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Page 1
CALCULUS
Important Series Expansion
a. ? ?
n
n
n
r 0
r
r
1 x C x
?
??
?
b. ? ?
1
2
1 x 1 x x ............
?
? ? ? ? ?
c. ? ? ? ?
23
23
x
xx
a 1 x log a xloga xloga ................
2! 3!
? ? ? ? ?
d.
35
xx
sinx x .................
3! 5!
? ? ?
e. ??
24
xx
cosx 1 + ......................
2! 4!
f. tan x =
3
5
2
x
x + x + .........
3!
15
?
g. log (1 + x) =
23
xx
x + + ............, x < 1
23
?
Important Limits
a.
lt
sinx
1
x0 x
?
?
b.
lt
tanx
1
x0 x
?
?
c. ? ?
1
n
x
lt
1 nx e
x0
??
?
d.
lt
cos x 1
x0
?
?
e. ? ?
1
x
lt
1 x e
x0
??
?
f.
? ?
??
??
x
lt
1
1 e
x
x
L – Hospitals Rule
If f (x) and g(x) are to function such that
? ?
lt
f x 0
xa
?
?
and ? ?
lt
g x 0
xa
?
?
Page 2
CALCULUS
Important Series Expansion
a. ? ?
n
n
n
r 0
r
r
1 x C x
?
??
?
b. ? ?
1
2
1 x 1 x x ............
?
? ? ? ? ?
c. ? ? ? ?
23
23
x
xx
a 1 x log a xloga xloga ................
2! 3!
? ? ? ? ?
d.
35
xx
sinx x .................
3! 5!
? ? ?
e. ??
24
xx
cosx 1 + ......................
2! 4!
f. tan x =
3
5
2
x
x + x + .........
3!
15
?
g. log (1 + x) =
23
xx
x + + ............, x < 1
23
?
Important Limits
a.
lt
sinx
1
x0 x
?
?
b.
lt
tanx
1
x0 x
?
?
c. ? ?
1
n
x
lt
1 nx e
x0
??
?
d.
lt
cos x 1
x0
?
?
e. ? ?
1
x
lt
1 x e
x0
??
?
f.
? ?
??
??
x
lt
1
1 e
x
x
L – Hospitals Rule
If f (x) and g(x) are to function such that
? ?
lt
f x 0
xa
?
?
and ? ?
lt
g x 0
xa
?
?
Then
? ?
? ?
? ?
? ?
lt lt f x f' x
x a x a g x g' x
?
??
If f’(x) and g’(x) are also zero asxa ? , then we can take successive derivatives till this
condition is violated.
For continuity, ? ? ? ?
lim
f x =f a
xa ?
For differentiability,
? ? ? ?
0 0
f x h f x lim
h0 h
??
??
??
?
??
??
exists and is equal to ? ?
0
f' x
If a function is differentiable at some point then it is continuous at that point but converse
may not be true.
Mean Value Theorems
? Rolle’s Theorem
If there is a function f(x) such that f(x) is continuous in closed interval a = x = b and f’(x)
is existing at every point in open interval a < x < b and f(a) = f(b).
Then, there exists a point ‘c’ such that f’(c) = 0 and a < c < b.
? Lagrange’s Mean value Theorem
If there is a function f(x) such that, f(x) is continuous in closed interval a = x = b; and f(x) is
differentiable in open interval (a, b) i.e., a < x < b,
Then there exists a point ‘c’, such that
? ?
? ? ? ?
? ?
f b f a
f' c
ba
?
?
?
Differentiation
Properties: (f + g)’ = f’ + g’ ; (f – g)’ = f’ – g’ ; (f g)’ = f’ g + f g’
Important derivatives
a.
n
x
? n
n 1
x
?
b.
1
nx
x
?
c.
? ?
?
aa
1
log x (log e)
x
d.
xx
e e ?
Page 3
CALCULUS
Important Series Expansion
a. ? ?
n
n
n
r 0
r
r
1 x C x
?
??
?
b. ? ?
1
2
1 x 1 x x ............
?
? ? ? ? ?
c. ? ? ? ?
23
23
x
xx
a 1 x log a xloga xloga ................
2! 3!
? ? ? ? ?
d.
35
xx
sinx x .................
3! 5!
? ? ?
e. ??
24
xx
cosx 1 + ......................
2! 4!
f. tan x =
3
5
2
x
x + x + .........
3!
15
?
g. log (1 + x) =
23
xx
x + + ............, x < 1
23
?
Important Limits
a.
lt
sinx
1
x0 x
?
?
b.
lt
tanx
1
x0 x
?
?
c. ? ?
1
n
x
lt
1 nx e
x0
??
?
d.
lt
cos x 1
x0
?
?
e. ? ?
1
x
lt
1 x e
x0
??
?
f.
? ?
??
??
x
lt
1
1 e
x
x
L – Hospitals Rule
If f (x) and g(x) are to function such that
? ?
lt
f x 0
xa
?
?
and ? ?
lt
g x 0
xa
?
?
Then
? ?
? ?
? ?
? ?
lt lt f x f' x
x a x a g x g' x
?
??
If f’(x) and g’(x) are also zero asxa ? , then we can take successive derivatives till this
condition is violated.
For continuity, ? ? ? ?
lim
f x =f a
xa ?
For differentiability,
? ? ? ?
0 0
f x h f x lim
h0 h
??
??
??
?
??
??
exists and is equal to ? ?
0
f' x
If a function is differentiable at some point then it is continuous at that point but converse
may not be true.
Mean Value Theorems
? Rolle’s Theorem
If there is a function f(x) such that f(x) is continuous in closed interval a = x = b and f’(x)
is existing at every point in open interval a < x < b and f(a) = f(b).
Then, there exists a point ‘c’ such that f’(c) = 0 and a < c < b.
? Lagrange’s Mean value Theorem
If there is a function f(x) such that, f(x) is continuous in closed interval a = x = b; and f(x) is
differentiable in open interval (a, b) i.e., a < x < b,
Then there exists a point ‘c’, such that
? ?
? ? ? ?
? ?
f b f a
f' c
ba
?
?
?
Differentiation
Properties: (f + g)’ = f’ + g’ ; (f – g)’ = f’ – g’ ; (f g)’ = f’ g + f g’
Important derivatives
a.
n
x
? n
n 1
x
?
b.
1
nx
x
?
c.
? ?
?
aa
1
log x (log e)
x
d.
xx
e e ?
e.
xx
e
a a log a ?
f. sin x ? cos x
g. cos x ? -sin x
h. tan x ?
2
sec x
i. sec x ? sec x tan x
j. cosec x ? - cosec x cot x
k. cot x ? - cosec
2
x
l. sin h x ? cos h x
m. cos h x ? sin h x
n.
?
?
1
2
1
sin x
1 - x
o.
2
1
-1
cos x
1x
?
?
?
p.
?
?
?
2
1
1
tan x
1x
q.
2
1
-1
cosec x
x x 1
?
?
?
r.
?
?
?
2
1
1
sec x
x x 1
s.
1
2
-1
cot x
1x
?
?
?
Increasing & Decreasing Functions
? ? ? f' x 0 V ? ? ? x a, b ? , then f is increasing in [a, b]
? ? ? f' x 0 V ? ? ? x a, b ? , then f is strictly increasing in [a, b]
? ? ? f' x 0 V ? ? ? x a, b ? , then f is decreasing in [a, b]
? ? ? f' x 0 V ? ? ? x a, b ? , then f is strictly decreasing in [a, b]
Page 4
CALCULUS
Important Series Expansion
a. ? ?
n
n
n
r 0
r
r
1 x C x
?
??
?
b. ? ?
1
2
1 x 1 x x ............
?
? ? ? ? ?
c. ? ? ? ?
23
23
x
xx
a 1 x log a xloga xloga ................
2! 3!
? ? ? ? ?
d.
35
xx
sinx x .................
3! 5!
? ? ?
e. ??
24
xx
cosx 1 + ......................
2! 4!
f. tan x =
3
5
2
x
x + x + .........
3!
15
?
g. log (1 + x) =
23
xx
x + + ............, x < 1
23
?
Important Limits
a.
lt
sinx
1
x0 x
?
?
b.
lt
tanx
1
x0 x
?
?
c. ? ?
1
n
x
lt
1 nx e
x0
??
?
d.
lt
cos x 1
x0
?
?
e. ? ?
1
x
lt
1 x e
x0
??
?
f.
? ?
??
??
x
lt
1
1 e
x
x
L – Hospitals Rule
If f (x) and g(x) are to function such that
? ?
lt
f x 0
xa
?
?
and ? ?
lt
g x 0
xa
?
?
Then
? ?
? ?
? ?
? ?
lt lt f x f' x
x a x a g x g' x
?
??
If f’(x) and g’(x) are also zero asxa ? , then we can take successive derivatives till this
condition is violated.
For continuity, ? ? ? ?
lim
f x =f a
xa ?
For differentiability,
? ? ? ?
0 0
f x h f x lim
h0 h
??
??
??
?
??
??
exists and is equal to ? ?
0
f' x
If a function is differentiable at some point then it is continuous at that point but converse
may not be true.
Mean Value Theorems
? Rolle’s Theorem
If there is a function f(x) such that f(x) is continuous in closed interval a = x = b and f’(x)
is existing at every point in open interval a < x < b and f(a) = f(b).
Then, there exists a point ‘c’ such that f’(c) = 0 and a < c < b.
? Lagrange’s Mean value Theorem
If there is a function f(x) such that, f(x) is continuous in closed interval a = x = b; and f(x) is
differentiable in open interval (a, b) i.e., a < x < b,
Then there exists a point ‘c’, such that
? ?
? ? ? ?
? ?
f b f a
f' c
ba
?
?
?
Differentiation
Properties: (f + g)’ = f’ + g’ ; (f – g)’ = f’ – g’ ; (f g)’ = f’ g + f g’
Important derivatives
a.
n
x
? n
n 1
x
?
b.
1
nx
x
?
c.
? ?
?
aa
1
log x (log e)
x
d.
xx
e e ?
e.
xx
e
a a log a ?
f. sin x ? cos x
g. cos x ? -sin x
h. tan x ?
2
sec x
i. sec x ? sec x tan x
j. cosec x ? - cosec x cot x
k. cot x ? - cosec
2
x
l. sin h x ? cos h x
m. cos h x ? sin h x
n.
?
?
1
2
1
sin x
1 - x
o.
2
1
-1
cos x
1x
?
?
?
p.
?
?
?
2
1
1
tan x
1x
q.
2
1
-1
cosec x
x x 1
?
?
?
r.
?
?
?
2
1
1
sec x
x x 1
s.
1
2
-1
cot x
1x
?
?
?
Increasing & Decreasing Functions
? ? ? f' x 0 V ? ? ? x a, b ? , then f is increasing in [a, b]
? ? ? f' x 0 V ? ? ? x a, b ? , then f is strictly increasing in [a, b]
? ? ? f' x 0 V ? ? ? x a, b ? , then f is decreasing in [a, b]
? ? ? f' x 0 V ? ? ? x a, b ? , then f is strictly decreasing in [a, b]
Maxima & Minima
Local maxima or minima
There is a maximum of f(x) at x = a if f’(a) = 0 and f”(a) is negative.
There is a minimum of f (x) at x = a, if f’(a) = 0 and f” (a) is positive.
To calculate maximum or minima, we find the point ‘a’ such that f’(a) = 0 and then decide
if it is maximum or minima by judging the sign of f”(a).
Global maxima & minima
We first find local maxima & minima & then calculate the value of ‘f’ at boundary points of
interval given eg. [a, b], we find f(a) & f(b) & compare it with the values of local maxima &
minima. The absolute maxima & minima can be decided then.
Taylor & Maclaurin series
? Taylor series
f(a + h) = f(a) + h f’(a) +
2
h
2
f”(a) + ………………..
? Maclaurin
f(x) = f(0) + x f’(0) +
2
x
2
f“(0)+……………..
Partial Derivative
If a derivative of a function of several independent variables be found with respect to any
one of them, keeping the others as constant, it is said to be a partial derivative.
Homogenous Function
n 2 2 n n 1 n
n
0 1 2
a x a x y a x y ............. a y
? ?
? ? ? ? is a homogenous function
of x & y, of degree ‘n’
=
? ? ? ? ? ?
2n
n
0 1 2
n y y y
x a a a .................... a
x x x
??
? ? ? ?
??
??
Euler’s Theorem
If u is a homogenous function of x & y of degree n, then
uu
x y nu
xy
?? ??
??
??
??
??
Page 5
CALCULUS
Important Series Expansion
a. ? ?
n
n
n
r 0
r
r
1 x C x
?
??
?
b. ? ?
1
2
1 x 1 x x ............
?
? ? ? ? ?
c. ? ? ? ?
23
23
x
xx
a 1 x log a xloga xloga ................
2! 3!
? ? ? ? ?
d.
35
xx
sinx x .................
3! 5!
? ? ?
e. ??
24
xx
cosx 1 + ......................
2! 4!
f. tan x =
3
5
2
x
x + x + .........
3!
15
?
g. log (1 + x) =
23
xx
x + + ............, x < 1
23
?
Important Limits
a.
lt
sinx
1
x0 x
?
?
b.
lt
tanx
1
x0 x
?
?
c. ? ?
1
n
x
lt
1 nx e
x0
??
?
d.
lt
cos x 1
x0
?
?
e. ? ?
1
x
lt
1 x e
x0
??
?
f.
? ?
??
??
x
lt
1
1 e
x
x
L – Hospitals Rule
If f (x) and g(x) are to function such that
? ?
lt
f x 0
xa
?
?
and ? ?
lt
g x 0
xa
?
?
Then
? ?
? ?
? ?
? ?
lt lt f x f' x
x a x a g x g' x
?
??
If f’(x) and g’(x) are also zero asxa ? , then we can take successive derivatives till this
condition is violated.
For continuity, ? ? ? ?
lim
f x =f a
xa ?
For differentiability,
? ? ? ?
0 0
f x h f x lim
h0 h
??
??
??
?
??
??
exists and is equal to ? ?
0
f' x
If a function is differentiable at some point then it is continuous at that point but converse
may not be true.
Mean Value Theorems
? Rolle’s Theorem
If there is a function f(x) such that f(x) is continuous in closed interval a = x = b and f’(x)
is existing at every point in open interval a < x < b and f(a) = f(b).
Then, there exists a point ‘c’ such that f’(c) = 0 and a < c < b.
? Lagrange’s Mean value Theorem
If there is a function f(x) such that, f(x) is continuous in closed interval a = x = b; and f(x) is
differentiable in open interval (a, b) i.e., a < x < b,
Then there exists a point ‘c’, such that
? ?
? ? ? ?
? ?
f b f a
f' c
ba
?
?
?
Differentiation
Properties: (f + g)’ = f’ + g’ ; (f – g)’ = f’ – g’ ; (f g)’ = f’ g + f g’
Important derivatives
a.
n
x
? n
n 1
x
?
b.
1
nx
x
?
c.
? ?
?
aa
1
log x (log e)
x
d.
xx
e e ?
e.
xx
e
a a log a ?
f. sin x ? cos x
g. cos x ? -sin x
h. tan x ?
2
sec x
i. sec x ? sec x tan x
j. cosec x ? - cosec x cot x
k. cot x ? - cosec
2
x
l. sin h x ? cos h x
m. cos h x ? sin h x
n.
?
?
1
2
1
sin x
1 - x
o.
2
1
-1
cos x
1x
?
?
?
p.
?
?
?
2
1
1
tan x
1x
q.
2
1
-1
cosec x
x x 1
?
?
?
r.
?
?
?
2
1
1
sec x
x x 1
s.
1
2
-1
cot x
1x
?
?
?
Increasing & Decreasing Functions
? ? ? f' x 0 V ? ? ? x a, b ? , then f is increasing in [a, b]
? ? ? f' x 0 V ? ? ? x a, b ? , then f is strictly increasing in [a, b]
? ? ? f' x 0 V ? ? ? x a, b ? , then f is decreasing in [a, b]
? ? ? f' x 0 V ? ? ? x a, b ? , then f is strictly decreasing in [a, b]
Maxima & Minima
Local maxima or minima
There is a maximum of f(x) at x = a if f’(a) = 0 and f”(a) is negative.
There is a minimum of f (x) at x = a, if f’(a) = 0 and f” (a) is positive.
To calculate maximum or minima, we find the point ‘a’ such that f’(a) = 0 and then decide
if it is maximum or minima by judging the sign of f”(a).
Global maxima & minima
We first find local maxima & minima & then calculate the value of ‘f’ at boundary points of
interval given eg. [a, b], we find f(a) & f(b) & compare it with the values of local maxima &
minima. The absolute maxima & minima can be decided then.
Taylor & Maclaurin series
? Taylor series
f(a + h) = f(a) + h f’(a) +
2
h
2
f”(a) + ………………..
? Maclaurin
f(x) = f(0) + x f’(0) +
2
x
2
f“(0)+……………..
Partial Derivative
If a derivative of a function of several independent variables be found with respect to any
one of them, keeping the others as constant, it is said to be a partial derivative.
Homogenous Function
n 2 2 n n 1 n
n
0 1 2
a x a x y a x y ............. a y
? ?
? ? ? ? is a homogenous function
of x & y, of degree ‘n’
=
? ? ? ? ? ?
2n
n
0 1 2
n y y y
x a a a .................... a
x x x
??
? ? ? ?
??
??
Euler’s Theorem
If u is a homogenous function of x & y of degree n, then
uu
x y nu
xy
?? ??
??
??
??
??
Maxima & minima of multi-variable function
2
2
xa
yb
f
let r
x
?
?
??
?
?
??
??
?
??
;
2
xa
yb
f
s
xy ?
?
??
?
?
??
??
??
??
;
2
2
xa
yb
f
t
y
?
?
??
?
?
??
??
?
??
? Maxima
rt >
2
s ; r < 0
? Minima
rt >
2
s ; r > 0
? Saddle point
rt <
2
s
Integration
Indefinite integrals are just opposite of derivatives and hence important derivatives must
always be remembered.
Properties of definite integral
a.
? ? ? ?
?
??
bb
aa
f x dx f t dt
b. ? ? ? ?
ba
a b
f x dx f x dx ??
??
c.
? ? ? ? ? ?
??
? ? ?
b cb
a a c
f x dx f x dx f x dx
d. ? ? ? ?
b b
aa
f x dx f a b x dx ? ? ?
??
e. ? ?
? ?
? ?
? ? ? ? ? ? ? ? ? ? ? ?
t
t
d
f x dx f t ' t f t ' t
dt
?
?
? ? ? ? ? ?
?
Vectors
? Addition of vector
ab ? of two vector a =
1 2 3
a ,a ,a ??
??
and b =
1 2 3
b ,b ,b ??
??
?? ? ? ? ?
?? 1 1 2 2 3 3
a + b = a b ,a b ,a b
? Scalar Multiplication
??
?? 1 2 3
ca = ca , ca , ca
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,Short Notes: Calculus | Short Notes for Electrical Engineering - Electrical Engineering (EE)
,Short Notes: Calculus | Short Notes for Electrical Engineering - Electrical Engineering (EE)
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Additional Information about Short Notes: Calculus for Electrical Engineering (EE) Preparation
Short Notes: Calculus Free PDF Download
The Short Notes: Calculus is an invaluable resource that delves deep into the core of the Electrical Engineering (EE) exam.
These study notes are curated by experts and cover all the essential topics and concepts, making your preparation more efficient and effective.
With the help of these notes, you can grasp complex subjects quickly, revise important points easily,
and reinforce your understanding of key concepts. The study notes are presented in a concise and easy-to-understand manner,
allowing you to optimize your learning process. Whether you're looking for best-recommended books, sample papers, study material,
or toppers' notes, this PDF has got you covered. Download the Short Notes: Calculus now and kickstart your journey towards success in the Electrical Engineering (EE) exam.
Importance of Short Notes: Calculus
The importance of Short Notes: Calculus cannot be overstated, especially for Electrical Engineering (EE) aspirants.
This document holds the key to success in the Electrical Engineering (EE) exam.
It offers a detailed understanding of the concept, providing invaluable insights into the topic.
By knowing the concepts well in advance, students can plan their preparation effectively.
Utilize this indispensable guide for a well-rounded preparation and achieve your desired results.
Short Notes: Calculus
Short Notes: Calculus Notes offer in-depth insights into the specific topic to help you master it with ease.
This comprehensive document covers all aspects related to Short Notes: Calculus.
It includes detailed information about the exam syllabus, recommended books, and study materials for a well-rounded preparation.
Practice papers and question papers enable you to assess your progress effectively.
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Access to Toppers' notes gives you an edge in understanding complex concepts.
Whether you're a beginner or aiming for advanced proficiency, Short Notes: Calculus Notes on EduRev are your ultimate resource for success.
Short Notes: Calculus Electrical Engineering (EE) Questions
The "Short Notes: Calculus Electrical Engineering (EE) Questions" guide is a valuable resource for all aspiring students preparing for the
Electrical Engineering (EE) exam. It focuses on providing a wide range of practice questions to help students gauge
their understanding of the exam topics. These questions cover the entire syllabus, ensuring comprehensive preparation.
The guide includes previous years' question papers for students to familiarize themselves with the exam's format and difficulty level.
Additionally, it offers subject-specific question banks, allowing students to focus on weak areas and improve their performance.
Study Short Notes: Calculus on the App
Students of Electrical Engineering (EE) can study Short Notes: Calculus alongwith tests & analysis from the EduRev app,
which will help them while preparing for their exam. Apart from the Short Notes: Calculus,
students can also utilize the EduRev App for other study materials such as previous year question papers, syllabus, important questions, etc.
The EduRev App will make your learning easier as you can access it from anywhere you want.
The content of Short Notes: Calculus is prepared as per the latest Electrical Engineering (EE) syllabus.
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