Can You Solve These Challenging Calculus Problems?

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    Calculus
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Discussion Overview

The discussion revolves around finding and developing challenging calculus problems suitable for a course project, specifically targeting first and second-year calculus topics. Participants share problem suggestions, methods for solving specific integrals, and approaches to limits, particularly the limit of sin(x)/x as x approaches 0.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested
  • Homework-related
  • Mathematical reasoning

Main Points Raised

  • One participant suggests focusing on problems involving trigonometric identities and substitutions, particularly in calculus II.
  • Another participant shares a resource from Harvey Mudd College with tutorials and quizzes on calculus topics.
  • Several participants propose specific integrals to solve, such as \(\int \sec\theta d\theta\) and \(\int_{0}^{\pi/2} \frac{dx}{1+(tan(x))^{\sqrt2}}\), discussing various methods for approaching these problems.
  • Multiple participants engage in a discussion about the limit \(\lim_{x\rightarrow 0} \frac{\sin x}{x}\), with differing methods proposed for proving it, including geometric interpretations and L'Hôpital's rule.
  • Some participants express uncertainty about the correctness of their methods and seek clarification on the reasoning behind the limit proofs.
  • Another participant introduces a problem involving maximizing a function, \(\max f(x) = \frac{1}{2^{x}} + \frac{1}{2^{1/x}}\), for \(x>0\).
  • Discussion includes the integral \(\int{x^x}dx\), with participants noting its complexity and the challenge of expressing it in elementary functions.

Areas of Agreement / Disagreement

Participants do not reach a consensus on the methods for proving the limit \(\lim_{x\rightarrow 0} \frac{\sin x}{x} = 1\), with multiple competing views and approaches presented. The discussion remains unresolved regarding the correctness of certain methods and interpretations.

Contextual Notes

Some participants reference specific calculus topics covered in their classes, such as improper integrals, parametric equations, and Taylor series, which may influence the types of problems they suggest. There is also mention of the limitations of certain methods, such as the need to avoid circular reasoning when using L'Hôpital's rule.

Who May Find This Useful

Students and educators interested in calculus problem-solving, particularly those looking for challenging problems or different approaches to limits and integrals.

  • #31
weagle2008 said:
Actually all you need is your trig identities:

sinx / x = cosx ; and since cos (0) = 1 ; sin (0) / 0 = 1

What? Take x = pi.

<br /> \frac{sin \pi}{\pi} = 0<br />

But...

<br /> cos \pi = -1<br />
 
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  • #32
Let t = time when this question was asked.

Find t.

:wink:
 
  • #33
l'Hôpital said:
What? Take x = pi.

<br /> \frac{sin \pi}{\pi} = 0<br />

But...

<br /> cos \pi = -1<br />

The question isn't as x approaches Pi it is as x approaches 0. As anyone knows Pi = 180 degrees. 2*Pi = 360 degrees which is = 0 in trig. Thus cos (Pi) = -1, but cos (0) = 1.
 
  • #34
weagle2008 said:
The question isn't as x approaches Pi it is as x approaches 0. As anyone knows Pi = 180 degrees. 2*Pi = 360 degrees which is = 0 in trig. Thus cos (Pi) = -1, but cos (0) = 1.

But you stated that (sinx)/x = cosx, which means that this is true for all x (which is wrong). This is what he was talking about.
sinx / x = cosx ; and since cos (0) = 1 ; sin (0) / 0 = 1

I would conjecture that you meant \lim_{x\rightarrow 0}\frac{sinx}{x} = \lim_{x\rightarrow 0} cosx. However, I'm not exactly sure how you got to that without L'H.
 

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