Solve Maclaurin Expansion Equation: cos(x)-2x2=0

In summary, the conversation discusses using the first two terms of the Maclaurin series expansion of cos(x) to solve the equation cos(x)-2x2=0. The solution is found to be +-0.816497, which is reasonably accurate compared to the actual answer of +-.63456. The first two non-vanishing terms are used in the expansion.
  • #1
pat666
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Homework Statement


Use the first two terms of the Maclaurin series expansion of cos(x) to solve the equation cos(x)-2x2=0 . Check its accuracy with a calculator ( is in radians).


Homework Equations



f(x)=f(0)+x(f'(0)) first two terms

The Attempt at a Solution


So I have found the Maclaurin expansion of cos(x) to be 1. This seems ridiculous to me so I'm wondering if its correct?
then:
1-2x2=0 so x=+-(sqrt(2)/2)
=+-.239755

I'm not sure what to say about the accuracy??
FYI this is an exam practice Q
Thanks
 
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  • #2
You want the first two non-vanishing terms. Go to higher order in the expansion.
 
  • #3
ok, non vanishing means not 0?
 
  • #4
now I get cos(x)=1-x^2/2
and the solution is +-0.816497
the actual answer is +-.63456 - seems reasonably accurate?
 

FAQ: Solve Maclaurin Expansion Equation: cos(x)-2x2=0

1. What is a Maclaurin Expansion equation?

A Maclaurin Expansion equation is a mathematical expression that represents a function as an infinite series of terms. It is named after Scottish mathematician Colin Maclaurin and is a special case of a Taylor series, where the series is centered at x = 0.

2. How is a Maclaurin Expansion equation solved?

To solve a Maclaurin Expansion equation, we use the Taylor series formula: f(x) = f(a) + f'(a)(x-a) + f''(a)(x-a)2/2! + f'''(a)(x-a)3/3! + ... + f(n)(a)(x-a)n/n! where a is the center of the series and n is the degree of the polynomial we want to approximate. In this case, we use the given equation cos(x)-2x2=0 and set a = 0.

3. What is the purpose of solving a Maclaurin Expansion equation?

The purpose of solving a Maclaurin Expansion equation is to approximate a function with a polynomial. This can be useful in simplifying complex functions and making calculations easier. It is also used in many areas of science, such as physics and engineering, to model real-world phenomena.

4. What is the relation between Maclaurin Expansion and the cosine function?

The Maclaurin Expansion of the cosine function is cos(x) = 1 - x2/2! + x4/4! - x6/6! + ... + (-1)n x2n/(2n)! where n is a positive integer. This means that the cosine function can be approximated by a polynomial with an infinite number of terms, where each term represents the contribution of a specific degree of x to the function.

5. Can a Maclaurin Expansion equation be used to solve other types of equations?

Yes, a Maclaurin Expansion equation can be used to solve various types of equations, such as trigonometric, exponential, and logarithmic equations. It can also be used to approximate non-polynomial functions, such as square roots and rational functions, as long as the series converges to the function at the given point.

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