Taylor Series Problems Help??

6)

f(x)= x^(-1)

f(1) = 1

f'(x) = (-1)x^(-2) = -1/x^2

f'(1) = -1

f''(x) = (-1)(-2)x^(-3) = 2/x^3

f''(1) = 2

f(x) = f(1) +(x-1) f'(1) + (x-1)^2 f''(1) / 2!

f(x) = 1 +(x-1)(-1) + (x-1)^2 (2)/(2!)

f(x) = 1 +1-x +(x^2-2x+1)

f(x) = x^2-3x+3

2)

f(x) = 3 + x + x^2

f(1) = 5

f'(x) = 1+2x

f'(1) = 3

f''(x) = 2

f''(1) = 2

f(x) = f(1) + (x-1) f'(1) +(x-1)^2 f''(1) /2!

f(x) = 5 +(x-1)(3) + (x-1)^2 (2/2)

f(x) = 5 +3x-3 +x^2-2x+1

f(x) = x^2+x+3

3)

f(x) = cos(2x)

f(pi) = cos(pi) = -1

f'(x) = -2sin(2x)

f'(pi) = 0

f''(x) = -4 cos(2x)

f''(pi) = -4 cos(2pi) = -4(1) = -4

f(x) = f(pi) + (x-pi) f'(pi) +(x-pi)^2 f''(pi) /2!

f(x) = -1 + (x-pi) (0) + (x-pi)^2 (-4/2)

f(x) = -1 -2 (x-pi)^2

4)

f(x)=sqrt(x)

f(4) = 2

f(x) = x^(1/2)

f'(x) = (1/2) x^(-1/2) = 1/(2x^(1/2)) = 1/(2sqrt(x))

f'(4) = 1/(2sqrt(4)) = 1/((2)(2)) = 1/4

f''(x) = (1/2)(-1/2) x^(-3/2) = -1/(4x^(3/2))

f''(4) = -1/(4 (4)^(3/2)) = -1/32

f(x) = f(4)+(x-4)f'(4) +(x-4)^2 f''(4)/2!

f(x) = 2 + (x-4) (1/4) + (x-4)^2 (-1/32)(1/2!)

f(x) = 2 + (1/4) x -1 +(x^2-8x+16)(-1/64)

f(x) = (-1/64)x^2 +x(1/4 +1/8) +2-1-1/4

f(x) = (1/64)x^2 +(3/8)x +3/4

General form of the Taylor Series:

f(a) * (x - a)^0 / 0! + f'(a) * (x - a)^1 / 1! + f''(a) * (x - a)^2 / 2! + f[3](a) * (x - a)^3 / 3! + .... + f[n](a) * (x - a)^n / n!

For instance, problem number 5

f(x) = 2 * cos(x)

a = pi/2

f(a) = 2 * cos(pi/2) = 2 * 0 = 0

f'(a) = -2 * sin(pi/2) = -2 * 1 = -2

f''(a) = -2 * cos(pi/2) = -2 * 0 = 0

f'''(a) = 2 * sin(pi/2) = 2 * 1 = 2

And it will repeat

0 + (-2) * (x - pi/2) / 1! + 0 * (x - pi/2)^2 / 2! + 2 * (x - pi/2)^3 / 3! + 0 * (x - pi/2)^4 / 4! + ..... =>

-2 * (x - pi/2) + 2 * (x - pi/2)^3 / 3! - 2 * (x - pi/2)^5 / 5! + 2 * (x - pi/2)^7 / 7! + .... =>

2 * (-1) * (x - pi/2) + 2 * (-1)^2 * (x - pi/2)^3 / 3! + 2 * (-1)^3 * (x - pi/2)^5 / 5! + 2 * (-1)^4 * (x - pi/2)^7 / 7! + ...

The infinite sum of: 2 * (-1)^n * (x - pi/2)^(2n - 1) / (2n - 1)! from n = 1 to n = infinity

Valuable advice: Its impolite to ask several questions per post. Ask just one per post. And ask one at a time & learn from it. Only then if u still need help, ask for further clarification.