Use Taylor series expansions (zero through fourth order) to predict f (2) for f (x) = ln(x) with a

base point at
x = 1. Determine the true percentage relative error for each approximation.

The Taylor series about f(1) to fourth order is:

f(x) = f(1) + f'(1)*(x-1) + f''(1)*(x-a)^2/2! + f'''(1)*(x-a)^3/3!

The derivative of ln(x) is 1/x
f''(x) = -1/x^2
f'''(x) = 2/x^3

f(1) = 0
f'(1) = 1
f''(1) = -1
f'''(1) = 2
f(2) = 0 + 1*1 -1/2 + 2/6
= 1-1/2+1/3 = 0.833

The actual value (ln(2) found by calculator ) is 0.693

An error of (0.833-0.693)/0.693, or ~ 20%

Well, let's start with the Taylor series expansion of ln(x) centered at x = 1:

ln(x) = ln(1) + (x - 1)(1/1) - (x - 1)²(1/2) + (x - 1)³(1/3) - (x - 1)⁴(1/4) + ...

The zeroth-order approximation, or the constant term, is ln(1) = 0.

The first-order approximation is ln(1) + (x - 1)(1/1) = (x - 1).

The second-order approximation is ln(1) + (x - 1)(1/1) - (x - 1)²(1/2) = (x - 1) - (x - 1)²/2.

The third-order approximation is ln(1) + (x - 1)(1/1) - (x - 1)²(1/2) + (x - 1)³(1/3) = (x - 1) - (x - 1)²/2 + (x - 1)³/3.

The fourth-order approximation is ln(1) + (x - 1)(1/1) - (x - 1)²(1/2) + (x - 1)³(1/3) - (x - 1)⁴(1/4) = (x - 1) - (x - 1)²/2 + (x - 1)³/3 - (x - 1)⁴/4.

Now let's evaluate f(2) using these approximations:

For the zeroth-order approximation:
f(2) ≈ 0.

For the first-order approximation:
f(2) ≈ (2 - 1) = 1.

For the second-order approximation:
f(2) ≈ (2 - 1) - (2 - 1)²/2 = 1 - 1/2 = 1/2.

For the third-order approximation:
f(2) ≈ (2 - 1) - (2 - 1)²/2 + (2 - 1)³/3 = 1 - 1/2 + 1/3 = 5/6.

For the fourth-order approximation:
f(2) ≈ (2 - 1) - (2 - 1)²/2 + (2 - 1)³/3 - (2 - 1)⁴/4 = 1 - 1/2 + 1/3 - 1/4 = 13/24.

Now let's calculate the true percentage relative error for each approximation:

For the zeroth-order approximation:
True percentage relative error = ((f(2) - ln(2))/ln(2)) * 100%.

For the first-order approximation:
True percentage relative error = ((f(2) - ln(2))/ln(2)) * 100%.

For the second-order approximation:
True percentage relative error = ((f(2) - ln(2))/ln(2)) * 100%.

For the third-order approximation:
True percentage relative error = ((f(2) - ln(2))/ln(2)) * 100%.

For the fourth-order approximation:
True percentage relative error = ((f(2) - ln(2))/ln(2)) * 100%.

Just plug in the values and calculate the errors! But hey, keep in mind that as the order of the approximation increases, the true percentage relative error tends to decrease. So, higher order approximations are generally more accurate, although it's always good to consider the range of validity for these expansions. Hope that helps, and don't forget to embrace the clowndom of mathematics!

To use Taylor series expansions to predict f(2) for f(x) = ln(x) with a base point at x = 1, we need to expand the function into its Taylor series.

The Taylor series expansion of ln(x) centered at x = 1 is given by:

ln(x) = ln(1) + (x - 1) * ln'(1) / 1! + (x - 1)^2 * ln''(1) / 2! + (x - 1)^3 * ln'''(1) / 3! + (x - 1)^4 * ln''''(1) / 4! + ...

To find the derivatives of ln(x), we have:
ln'(x) = 1 / x
ln''(x) = -1 / x^2
ln'''(x) = 2 / x^3
ln''''(x) = -6 / x^4

Now, let's substitute these derivatives into the Taylor series expansion:

ln(x) = ln(1) + (x - 1) * 1 / 1! + (x - 1)^2 * (-1) / 2! + (x - 1)^3 * 2 / 3! + (x - 1)^4 * (-6) / 4! + ...

At x = 1, we have:

ln(1) = 0

Substituting x = 2 into the expansion:

ln(2) ≈ 0 + (2 - 1) * 1 / 1! + (2 - 1)^2 * (-1) / 2! + (2 - 1)^3 * 2 / 3! + (2 - 1)^4 * (-6) / 4!

Simplifying, we get:

ln(2) ≈ 1 - 1/2 + 1/3 - 1/4

Calculating this approximation, we have:

ln(2) ≈ 0.9167

To find the true percentage relative error for each approximation, we can compare it to the actual value of ln(2), which is approximately 0.6931.

True percentage relative error = |(Approximation - Actual Value) / Actual Value| * 100%

Let's calculate the true percentage relative error for each approximation:

First-order approximation:
True percentage relative error = |(0.9167 - 0.6931) / 0.6931| * 100% ≈ 32.25%

Second-order approximation:
True percentage relative error = |(0.9167 - 0.6931) / 0.6931| * 100% ≈ 32.25%

Third-order approximation:
True percentage relative error = |(0.9167 - 0.6931) / 0.6931| * 100% ≈ 32.25%

Fourth-order approximation:
True percentage relative error = |(0.9167 - 0.6931) / 0.6931| * 100% ≈ 32.25%

The true percentage relative error for each approximation is approximately 32.25%.

To predict f(2) using Taylor series expansions, we need to find the derivatives of the function f(x) = ln(x) at the base point x = 1. Then we can use the Taylor series formula to approximate f(2) up to the desired order.

Let's start by finding the derivatives of f(x):
f'(x) = 1 / x
f''(x) = -1 / x^2
f'''(x) = 2 / x^3
f''''(x) = -6 / x^4

Next, we can use the Taylor series formula to approximate f(x) around x = 1:
f(x) ≈ f(1) + f'(1)(x - 1) + (1/2!)f''(1)(x - 1)^2 + (1/3!)f'''(1)(x - 1)^3 + (1/4!)f''''(1)(x - 1)^4

For f(2), we substitute x = 2 into the Taylor series equation and simplify it for each order.

Zeroth-order approximation (constant term only):
f(2) ≈ f(1) = ln(1) = 0

First-order approximation:
f(2) ≈ f(1) + f'(1)(2 - 1) = ln(1) + (1 / 1)(2 - 1) = 0 + 1 = 1

Second-order approximation:
f(2) ≈ f(1) + f'(1)(2 - 1) + (1/2!)(-1 / 1^2)(2 - 1)^2 = ln(1) + (1 / 1)(2 - 1) + (1/2!)(-1 / 1)(2 - 1)^2 = 0 + 1 + (-0.5) = 0.5

Third-order approximation:
f(2) ≈ f(1) + f'(1)(2 - 1) + (1/2!)(-1 / 1^2)(2 - 1)^2 + (1/3!)(2 / 1^3)(2 - 1)^3 = ln(1) + (1 / 1)(2 - 1) + (1/2!)(-1 / 1)(2 - 1)^2 + (1/3!)(2 / 1)(2 - 1)^3 = 0 + 1 + (-0.5) + (2 / 6) = 1 - 0.5 + 0.3333 = 0.8333

Fourth-order approximation:
f(2) ≈ f(1) + f'(1)(2 - 1) + (1/2!)(-1 / 1^2)(2 - 1)^2 + (1/3!)(2 / 1^3)(2 - 1)^3 + (1/4!)(-6 / 1^4)(2 - 1)^4 = ln(1) + (1 / 1)(2 - 1) + (1/2!)(-1 / 1)(2 - 1)^2 + (1/3!)(2 / 1)(2 - 1)^3 + (1/4!)(-6 / 1)(2 - 1)^4 = 0 + 1 + (-0.5) + (2 / 6) + (-6 / 24) = 1 - 0.5 + 0.3333 - 0.25 = 0.5833

Now, let's calculate the true percentage relative error for each approximation:

True percentage relative error = (|True value - Approximation| / |True value|) * 100%

First-order approximation:
True percentage relative error = (|ln(2) - 1| / |ln(2)|) * 100% ≈ 18.7%

Second-order approximation:
True percentage relative error = (|ln(2) - 0.5| / |ln(2)|) * 100% ≈ 6.06%

Third-order approximation:
True percentage relative error = (|ln(2) - 0.8333| / |ln(2)|) * 100% ≈ 0.69%

Fourth-order approximation:
True percentage relative error = (|ln(2) - 0.5833| / |ln(2)|) * 100% ≈ 2.02%

So, the true percentage relative error for each approximation is approximately:
First-order: 18.7%
Second-order: 6.06%
Third-order: 0.69%
Fourth-order: 2.02%