Estimating (16.1)1/4 using Taylor's Expansion at x=16

In summary, the Taylor Expansion Problem is a mathematical concept that uses polynomials to approximate a function at a given point. It is solved using the Taylor Series and has many applications in mathematics, physics, and engineering. Some common errors when using this method include using an incorrect number of terms, using an invalid series, and not considering the remainder term. It is important to carefully consider the domain and accuracy of the series when using it to approximate a function.
  • #1
ruby_duby
46
0

Homework Statement



Use the taylor's expansion of f(x)= x1/4 about x= 16 to estimate (16.1)1/4

Homework Equations


Taylors formula: f(a) + f'(a) (x-a) + (f''(a)/2!) (x-a)2+...

The Attempt at a Solution



Ok I have calculate the taylor expansion to be: 2 + (1/32) (x-16)-(3/320) (x-16)2+ (7/262144) (x-16)3

I just don't know what to do after this - do i just substitute 16.1 into the value for x in the taylor expansion I have just found?
 
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  • #2
yep - then check whether you're close with a caclulator
 
  • #3
thanks it works
 

FAQ: Estimating (16.1)1/4 using Taylor's Expansion at x=16

What is the Taylor Expansion Problem?

The Taylor Expansion Problem is a mathematical concept that involves approximating a function using a series of polynomials. It is used to find the value of a function at a point by using the values of its derivatives at that point.

How is the Taylor Expansion Problem solved?

The Taylor Expansion Problem is solved by using the Taylor Series, which is a representation of a function as an infinite sum of terms, each of which is a multiple of a successive power of the variable.

What is the purpose of the Taylor Expansion Problem?

The purpose of the Taylor Expansion Problem is to approximate a function at a given point, especially when the function is difficult or impossible to evaluate directly. It is also used to improve the accuracy of numerical methods for solving differential equations.

What are the applications of the Taylor Expansion Problem?

The Taylor Expansion Problem has many applications in mathematics, physics, and engineering. It is used in calculus to solve problems involving rates of change, in numerical analysis to approximate functions, and in mechanics to model the motion of objects.

What are some common errors when using the Taylor Expansion Problem?

Some common errors when using the Taylor Expansion Problem include using an incorrect number of terms in the series, using a series that is only valid for a certain range of values, and not accounting for the remainder term in the approximation. It is important to carefully consider the domain and accuracy of the series when using it to approximate a function.

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