# If #f(x) = ln(1+2x)#, where a = 2 and n = 3 how do you approximate f by a Taylor polynomial with degree n at the number a?

If that is the case, you can apply the Taylor Series formulation to third degree:

And so we need these:

We can then say that:

By signing up, you agree to our Terms of Service and Privacy Policy

To approximate ( f(x) = \ln(1 + 2x) ) by a Taylor polynomial with degree ( n = 3 ) at the number ( a = 2 ), we need to compute the first three derivatives of ( f(x) ) and evaluate them at ( x = 2 ).

- ( f(x) = \ln(1 + 2x) )
- First derivative: ( f'(x) = \frac{1}{1 + 2x} )
- Second derivative: ( f''(x) = -\frac{2}{(1 + 2x)^2} )
- Third derivative: ( f'''(x) = \frac{8}{(1 + 2x)^3} )

Now, evaluate these derivatives at ( x = 2 ):

- ( f(2) = \ln(1 + 2 \times 2) = \ln(5) )
- ( f'(2) = \frac{1}{1 + 2 \times 2} = \frac{1}{5} )
- ( f''(2) = -\frac{2}{(1 + 2 \times 2)^2} = -\frac{2}{25} )
- ( f'''(2) = \frac{8}{(1 + 2 \times 2)^3} = \frac{8}{125} )

The Taylor polynomial of degree ( n = 3 ) at ( a = 2 ) is given by:

[ P_3(x) = f(a) + f'(a)(x - a) + \frac{f''(a)}{2!}(x - a)^2 + \frac{f'''(a)}{3!}(x - a)^3 ]

Substituting the values we found:

[ P_3(x) = \ln(5) + \frac{1}{5}(x - 2) - \frac{1}{25}(x - 2)^2 + \frac{1}{125}(x - 2)^3 ]

This polynomial approximates ( f(x) = \ln(1 + 2x) ) near ( x = 2 ) with an error that decreases as ( n ) increases.

By signing up, you agree to our Terms of Service and Privacy Policy

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

- How do you find the maclaurin series expansion of #cos(x^3)#?
- Systems of ODEs. y'1=2y1+2y2 y'2=5y1-y2 By:y1(0)=0, y2(0)=7 (How about y. y=?)
- How do you find a power series representation for #ln(8-x)# and what is the radius of convergence?
- Find the first four terms of the Taylor Series: #f(x)=xe^x# given #a=0#?
- How can you find the taylor expansion of #f(x) = x^2 + 2x + 5# about x=3?

- 98% accuracy study help
- Covers math, physics, chemistry, biology, and more
- Step-by-step, in-depth guides
- Readily available 24/7