Taylor`s expansions - Proposed Exercises

Taylor`s expansions - Proposed Exercises

Taylor’s expansions - Proposed Exercises
1. Compute the Taylor’s expansion of order n in x0 of the following functions:
(a) f (x) = 2x
(n = 4, x0 = 0)
(b) f (x) = log (4 − x2 )
2
(n = 3, x0 = 1)
(c) f (x) = sin x − sin x2
(n = 4, x0 = 0)
n = 4, x0 = π2
n = 2 , x0 = π3 .
(d) f (x) = sin x
(e) f (x) = sin x
2. Compute the principal part with respect to the sample 1/x or x:
1
1
(a) f (x) = e x − esin x ,
x → +∞
(b) f (x) = sin (sin x) − x cos x,
x→0
(c) f (x) = sin x cos 2x + sin 2x − 1 ,
x→0
(d) f (x) = sin x x − log(1 + x) ,
x → 0.
3. Compute the following limits using Taylor’s expansions:
2
ex − cos x − 23 x2
x→0
x4
2
sin x − sin x2
(b) lim 2
x→0 x log (cos x)
(a) lim
2
51+tan x − 5
(c) lim
x→0
1 − cos x
1
(1 + x) x − e
x→0
x
1
sin x x2
(e) lim
x→0
x
1
2
(f) lim x − x log 1 + sin
x→+∞
x
(d) lim
ex − 1 + log (1 − x)
x→0
tan x − x
1
1
1
−
(h) lim
.
x→0 x
sin x x
(g) lim
1
Solutions
1
1
1
1. (a) f (x) = 1 + x log 2 + x2 log2 2 + x3 log3 2 + x4 log4 2 + o x4 , x → 0;
2
24
6
(b) f (x) = 5 + 4(x − 1) + o (x − 1)2 , x → 1;
1
(c) f (x) = − x4 + o x4 , x → 0;
3 π 2
π 4
π
1 π 4
1
x−
x−
, x→ ;
+
+o
x−
(d) f (x) = 1 −
2
2
√ 2 2 24√ 2 3 1
3
π
π 2
π
π 2
x−
−
x−
, x→ .
+o x−
+
(e) f (x) =
2
2
3
4
3
3
3
2. (a)
1 1
,
6 x3
3. (a)
11
1
2
e
1
1
1
; (b) ; (c) 10 log 5; (d) − ; (e) e− 6 ; (f) ; (g) − ; (h) .
24
3
2
2
2
6
x → +∞; (b)
1 3
x ,
6
x → 0; (c) 2x2 ,
x → 0; (d)
2
1 3
x ,
2
x → 0.