt = \frac {1}{3}, $ find $ dx/dt $ when $ x = 2 $ and $ y = \frac {2}{3} \sqrt 5. $
(b) If $ dx/dt = 3, $ find $ dy/dt $ when $ x = -2 $ and $ y = \frac {2}{3} \sqrt 5. $
Answer
a) $\frac{d x}{d t}=-\frac{\sqrt{5}}{4}$
b) $\frac{d y}{d t}=\frac{4 \sqrt{5}}{5}$
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Video Transcript
Why did you say we're told that four X squared plus nine y squared equals 36 on the show? So this is the equation of an ellipse where X and y are both functions of t Get this. Pardon me. We're told that dy DT equals one third and ratifying the x tt when X equals two. Yes. Cigarette. Yeah. And why equals two thirds of route five? Yes, disappear. So in order to find the x c t, I'm going to use implicit differentiation on our lips equation. So differentiating both sides of respect the teeth. So we have two times four is eight x times The expertise plus two times nine is 18 times y times dy DT and the right hand side is simply zero solving for the execution, the expert C is equal to negative 18 y times dy DT over eight x which you could simplify to negative nine wide dy DT over guys, we're we're at X plugging in our values. You get negative nine times two thirds route five times one third over four times two it's ms is equal to negative. See the 9/9 minutes to five or four times two, which is Route 5/4. All right, please. This is DX DT Negative. Route 5/4, actually, Sharp party. We're told that the X 15 equals three and rectifying the white tea when X equals negative two. And why it was two thirds or five. Said was plan now, just like in part a use implicit differentiation. And we get the same equation and I want to solve for dy DT instead of the expertise. So we have D y t t is equal to negative eight x the x 30 over 18 y couldn't you just lie up? This simplifies to negative four x over nine y the x 30 substituting our values. This is native four times negative. Two over nine times two thirds Route five Nice three is equal to the negatives. Cancel out 8%. Now this is eight over 19, 0 to 5, which is 4/5. And if you rationalize the denominator, this is equal to four. Route 5/5. Oh,
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