Deriving the 3rd Taylor Polynomial for sin(4x^2)
The Taylor polynomial is a powerful mathematical tool used to approximate functions. It is particularly useful when exact calculations are too complex. In this article, we will derive the 3rd Taylor polynomial for the function sin(4x^2) centered at x 0. We will do this by calculating the derivatives of the function and applying the Taylor series expansion formula.
Understanding the Taylor Series Expansion
The n-th Taylor polynomial, P_n(x), of a function f(x) at a point x a is given by:
P_n(x) f(a) f'(a)(x-a) frac{f''(a)}{2!}(x-a)^2 frac{f'''(a)}{3!}(x-a)^3 ...
For this example, we are centering our polynomial at a 0, which simplifies the formula to:
P_n(x) f(0) f'(0)x frac{f''(0)}{2!}x^2 frac{f'''(0)}{3!}x^3
Step-by-Step Derivation of the 3rd Taylor Polynomial
We will follow these steps:
Calculate the derivatives of sin(4x^2) at x 0. Construct the 3rd Taylor polynomial using the values obtained.Step 1: Calculate the Derivatives at x 0
Zeroth Derivative:
f(0) sin(4*0^2) sin(0) 0
First Derivative:
f'(x) frac{d}{dx}sin(4x^2) 8x cos(4x^2)
f'(0) 8*0 * cos(0) 0
Second Derivative:
f''(x) frac{d}{dx}[8x cos(4x^2)] 8 cos(4x^2) - 64x^2 sin(4x^2)
f''(0) 8 cos(0) - 64*0^2 * sin(0) 8
Third Derivative:
f'''(x) frac{d}{dx}[8 cos(4x^2) - 64x^2 sin(4x^2)] -32x sin(4x^2) - 128x sin(4x^2) - 512x^3 cos(4x^2) -64x sin(4x^2) - 128x sin(4x^2) - 512x^3 cos(4x^2)
f'''(0) -64*0 * sin(0) - 128*0 * sin(0) - 512*0^3 * cos(0) 0
Step 2: Construct the 3rd Taylor Polynomial
Now, we use the values we calculated to construct the 3rd Taylor polynomial, P_3(x):
P_3(x) f(0) f'(0)x frac{f''(0)}{2!}x^2 frac{f'''(0)}{3!}x^3
Substituting the values:
P_3(x) 0 frac{8}{2}x^2 0 4x^2
Final Answer
The 3rd Taylor polynomial of sin(4x^2) centered at x 0 is:
P_3(x) 4x^2
Alternative Methods
Another method involves using the Taylor series for sin(x) and substituting 4x^2 instead of x. However, this requires a bit more algebraic manipulation.
In summary, the 3rd Taylor polynomial for sin(4x^2) centered at x 0 is 4x^2.