Taylor Series

Question 1

Taylor vs Maclaurin

Answer 1

Taylor approximates the value of a function around x = a, Maclaurin only around x = 0

Question 2

f(x+a) Taylor formula

Answer 2

f(a) + f’(a) x + (f’‘(a)/2!)x^2 + (f’’‘(a)/3!)x^3 + …

Maclaurin but replace any f(0) with f(a)

Question 3

f(x) Taylor formula

Answer 3

f(a) + f’(a) (x-a) + (f’‘(a)/2!)(x-a)^2 + (f’’‘(a)/3!)(x-a)^3 + …

Question 4

lim x->c (f(x) + g(x))

Answer 4

lim x->c f(x) + lim x->c g(x)

Question 5

lim x->c (f(x) / g(x))

Answer 5

lim x->c g(x)

Question 6

lim x->c (f(x) * g(x))

Answer 6

lim x->c f(x) * lim x->c g(x)

Question 7

lim x->c af(x)

Answer 7

a(lim x->c f(x))

Question 8

Indeterminate forms tactic

Answer 8

• Replace with a Taylor expansion where possible

- When you have f(x)g(x) giving 0*∞, use f(x)/(1/g(x))

Question 9

Differential equations Taylor

Answer 9

Find values of y, dy/dx, d^2y/dx^2 etc for x = x0 by differentiating and substitute into the f(x) Taylor formula using (x-x0)^n