Taylors Polynomials
- Author:
- Robert Sneddon
- Topic:
- Algebra, Integral Calculus
Intro
Today we will be looking at Taylors Polynomials, what they are and what there used for.
To start, I would like you to complete the following questions:
Post Question Reflection
What you should have found is that sometimes we have integrals that either really really hard or just don't have a solution, such as example 3, and this is where Taylors Polynomials can help us out!
So what is a Taylor Polynomial I hear you ask well...
Taylor polynomials are approximations of a function. They turn any differentiable function from something nasty into a regular polynomial. Something which can look quite nasty such as is actually approximately equal to . The higher our degree of the Taylors Polynomial the better the approximation (this can be seen if we plot the two functions on the same set of axis).
How can we use this?
So first I will tackle how we can apply Taylors Polynomial and then how we compute Taylors Polynomial.
Take the question we posed earlier applying a Taylors Polynomial we can say that approximated to order 4. So our question simply becomes which we can easily compute to be . This isn't a very true approximation as our Taylors Polynomial skews away from our original curve. We could make our approximation more accurate just by increasing the order or our Taylor Polynomial.
How do I calculate a Taylors Polynomial?
Now you see the use of a Taylors Polynomial it's finally time to calculate some.
For this we will need to reach back into our minds and remember a little bit of Power Series.
Step 1) Calculate the first derivative of the function we are wanting to turn into a Taylors Polynomial,
Step 2) Substitute the value of x=a that your given in the question into the first derivative,
Step 3) Times that number by as we are working with the first derivative,
Step 4) Calculate the second derivative of the function we want to turn into a Taylors Polynomial,
Step 5) Substitute the value of a that your given into the second derivative,
Step 6) Times that number by as we are working with the second derivative.
And so on to your required degree of accuracy if this isn't sated calculate up to and including the 4th derivative.
As a formula this looks like.
Or as a series
Example Questions:
Find the Taylor Polynomial of near x=0 to degree 3.
Estimate the solution to near x=1 to degree 3.
Questions
1)
a) Find the Taylor Series of f(x)=ln(3+4x) about x=0
b) Find the Taylor Series of f(x)=exp(2x) about x=4
2) For this part integrate both f(x)'s between 0 and 1 with respect to x
3) Prove that the integral of exp(-(x^2)) between 0 and 3 with respect to x is approximately equal too 18.3