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**Unformatted text preview: **Math 210
Kouba
Discussion Sheet 11 1.) Recall that if y = f(ac) is a function and
a0 + a1(:1: — a) + a2(a: — a)2 + (13(3) — (1)3 + a4(a: -— (1)4 + = z an(a7 — a)" is the Taylor Series (or Maclaurin series if a = 0 ) centered at :1: = a for y = f (11:) , then (n) a
an 2 f n'( ) . Use this formula to compute the ﬁrst four nonzero terms and the general
formula for the Taylor series expansion for each function about the given value of a . a.) f(a:) : em centered at as = 0 b.) f(:1:) = 6”” centered at x = ln2
1
1 — a: e.) f(:c) = l centered at a: = 1 f.) f(:v) 2 Va: + 5 centered at x = —1 centered at a: = 0 d.) f(ac) = sinx centered at a: = 0 2.) Use the suggested method to ﬁnd the ﬁrst four nonzero terms of the Maclaurin series
for each function. 1 1
a.) f (1:) = 2 (Substitute ~x2 into the Maclaurin series for 1 a: .)
x _
b.) f (:12) : $3639” (Substitute -—3a: into the Maclaurin series for e”8 and then
multiply by 3:3 .)
m 1 1
c.) f (x) = e : 6”” (Multiply the Maclaurin series for e” and term
1 — a: 1 — a: 1 — :5
by term and then group like powers of x .)
$
(1.) f (:13) = 1 (Use polynomial division. Divide the Maclaurin series for can by
— a:
1 ~— x ) e.) f (3:) = 3x2 cos(:c3) (Substitute :33 into the Maclaurin series for sinx then differ— entiate term by term.) 1
f.) f(:r) = arctanw (Integrate the Maclaurin series for 1 + t2 from 0 to as.) 1
3) The Maclaurin series for f(x) 2 1+ :3 is 1 — x + 3:2 — 51:3 + 3:4 — 3:5 +
a.) Show that f(a:) = 1: and 1 — a: + 1:2 — x3 + $4 —— x5 + have the same value
at
at x = 0 . 1
b.) Show that f(:1:) = 1 + and 1 — a: + x2 — $3 + 1:4 — $5 + have the same value
a: at :c = 1/2.
c.) Show that f(m) = value at :1: = 1 .
d.) For what x—values is f (x) = 1 1+2: and 1—x+m2——x3+x4—x5+ donot have the same 1 deﬁned ?
1 + a; e.) For what x—values is the Maclaurin series 1 — a: + x2 — x3 + x4 - x5 + deﬁned ? NOTE : It can be shown that f (2:) = 1 + a: x4 — x5 + are equal on the interval (—1,1). and its Maclaurin series 1 -— a: + x2 -— x3 + 4.) The following deﬁnite integral cannot be evaluated using the Fundamental Theorem of
calculus. Use the Maclaurin series for cosa: and the absloute error |R,,| for an alternating 1
series to estimate the value of this integral with error at most 0.0001 : / cos(a:2) d1:
0 5.) Write each Maclaurin series as an ordinary function. (3903 (393)5 (356V (306)9 a.) (3m) —— 3! + 5! — 7! + 9! -— ~- (HINT: Use sinm.)
b.) 172 — x3 + x4 — 11:5 + x6 — . -- (HINT: Factor.)
1 9:2 9:3 x4 w
C.)—-+3—$!+74—'+-5+%+W (HINTZUSQe.)
d.) x + 219+ 32:3 + 4174+ 52:5 + - .. (Challenging) -1 2 .
6.) (Challenging) Consider the function f (11;) = {e O/m , ‘f 1f :1: 35 0
, 1 a: = . a) Determine limf f.(:t:)
:c-eoo b.) Determine 32mm f(x). 0.) Determine lim f (as) d.) Use a graphing calculator to graph this function. e ) Compute f’ (0). You need to use the limit deﬁnition of the derivative.
f.) Compute f ’ ' (0). You need to use the limit deﬁnition of the derivative.
g.) It can be shown that f‘")(0) = 0 for n = 0, 1,2,3,4, h.) Determine the Maclaurin series for f (x) . i.) For what x—values is f (x) deﬁned ? j.) For what x-values is its Maclaurin deﬁned ? k.) For what x—values are f (x) and its Maclaurin series equal ? “In mathematics you don’t understand things. You just get used to them.” — Johann von
N eumann ...

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