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Implicit Differentiation
Not all equations can be written like y = f(x), so taking the derivative can be tricky. Save the mess and do it directly with implicit differentiation.
In a fairy tale, a wizard rides a cloud which is moving to the right at a speed of \(15\text{ m/s.}\) The wizard throws a ball vertically upward with a speed of \(3\text{ m/s}\) relative to the cloud. When the cloud has moved to the right \(30 \text{ m}\) following the throw, what is the velocity of the ball as seen by people on the ground?
Assume that air resistance is negligible and ignore the effect of gravity.
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Two people \(A\) and \(B\) start off at the point \((0,0)\) at the same time and walk east and north at \(10\text{ m/min}\) and \(6\text{ m/min},\) respectively. What is the rate at which their distance increases after \(3\text{ min}?\)
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Note: The above diagram is not drawn to scale.
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An aircraft is flying 8000 feet above the ground. It is moving due east at a velocity of 900 feet per second and is located due east of an observation tower. At a certain moment, the angle of elevation is \( \frac{ \pi}{4} \). At that time, what is the rate of increase of the angle of elevation (in radians per second)?
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