The McLaurin series for a function f(x) is given by 1 d f (0), then f...

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The McLaurin series for a function f(x) is given by 1 d" f (0), then f (x) = ax, where a == k-0 Answer two of the following three parts. Each part carries 10 marks. n! dx" a) The function f(x) is given by the expression d" f dx" c) The variables y and x satisfy the equation -(0)=n!a,. f(x)=xe *. Find the first six terms of the McLaurin series for the function. Check your result by using the McLaurin series for the exponential function. b) Show that the function f(x) = xcos(x) satisfies the differential equation d f 2 dx -f- 2 sin(x). With this result, find the first three non-zero terms of the McLaurin series for f(x) as a function of x, by finding the derivatives of f(x) at x = 0 using the differential equation. dy dy +2 +9y=0, with = 0 and y = 2 when x = 0. dx dx dy dx Set up a recurrence relation for the values of the derivatives of y at x = 0, and from this construct a recurrence relation for the coefficients of the McLaurin series for y(x). Find the first four non-zero terms of the McLaurin series for y(x) and state how these coefficients will behave as n increases. The McLaurin series for a function f(x) is given by 1 d" f (0), then f (x) = ax, where a == k-0 Answer two of the following three parts. Each part carries 10 marks. n! dx" a) The function f(x) is given by the expression d" f dx" c) The variables y and x satisfy the equation -(0)=n!a,. f(x)=xe *. Find the first six terms of the McLaurin series for the function. Check your result by using the McLaurin series for the exponential function. b) Show that the function f(x) = xcos(x) satisfies the differential equation d f 2 dx -f- 2 sin(x). With this result, find the first three non-zero terms of the McLaurin series for f(x) as a function of x, by finding the derivatives of f(x) at x = 0 using the differential equation. dy dy +2 +9y=0, with = 0 and y = 2 when x = 0. dx dx dy dx Set up a recurrence relation for the values of the derivatives of y at x = 0, and from this construct a recurrence relation for the coefficients of the McLaurin series for y(x). Find the first four non-zero terms of the McLaurin series for y(x) and state how these coefficients will behave as n increases.