Formula Sheets: Calculus | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE) PDF Download

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Engineering Mathematics: Calculus F orm ula Sheet
for Electrical GA TE
Limits
• Limit Definition :
lim
x?a
f(x) = L if ?? > 0,?d > 0 suc h that 0 <|x-a| < d =? |f(x)-L| < ?
• L’Hôpital’s R ule : F or indeterminate forms
0
0
or
8
8
.
lim
x?a
f(x)
g(x)
= lim
x?a
f
'
(x)
g
'
(x)
Differen tiation
• Deriv ativ e Definition :
f
'
(x) = lim
h?0
f(x+h)-f(x)
h
• Common Deriv ativ es :
d
dx
(x
n
) = nx
n-1
,
d
dx
(sinx) = cosx,
d
dx
(cosx) =-sinx
d
dx
(e
x
) = e
x
,
d
dx
(lnx) =
1
x
• Chain R ule :
d
dx
f(g(x)) = f
'
(g(x))·g
'
(x)
• Pro duct R ule :
d
dx
(uv) = u
'
v +uv
'
• Quotien t R ule :
d
dx
(
u
v
)
=
u
'
v-uv
'
v
2
In tegration
• Definite In tegral :
?
b
a
f(x)dx = F(b)-F(a), where F
'
(x) = f(x)
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Page 2


Engineering Mathematics: Calculus F orm ula Sheet
for Electrical GA TE
Limits
• Limit Definition :
lim
x?a
f(x) = L if ?? > 0,?d > 0 suc h that 0 <|x-a| < d =? |f(x)-L| < ?
• L’Hôpital’s R ule : F or indeterminate forms
0
0
or
8
8
.
lim
x?a
f(x)
g(x)
= lim
x?a
f
'
(x)
g
'
(x)
Differen tiation
• Deriv ativ e Definition :
f
'
(x) = lim
h?0
f(x+h)-f(x)
h
• Common Deriv ativ es :
d
dx
(x
n
) = nx
n-1
,
d
dx
(sinx) = cosx,
d
dx
(cosx) =-sinx
d
dx
(e
x
) = e
x
,
d
dx
(lnx) =
1
x
• Chain R ule :
d
dx
f(g(x)) = f
'
(g(x))·g
'
(x)
• Pro duct R ule :
d
dx
(uv) = u
'
v +uv
'
• Quotien t R ule :
d
dx
(
u
v
)
=
u
'
v-uv
'
v
2
In tegration
• Definite In tegral :
?
b
a
f(x)dx = F(b)-F(a), where F
'
(x) = f(x)
1
• Common In tegrals :
?
x
n
dx =
x
n+1
n+1
+C (n?=-1),
?
1
x
dx = ln|x|+C
?
sinxdx =-cosx+C,
?
cosxdx = sinx+C,
?
e
x
dx = e
x
+C
• In tegration b y P arts :
?
udv = uv-
?
vdu
• Substitution R ule :
?
f(g(x))g
'
(x)dx =
?
f(u)du, where u = g(x)
Multiv ariable Calculus
• P artial Deriv ativ es :
?f
?x
= lim
h?0
f(x+h,y)-f(x,y)
h
• Gradien t :
?f =
(
?f
?x
,
?f
?y
)
• Directional Deriv ativ e :
D
u
f =?f ·u, where u is a unit v ector
• Double In tegral :
??
D
f(x,y)dA =
?
b
a
?
d
c
f(x,y)dydx
Applications
• Maxima/Minima (Single V ariable) :
f
'
(x) = 0 (critical p oin ts), f
''
(x) > 0 (minima), finky
''
(x) < 0 (maxima)
• Maxima/Minima (Multiv ariable) :
?f = 0, D = f
xx
f
yy
-(f
xy
)
2
, D > 0 and f
xx
> 0 (minima), D > 0 and f
xx
< 0 (maxima)
• Line In tegral :
?
C
F·dr =
?
b
a
F(r(t))·r
'
(t)dt
• Area Under Curv e :
A =
?
b
a
f(x)dx
2
Page 3


Engineering Mathematics: Calculus F orm ula Sheet
for Electrical GA TE
Limits
• Limit Definition :
lim
x?a
f(x) = L if ?? > 0,?d > 0 suc h that 0 <|x-a| < d =? |f(x)-L| < ?
• L’Hôpital’s R ule : F or indeterminate forms
0
0
or
8
8
.
lim
x?a
f(x)
g(x)
= lim
x?a
f
'
(x)
g
'
(x)
Differen tiation
• Deriv ativ e Definition :
f
'
(x) = lim
h?0
f(x+h)-f(x)
h
• Common Deriv ativ es :
d
dx
(x
n
) = nx
n-1
,
d
dx
(sinx) = cosx,
d
dx
(cosx) =-sinx
d
dx
(e
x
) = e
x
,
d
dx
(lnx) =
1
x
• Chain R ule :
d
dx
f(g(x)) = f
'
(g(x))·g
'
(x)
• Pro duct R ule :
d
dx
(uv) = u
'
v +uv
'
• Quotien t R ule :
d
dx
(
u
v
)
=
u
'
v-uv
'
v
2
In tegration
• Definite In tegral :
?
b
a
f(x)dx = F(b)-F(a), where F
'
(x) = f(x)
1
• Common In tegrals :
?
x
n
dx =
x
n+1
n+1
+C (n?=-1),
?
1
x
dx = ln|x|+C
?
sinxdx =-cosx+C,
?
cosxdx = sinx+C,
?
e
x
dx = e
x
+C
• In tegration b y P arts :
?
udv = uv-
?
vdu
• Substitution R ule :
?
f(g(x))g
'
(x)dx =
?
f(u)du, where u = g(x)
Multiv ariable Calculus
• P artial Deriv ativ es :
?f
?x
= lim
h?0
f(x+h,y)-f(x,y)
h
• Gradien t :
?f =
(
?f
?x
,
?f
?y
)
• Directional Deriv ativ e :
D
u
f =?f ·u, where u is a unit v ector
• Double In tegral :
??
D
f(x,y)dA =
?
b
a
?
d
c
f(x,y)dydx
Applications
• Maxima/Minima (Single V ariable) :
f
'
(x) = 0 (critical p oin ts), f
''
(x) > 0 (minima), finky
''
(x) < 0 (maxima)
• Maxima/Minima (Multiv ariable) :
?f = 0, D = f
xx
f
yy
-(f
xy
)
2
, D > 0 and f
xx
> 0 (minima), D > 0 and f
xx
< 0 (maxima)
• Line In tegral :
?
C
F·dr =
?
b
a
F(r(t))·r
'
(t)dt
• Area Under Curv e :
A =
?
b
a
f(x)dx
2
Series
• T a ylor Series :
f(x) =
8
?
n=0
f
(n)
(a)
n!
(x-a)
n
• Maclaurin Series : T a ylor series at a = 0 .
f(x) =
8
?
n=0
f
(n)
(0)
n!
x
n
• Common Series :
e
x
=
8
?
n=0
x
n
n!
, sinx =
8
?
n=0
(-1)
n
x
2n+1
(2n+1)!
, cosx =
8
?
n=0
(-1)
n
x
2n
(2n)!
Key Notes
• GA TE F o cus : Solv e limits using L’Hôpital’s rule, compute deriv ativ es/in tegrals, find
maxima/minima, ev aluate line/double in tegrals.
• Multiv ariable Calculus : F o cus on gradien t and critical p oin ts for optimization prob-
lems.
• Series : Use T a ylor/Maclaurin for appro ximations in electrical systems.
• Error A v oidance : Chec k indeterminate forms, ensure prop er substitution, v erify
critical p oin ts.
• Units : Dimensionless for pure math; use SI units if applied (e.g., , ).
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