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If an object with mass $ m $ is dropped from rest, one model for its speed $ v $ after $ t $ seconds, taking air resistance into account, is $$ v = \frac{mg}{c}(1 - e^{-ct/m}) $$where $ g $ is the acceleration due to gravity and $ c $ is a positive constant. (In Chapter 9 we will be able to deduce this equation from the assumption that the air resistance is proportional to the speed of the object; $ c $ is the proportionality constant.)(a) Calculate $ lim_{t\to \infty} v $. What is the meaning of this limit?(b) For fixed $ t $, use l'Hospital's Rule to calculate $ lim_{c\to 0^+} v $. What can you conclude about the velocity of a falling object in a vacuum?
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02:27
Wen Zheng
01:20
Amrita Bhasin
Calculus 1 / AB
Calculus 2 / BC
Chapter 4
Applications of Differentiation
Section 4
Indeterminate Forms and l'Hospital's Rule
Derivatives
Differentiation
Volume
Campbell University
Baylor University
Idaho State University
Lectures
04:35
In mathematics, the volume of a solid object is the amount of three-dimensional space enclosed by the boundaries of the object. The volume of a solid of revolution (such as a sphere or cylinder) is calculated by multiplying the area of the base by the height of the solid.
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A review is a form of evaluation, analysis, and judgment of a body of work, such as a book, movie, album, play, software application, video game, or scientific research. Reviews may be used to assess the value of a resource, or to provide a summary of the content of the resource, or to judge the importance of the resource.
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So for this problem we're having tea, go to infinity. Um and we see that when T goes to infinity, we end up getting things cancel out, and we get MG oversee that's the first portion. But then for part B, what we're gonna do is we're letting when we have the indeterminate form As C approaches zero from the right, what we end up having is to take the derivative of the top and bottom. So once we use the hotel's rule, we end up getting by treating see as a mhm variable, we end up getting the limit. Uh see approaches zero of energy times T over em of E. To the negative C. T. Over end. Then as we when we plug in uh C two B zero, we end up getting just G. T. As the final answer for that.
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