Find Degree 3 Taylor Polynomial Approximation of 5ln(sec(x))

In summary, the degree 3 Taylor polynomial approximation to the function f(x)=5ln(sec(x)) about x=0 can be found by using the Taylor polynomial equation and finding the derivatives of the function. When plugging in 0 for the x's, make sure to use the correct trigonometric function as sec 0 is not undefined.
  • #1
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Homework Statement



Find the degree 3 Taylor polynomial approximation to the function f(x)=5ln(sec(x)) about x=0.


Homework Equations



the taylor polynomial equation

The Attempt at a Solution



Here are my derivatives
f(x)=5ln(secx)
f'(x)=5tanx
f''(x)5sec^2(x)
f'''(x)=10sec^2(x)tanx

Please let me know if any of the above are wrong

When I try to plug in 0 for the x's above I end up with a whole lot of undefined answers because sec0=undefined. How can I get around this?

thanks
 
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  • #2
Are you sure that [itex]sec 0 = \frac{1}{cos 0}[/itex] is undefined?
 
  • #3
Fightfish said:
Are you sure that [itex]sec 0 = \frac{1}{cos 0}[/itex] is undefined?

oops, got sec and csc confused. Thanks
 

1. What is a degree 3 Taylor polynomial approximation?

A degree 3 Taylor polynomial approximation is a mathematical function that is used to approximate a more complex function by using its first, second, and third derivatives at a specific point. It is a type of power series expansion that can be used to estimate the value of a function at a certain point.

2. How is a degree 3 Taylor polynomial approximation calculated?

The degree 3 Taylor polynomial approximation is calculated using the formula:
f(x) ≈ f(a) + f'(a)(x-a) + (f''(a)/2!)(x-a)^2 + (f'''(a)/3!)(x-a)^3
where f(a) is the value of the function at a specific point, f'(a) is the first derivative of the function at that point, f''(a) is the second derivative, and f'''(a) is the third derivative. These values are then substituted into the formula to get the approximation.

3. What is the purpose of using a Taylor polynomial approximation?

The purpose of using a Taylor polynomial approximation is to estimate the value of a function at a specific point without needing to evaluate the function directly. It can also be used to approximate the behavior of a function near a given point, providing a simplified representation of the function's behavior.

4. What is the degree 3 Taylor polynomial approximation of 5ln(sec(x))?

The degree 3 Taylor polynomial approximation of 5ln(sec(x)) is:
5x - (5x^3)/6

5. How accurate is a degree 3 Taylor polynomial approximation?

The accuracy of a degree 3 Taylor polynomial approximation depends on the function and the point at which it is being approximated. Generally, the more derivatives that are used in the calculation, the more accurate the approximation will be. However, as the degree of the polynomial increases, the complexity of the calculations also increases. In some cases, a higher degree polynomial may be needed to achieve a desired level of accuracy.

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