Taylor series problem (non-direct differentiation?)

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The discussion revolves around solving a Taylor series problem without direct differentiation. The original poster seeks an alternative method as direct differentiation is deemed too lengthy. A suggested solution involves multiplying the Taylor series of sin(7x) by x^3. This approach simplifies the problem significantly. The poster expresses gratitude for discovering this straightforward solution.
Loopas
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I attached a picture of the problem from my online HW. I know how to solve the problem through direct differentiation, but that would too long to find the derivatives for this problem, and the problem actually suggests that I find another way. So my question is, what's the best way to solve this?
 

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Loopas said:
I attached a picture of the problem from my online HW. I know how to solve the problem through direct differentiation, but that would too long to find the derivatives for this problem, and the problem actually suggests that I find another way. So my question is, what's the best way to solve this?

Multiply the Taylor series of \sin(7x) by x^3.
 
Thank you... wouldve never guessed it was so simple.
 

Thread 'Distance between a Clock's hands when the distance is increasing most rapidly'

Question: A clock's minute hand has length 4 and its hour hand has length 3. What is the distance between the tips at the moment when it is increasing most rapidly?(Putnam Exam Question) Answer: Making assumption that both the hands moves at constant angular velocities, the answer is ## \sqrt{7} .## But don't you think this assumption is somewhat doubtful and wrong?
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