Question

Find an approximation to ex(x)sin(x) for small x. O 1.x-(x 2)/2+(xA3)16 О 2. -(x^2)+(x^3) O 3.x+(x 2)+(x3)2 о 4. xt(x2)+(x^3)/3 O 5.x+(x*2)-(xA3)/6

Answer 1

Write the formulae for x and e^x from basics and multiply them by neglecting less values from Small values of x as shown below by considering up to cube of x

Find e^x (sinx) approximation for small 'x' Step (1) e^x = 1 + x/1! + x^2/2! + x^3/3! + x^4/4! + x^5/5! + sinx = x - x^3/3! + x^5/5! - x^7/7! + x^9/9! ... step (2) e^x sinx = (1 + x/1! + x^2/2! + x^3/3! + ...) (x - x^3/3! + x^5/5! - x^7/7! +) e^sinx = [x - x^3/3! + x^5/5! - x^7/7! + ....] + [x^2 - x^4/3! + x^6 +] + [x^3/2! - x^5/12 ...] + [x^2 - x^4/3! + x^6/5! ..] + [x^3/2! - x^5/12 ...] + [x^4/6 - x^6/36 ...] = x - x^3/3! + x^2