Q 3. Find the Taylor series of f at the point c for : (a) f(x) = e", c= 2 (c) f(x) = sin x, c= T (b) f(x) = }, c = 1 (d) f(x) = cos x, c= 5

1/2 O O The Taylor polynomial for the function f(x)=- cosx; x, = in sigma notation (-1)* (2*+1)! (-1)* (x- (2 k+1)! None

Let f(x) = cos (3x²) - 1 x² = Evaluate the 10th derivative of f at x = 0. f(10) (0) = Free-form Snip Hint: Build a Maclaurin series for f(x) from the series for cos(x).

Let f(x) = cos (4x³) - 1 x5 Evaluate the 7th derivative of f at x = 0. f(7) (0) = 0 Hint: Build a Maclaurin series for f(x) from the series for cos(x).

Consider the function f(x) = Vx + I. Compute f'(x) and f"(x). • (a) • (b) ) Write down the Taylor polynomials T1 and T2 of the function f at the point a = 0. • (c) DWhich value of b makes f(b) equal to V1.4? • (d) ) What are the estimates of V1.4 given by Tj and T2 from part (b)? (The actual value of V1.4 is about 1. 1832.)

7. (a) Find the Taylor series of the function f(x) = x³ – 2.x + 4 at x = 2 - (b) Find the Maclaurin series of the function f(x) = e

Find the series solution to 2xy'' – xy' + x (x + 1)y = 0 about x = 0.