Taylor polynom and some functionproblem.

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SUMMARY

This discussion focuses on solving calculus problems involving derivatives and Taylor polynomials. The first problem requires finding the derivative y'(0) for the equation y² - e^sin(x) + xy = sin(x)cos(y) + 3, assuming y = y(x). The second problem involves calculating the third-degree Taylor polynomial for the function x ln(x) at x = 1 and using it to approximate 2 ln(2). The third problem examines the function g(x) = sin(x) / cos³(x), demonstrating that it has an inverse function in the interval x ∈ (-π/2, π/2) and finding the derivative of the inverse at x = 2.

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  • Understanding of calculus concepts, specifically derivatives and Taylor series.
  • Familiarity with inverse functions and their properties.
  • Knowledge of trigonometric functions and their behavior in specified intervals.
  • Ability to perform function approximations using Taylor polynomials.
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  • Study the process of finding derivatives of implicit functions.
  • Learn how to derive Taylor series expansions for various functions.
  • Explore the properties and applications of inverse functions in calculus.
  • Investigate the behavior of trigonometric functions and their inverses in specific intervals.
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Students and educators in mathematics, particularly those studying calculus, as well as anyone needing assistance with derivatives, Taylor series, and inverse functions.

NorwegianStud
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Anyone bored enough to want to help me out with some calculus? I got to deliver this in 6 hours and can't work these out. Help would be SO much appreciated, I've been at it all night and can't make it out.

1. y^2 - e^sin(x) + xy = sin (x)* cos (y) +3
assume y= y(x) and find y ' (0)

2. Find Taylor polynom of 3rd degree(correct English word?) for x ln x IF x = 1 AND use this to approximate 2 Ln 2. Use the estimate in E3 to find an intervall that contains 2 Ln 2.

3. g(x) = Sin (x) / Cos3 (x).
Show that g(x) have an inverse function in the interval x∈ ( -π/2, π/2). What is D(g^-1)? (Looks like a D, can't find the right one)

AND find the (d/dx)g^-1(x) in the point x = 2. (hint: g(π/4 = 2) )
 
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Bump. Please, anyone. I really need this to get my paper approved, and I don't know where else to ask.

REALLY would be appreciated. If you only can be bothered to solve one that is perfectly fine.
 
What have you tried so far?
 

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