Power Series

Question 1

What do we use Taylor Polynomials for?

Answer 1

to approximate f(x) for an arbitrary function f close to a point x0

Question 2

Taylor Polynomials

P1(x)

Answer 2

P1(x) = f(x0) + f’(x0)*(x-x0)

Question 3

Taylor Polynomials

P2(x)

Answer 3

P2(x) = P1(x) + c2*(x-x0)^2

Question 4

Taylor Polynomials

P3(x)

Answer 4

P3(x) = P2(x) + c3*(x-x0)^3

Question 5

Taylor Polynomials

Pn(x)

Answer 5

Pn(x) = f(x0) + f’(x0)*(x-x0) + f’‘(x0)/2! *(x-x0)^2 +
f’’‘(x0)/3! * (x-x0)^3 + …
OR
Pn(x) = (r=0->n)Σ f’r(x0)/r! * (x-x0)^r

Question 6

Taylor’s Theorem

Answer 6

f(x) = Pn(x) + Rn(x)
where Rn(x) is the remainder or error

Question 7

Taylor’s Theorem

Rn(x)

Answer 7

Rn(x) = [f’(n+1)(c) / (n+1)! ]*(x-x0)^(n+1)

where c is an unknown number between x0 and x

Question 8

Maxima and Minima

Answer 8

• find the first non zero derivative of f
• if for this nth derivative, n is even:
• -if the derivative is greater than 0 then the point is a minima
• -if the derivative is less than 0 then the point is a maxima
• if the first non zero derivative is an odd derivative then it is a point of inflection

Question 9

Power Series

Answer 9

(n=0->∞) Σ an (x-x0)^n

where an are real constants

Question 10

Power Series Convergence

Answer 10

R = (n->∞)lim |an/a(n+1)|

-if R exists and is non-zero then the series converges for (x0-R)