find taylor series...
a.f(x)=x^3-2x+4,a=2
To find the Taylor series of a function f(x) centered at a specific value, the general formula is:
f(x) = f(a) + f'(a)(x-a) + f''(a)(x-a)^2/2! + f'''(a)(x-a)^3/3! + ...
In this case, we have f(x) = x^3 - 2x + 4 and a = 2.
Step 1: Find the value of f(a)
Substitute x = 2 into the function:
f(2) = (2)^3 - 2(2) + 4
f(2) = 8 - 4 + 4
f(2) = 8
So, f(a) = f(2) = 8.
Step 2: Find the derivative of f(x)
f'(x) = 3x^2 - 2
Step 3: Find the value of f'(a)
Substitute x = 2 into the derivative:
f'(2) = 3(2)^2 - 2
f'(2) = 12 - 2
f'(2) = 10
So, f'(a) = f'(2) = 10.
Step 4: Find the second derivative of f(x)
f''(x) = 6x
Step 5: Find the value of f''(a)
Substitute x = 2 into the second derivative:
f''(2) = 6(2)
f''(2) = 12
So, f''(a) = f''(2) = 12.
Now we can start building the Taylor series:
f(x) = f(a) + f'(a)(x-a) + f''(a)(x-a)^2/2! + ...
f(x) = 8 + 10(x-2) + 12(x-2)^2/2! + ...
Simplifying further is possible, but this is the general form of the Taylor series of f(x) centered at a = 2.
To find the Taylor series of a function, we need to find its derivatives at a given point and use them to determine the coefficients of the series. In this case, we need to find the Taylor series of the function f(x) = x^3 - 2x + 4, centered at a = 2.
Step 1: Find the derivatives of the function f(x).
f'(x) = 3x^2 - 2, f''(x) = 6x, f'''(x) = 6
Step 2: Evaluate the derivatives at the point a = 2.
f'(2) = 3(2)^2 - 2 = 10
f''(2) = 6(2) = 12
f'''(2) = 6
Step 3: Write out the general form of the Taylor series terms.
The general form of the Taylor series for a function f(x) centered at a can be represented as follows:
f(x) = f(a) + f'(a)(x - a) + (f''(a)/2!)(x - a)^2 + (f'''(a)/3!)(x - a)^3 + ...
Step 4: Plug in the values of f(a) and the derivatives at a into the series.
f(x) = f(2) + f'(2)(x - 2) + (f''(2)/2!)(x - 2)^2 + (f'''(2)/3!)(x - 2)^3 + ...
= (2^3 - 2(2) + 4) + (10)(x - 2) + (12/2!)(x - 2)^2 + (6/3!)(x - 2)^3 + ...
Simplifying, we get:
f(x) = x^3 - 2x + 4 + 10(x - 2) + 6(x - 2)^2 + (x - 2)^3 + ...
Therefore, the Taylor series of f(x) centered at a = 2 is:
f(x) = x^3 - 2x + 4 + 10(x - 2) + 6(x - 2)^2 + (x - 2)^3 + ...