1.
Expand
e
x
{\displaystyle e^{x}}
x
−
2
{\displaystyle x-2}
Ans.
e
x
=
e
2
+
e
2
(
x
−
2
)
+
e
2
2
!
(
x
−
2
)
2
+
⋯
.
{\displaystyle e^{x}=e^{2}+e^{2}(x-2)+{\tfrac {e^{2}}{2!}}(x-2)^{2}+\cdots .}
2.
Expand
x
3
−
2
x
2
+
5
x
−
7
{\displaystyle x^{3}-2x^{2}+5x-7}
x
−
1
{\displaystyle x-1}
Ans.
−
3
+
4
(
x
−
1
)
+
(
x
−
1
)
2
+
(
x
−
1
)
3
.
{\displaystyle -3+4(x-1)+(x-1)^{2}+(x-1)^{3}.}
3.
Expand
3
y
2
−
14
y
+
7
{\displaystyle 3y^{2}-14y+7}
y
−
3
{\displaystyle y-3}
Ans.
−
8
+
4
(
y
−
3
)
+
2
(
y
−
3
)
2
.
{\displaystyle -8+4(y-3)+2(y-3)^{2}.}
4.
Expand
5
z
2
+
7
z
+
3
{\displaystyle 5z^{2}+7z+3}
z
−
2
{\displaystyle z-2}
Ans.
37
+
27
(
z
−
2
)
+
5
(
z
−
2
)
2
.
{\displaystyle 37+27(z-2)+5(z-2)^{2}.}
5.
Expand
4
x
3
−
17
x
2
+
11
x
+
2
{\displaystyle 4x^{3}-17x^{2}+11x+2}
x
−
4
{\displaystyle x-4}
6.
Expand
5
y
3
+
6
y
3
−
17
y
2
+
18
y
−
20
{\displaystyle 5y^{3}+6y^{3}-17y^{2}+18y-20}
y
+
4.
{\displaystyle y+4.}
7.
Expand
e
x
{\displaystyle e^{x}}
x
+
1
{\displaystyle x+1}
8.
Expand
sin
x
{\displaystyle \sin x}
x
−
α
{\displaystyle x-\alpha }
9.
Expand
cos
x
{\displaystyle \cos x}
x
−
α
{\displaystyle x-\alpha }
10.
Expand
cos
(
a
+
x
)
{\displaystyle \cos(a+x)}
x
{\displaystyle x}
Ans.
cos
(
a
+
x
)
=
cos
a
−
x
sin
a
−
x
2
2
!
+
x
3
3
!
sin
a
+
⋯
.
{\displaystyle \cos(a+x)=\cos a-x\sin a-{\tfrac {x^{2}}{2!}}+{\tfrac {x^{3}}{3!}}\sin a+\cdots .}
11.
Expand
log
(
x
+
h
)
{\displaystyle \log(x+h)}
x
.
{\displaystyle x.}
Ans.
log
(
x
+
h
)
=
log
h
+
x
h
+
x
2
2
h
2
+
x
3
3
h
3
+
⋯
.
{\displaystyle \log(x+h)=\log h+{\tfrac {x}{h}}+{\tfrac {x^{2}}{2h^{2}}}+{\tfrac {x^{3}}{3h^{3}}}+\cdots .}
12.
Expand
tan
(
x
+
h
)
{\displaystyle \tan(x+h)}
h
.
{\displaystyle h.}
Ans.
tan
(
x
+
h
)
=
tan
h
+
h
sec
2
x
+
h
2
sec
2
x
tan
x
+
⋯
.
{\displaystyle \tan(x+h)=\tan h+h\sec ^{2}x+h^{2}\sec ^{2}x\tan x+\cdots .}
13.
Expand the following in in powers of
h
.
{\displaystyle h.}
(a)
(
x
+
h
)
n
=
x
n
+
n
x
n
−
1
h
+
n
(
n
−
1
)
2
!
x
n
−
2
h
2
+
n
(
n
−
1
)
(
n
−
2
)
3
!
x
n
−
3
h
3
+
⋯
.
{\displaystyle (x+h)^{n}=x^{n}+nx^{n-1}h+{\tfrac {n(n-1)}{2!}}x^{n-2}h^{2}+{\tfrac {n(n-1)(n-2)}{3!}}x^{n-3}h^{3}+\cdots .}
(b)
e
x
+
h
=
e
x
(
1
+
h
+
h
2
2
!
+
h
3
3
!
+
⋯
)
.
{\displaystyle e^{x+h}=e^{x}\left(1+h+{\tfrac {h^{2}}{2!}}+{\tfrac {h^{3}}{3!}}+\cdots \right).}
145. Maclaurin's Theorem and Maclaurin's Series.
a
=
0
{\displaystyle a=0}
(61 ), §144 , giving
(64 )
f
(
x
)
=
f
(
0
)
+
x
)
1
!
f
′
(
0
)
+
x
2
2
!
f
″
(
0
)
+
x
3
3
!
f
‴
(
0
)
+
⋯
{\displaystyle f(x)=f(0)+{\frac {x)}{1!}}f'\left(0\right)+{\frac {x^{2}}{2!}}f''\left(0\right)+{\frac {x^{3}}{3!}}f'''\left(0\right)+\cdots }
+
x
n
−
1
(
n
−
1
)
!
f
(
n
−
1
)
(
0
)
+
x
n
(
x
−
1
)
!
f
(
n
)
(
x
1
)
,
{\displaystyle +{\frac {x^{n-1}}{\left(n-1\right)!}}f^{\left(n-1\right)}\left(0\right)+{\frac {x^{n}}{\left(x-1\right)!}}f^{\left(n\right)}\left(x_{1}\right),}
where
x
1
{\displaystyle x_{1}}
x
{\displaystyle x}
(64 ) is called Maclaurin's Theorem. The right-hand member is evidently a series in
x
{\displaystyle x}
(62 ), §144 , is a series in
x
−
a
{\displaystyle x-a}
Placing
a
=
0
{\displaystyle a=0}
(62 ), §144 we get Maclaurin's Series' [ 2]
(65 )
f
(
x
)
=
f
(
0
)
+
x
)
1
!
f
′
(
0
)
+
x
2
2
!
f
″
(
0
)
+
x
3
3
!
f
‴
(
0
)
+
⋯
,
{\displaystyle f(x)=f(0)+{\frac {x)}{1!}}f'\left(0\right)+{\frac {x^{2}}{2!}}f''\left(0\right)+{\frac {x^{3}}{3!}}f'''\left(0\right)+\cdots ,}
↑ In these examples assume that the functions can be developed into a power series.
↑ Named after [Maclaurin ] (1698-1746), being first published in his [Treatise of Fluxions [1] ] (1692-1770).
