Final answer:
The Taylor series expansion of f(x) = sin(πx/2) about x_=0 is determined by calculating the derivatives of the function. The remainder of the Taylor series can be calculated using the Lagrange form. To estimate the number of terms needed for accuracy, the Lagrange form can be used to find an upper bound. The Taylor polynomials can be plotted by substituting x-values into the polynomials.
Step-by-step explanation:
a) To expand the function f(x) = sin(πx/2) in a Taylor series about the point x_=0, we can start by calculating the derivatives of the function. The derivatives of sin(x) are: f'(x) = cos(x), f''(x) = -sin(x), f'''(x) = -cos(x), and so on. For this expansion, we only need the derivatives up to f^(7)(x).
Using the Taylor series formula, the expansion of f(x) about x_=0 becomes: f(x) = f(0) + f'(0)x + (f''(0)x²/2!) + (f'''(0)x³/3!) + ... + (f^(7)(0)x⁷/7!).
b) The remainder of the Taylor series can be calculated using the Lagrange form of the remainder: R_n(x) = (f^(n+1)(c)x^(n+1))/(n+1)!, where c is some value between x and 0.
c) To estimate the number of terms needed to guarantee accuracy within 10⁻⁵ for all x in the interval [-1,1], we can use the Lagrange form of the remainder again. By finding an upper bound for |f^(n+1)(c)| in the interval [-1,1], we can determine the minimum value for n that satisfies the condition.
d) To plot f(x) and its 1st, 3rd, 5th, and 7th degree Taylor polynomials over [-2,2], we can substitute the x-values in the interval into the respective polynomials and plot the resulting points.