How Do You Graph xcos(1/x) by Hand?

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SUMMARY

The discussion focuses on graphing the function xcos(1/x) by hand, particularly for exam preparation without calculators. The user differentiated the function using the product rule, resulting in the expression cos(1/x) + sin(1/x)/x. Key insights include the bounds of the function, specifically that -x ≤ xcos(1/x) ≤ x, and the behavior of cos(1/x) as x approaches infinity, where it approaches 1. Additionally, the oscillation of cos(1/x) as x approaches 0 is highlighted as a critical consideration for accurate graphing.

PREREQUISITES
  • Understanding of calculus, specifically differentiation and the product rule.
  • Familiarity with trigonometric functions and their properties.
  • Knowledge of limits and asymptotic behavior of functions.
  • Graphing techniques for non-standard functions without technological aids.
NEXT STEPS
  • Study the properties of the function xcos(1/x) in detail.
  • Learn about the behavior of trigonometric functions near zero and their oscillations.
  • Practice graphing functions that involve limits and asymptotic analysis.
  • Explore techniques for sketching graphs of functions without calculators.
USEFUL FOR

Students preparing for calculus exams, educators teaching graphing techniques, and anyone interested in understanding the behavior of oscillatory functions in mathematical analysis.

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Homework Statement


Draw a graph of xcos(1/x)2. The attempt at a solution

I've differentiated the equation using the product rule.

This got me to:

cos(1/x) + sin(1/x)/x

What I don't know how to do is actually go about drawing the graph. This is also a past paper question for an exam where I won't be allowed calculators.

Thanks!
 
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JamesBwoii said:

Homework Statement


Draw a graph of xcos(1/x)


2. The attempt at a solution

I've differentiated the equation using the product rule.

This got me to:

cos(1/x) + sin(1/x)/x

What I don't know how to do is actually go about drawing the graph. This is also a past paper question for an exam where I won't be allowed calculators.

Thanks!

Calculators won't help you with that.

(1) [itex]-x \leq x\cos(1/x) \leq x[/itex].
(2) [itex]\cos(1/x) \to 1[/itex] as [itex]x \to \infty[/itex].
(3) How fast does [itex]\cos(1/x)[/itex] oscillate at [itex]x \to 0[/itex]?
 

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