UNIT 10] power & taylor series

Question 1

what is the purpose of finding several derivates at one point of a function?

Answer 1

it gives a better approximation polynomial for the given function

[with each derivative a new property is identified (ex. first = increasing/decreasing, second = concavity), as more proparities are identified the entire function can be approximated better.]

Question 2

what is the center of the polynomial?

Answer 2

where the polynomial is expanded about

Question 3

what is the polynomial referred to if the center is at 0?

Answer 3

maclaurin polynomial

Question 4

what are 2 ways taylor polynomials are usually denoted?

Answer 4

(P_n)(x) or (T_n)(x)

Question 5

if f(x) has n derivatives at c (center), then what is the polynomial called?

Answer 5

the nth taylor polynomial for f(x) at c

Question 6

if f(x) has n derivatives at c (center) = 0, then what is the polynomial called?

Answer 6

the nth maclaurin polynomial for f(x) at c

Question 7

what do the a_n values have to match up with?

Answer 7

the derivatives of f(x)
ex. a_0 = f(x)
a_1 = f’(x)
a_2 = f’‘(x)
a_3 = f’’‘(x)

Question 8

what does the nth taylor polynomial for f(x) at c look like?

Answer 8

(P_n)(x) = f(c) + f’(c)(x-c) + ((f’‘(c))((x-c)^2))/(2!) + … + ((f’n’(c))((x-c)^n))/(n!)

Question 9

what does the nth maclaurin polynomial for f(x) at c look like?

Answer 9

(P_n)(x) = f(0) + f’(0)(x) + ((f’‘(0))(x^2))/(2!) + … + ((f’n’(0))(x^n))/(n!)

Question 10

what does “_____ degree taylor polynomial” mean?

Answer 10

the degree to which the last term of the polynomial will be raised, the n value

Question 11

what does a taylor polynomial that includes infinitely many terms become?

Answer 11

a taylor series that converges to the given function within the intervals of convergence

Question 12

what taylor series is “1 + x + (x^2)/(2!) + (x^3)/(3!) + … + (x^n)/(n!) + …” ?

Answer 12

the taylor series of e^x

Question 13

what taylor series is “x - (x^3)/(3!) + (x^5)/(5!) - (x^7)/(7!) + … + (((-1)^n)(x^(2n+1)))/(2n+1)! + …” ?

Answer 13

the taylor series of sin(x)

Question 14

what taylor series is “x - (x^2)/(2!) + (x^4)/(4!) - (x^6)/(6!) + … + (((-1)^n)(x^(2n)))/(2n)! + …” ?

Answer 14

the taylor series of cos(x)

Question 15

what does a geometric series look like?

Answer 15

∑ ∞ n=0 (a((r)^n)

Question 16

when does a geometric series converge?

Answer 16

when |r|<1

Question 17

how do you find the sum (S) of a geometric series that converges?

Answer 17

(a)/(1-r) = S

Question 18

what is the first step to finding a taylor series for a function?

Answer 18

ensure the denominator starts with a 1

Question 19

what is the second step to finding a taylor series for a function?

Answer 19

ensure the r value is being subtracted from 1 (1-r)

Question 20

what is the third step to finding a taylor series for a function?

Answer 20

determine a and r

Question 21

what is the fourth and last step to finding a taylor series for a function?

Answer 21

write the power series as h(x) = ∑ ∞ n=0 (a((r)^n)

Question 22

how do you find the general term of a series’ derivative?

Answer 22

the original general term is given to you by the original series, simply take the derivative

Question 23

do you differentiate n, x, or both?

Answer 23

only x, because n is a constant