Question

Find the taylor series about 0 for each of the functions below. give the first three non-zero terms for each.

x⁴ sin (x²) - x⁶ = ____

Answer 1

The Taylor series expansion of the given function about 0 is 0.

Step-by-step explanation:

The Taylor series expansion of a function about a point gives an approximation of the function using a polynomial. To find the Taylor series about 0 for the given function, we need to find the derivatives of the function and evaluate them at x = 0.

Let's start with the function f(x) = x⁴sin(x²) - x⁶. The derivatives of f(x) are: f'(x) = 4x³sin(x²) + 2x⁵cos(x²), f''(x) = 12x²sin(x²) + 20x⁴cos(x²) - 4x⁴sin(x²), and f'''(x) = 24xsin(x²) + 60x³cos(x²) - 16x³sin(x²).

Now let's evaluate these derivatives at x = 0:

f'(0) = 0, f''(0) = 0, f'''(0) = 0.

Since all the derivatives evaluated at x = 0 are 0, the Taylor series about 0 for the given function starts with the constant term. Therefore, the first three nonzero terms are:

f(x) ≈ f(0) + f'(0)x + f''(0)x²/2! = 0 + 0x + 0x²/2! = 0.T

he Taylor series or Taylor expansion of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the function and the sum of its Taylor series are equal near this point.