Find the taylor polynomial of degree 3 for f(x)=1/x+2
F(x)=-1/x+2)
well,
f = 1/(x+2)
f' = -1(x+2)^2
f" = 2/(x+2)^3
f(3) = -6/(x+3)^4
So, the Taylor series at x=0 is
1/2 - x/4 + x^2/8 - x^3/16 + ...
To find the Taylor polynomial of degree 3 for the function f(x) = 1/(x+2), we first need to find the derivatives of f(x) at a specific point.
The function f(x) = 1/(x+2) can be written as f(x) = (x+2)^(-1).
Taking the first derivative:
f'(x) = (-1)(x+2)^(-2) = -1/(x+2)^2
Taking the second derivative:
f''(x) = (-1)(-2)(x+2)^(-3) = 2/(x+2)^3
Taking the third derivative:
f'''(x) = (-1)(-2)(-3)(x+2)^(-4) = -6/(x+2)^4
Now, we need to evaluate these derivatives at a specific point, often chosen to be 0 when working with Taylor polynomials.
f(0) = 1/(0+2) = 1/2
f'(0) = -1/(0+2)^2 = -1/4
f''(0) = 2/(0+2)^3 = 1/4
f'''(0) = -6/(0+2)^4 = -3/8
Using these evaluated derivatives, we can write the Taylor polynomial of degree 3 for f(x) as follows:
P3(x) = f(0) + f'(0)x + (f''(0)/2!)x^2 + (f'''(0)/3!)x^3
P3(x) = (1/2) + (-1/4)x + (1/4)(x^2/2) + (-3/8)(x^3/6)
Simplifying this polynomial, we get:
P3(x) = 1/2 - 1/4x + 1/8x^2 - 1/16x^3
Therefore, the Taylor polynomial of degree 3 for f(x) = 1/(x+2) is P3(x) = 1/2 - 1/4x + 1/8x^2 - 1/16x^3.
To find the Taylor polynomial of degree 3 for the function f(x) = 1/(x + 2), we can use the formula for the Taylor series expansion. Let's go through each step to compute the coefficients.
Step 1: Determine the value for which we want to expand the Taylor series. In this case, let's choose the value a = 0.
Step 2: Calculate the derivatives of the function f(x) up to the desired degree. In this case, we need to calculate the derivatives up to the third degree. Let's find the derivatives:
f(x) = 1/(x + 2)
f'(x) = -1/(x + 2)^2
f''(x) = 2/(x + 2)^3
f'''(x) = -6/(x + 2)^4
Step 3: Evaluate the derivatives at x = a. Since a = 0, we evaluate the derivatives at x = 0:
f(0) = 1/(0 + 2) = 1/2
f'(0) = -1/(0 + 2)^2 = -1/4
f''(0) = 2/(0 + 2)^3 = 1/2
f'''(0) = -6/(0 + 2)^4 = -3/4
Step 4: Compute the coefficients of the Taylor polynomial using the derivatives and the powers of (x - a):
The Taylor polynomial of degree n for a function f(x) is given by the formula:
P(x) = f(a) + f'(a)(x - a) + f''(a)(x - a)^2/2! + f'''(a)(x - a)^3/3! + ...
Using the values we found in step 3, we can write the Taylor polynomial of degree 3 for f(x) as follows:
P(x) = f(0) + f'(0)(x - 0) + f''(0)(x - 0)^2/2! + f'''(0)(x - 0)^3/3!
P(x) = 1/2 - 1/4x + 1/4x^2/2! - 3/4x^3/3!
Simplifying the expression gives:
P(x) = 1/2 - 1/4x + 1/8x^2 - 1/32x^3