UPSC Exam > UPSC Notes > Mathematics Optional Notes for UPSC > Function of a Real Variable
Function of a Real Variable | Mathematics Optional Notes for UPSC PDF Download
| Download, print and study this document offline |
Please wait while the PDF view is loading
Page 1
Edurev123
1. Function of a Real Variable
1.1 Suppose that ?? ''
is continuous on [?? ,?? ] and that ?? has three zeros in the
interval (?? ,?? ) . Shov that ?? ''
has at least one zero in the interval (?? ,?? ) .
(2009 : 12 . Marks)
Solution:
Insight: This question uses the fact that continuity of any derivative of a function ensures
continuity and differentiability of lower order derivatives and the Rolle's theorem.
?? ''
is continuous on [1,2]
??? is continuous and differentiable on [1,2]
??? is continuous and differentiable on [1,2]
?? has three zeros in (1,2) . Let them be ?? 1
,?? 2
,?? 3
with ?? 1
<?? 2
<?? 3
.
Inflection Point
In the interval [?? 1
,?? 2
], applying Rolle's theorem.
?? is continuous on [?? 1
,?? 2
].
?? is differentiable on (?? 1
,?? 2
) .
?? (?? 1
)=?? (?? 2
)=0
???? ?(?? 1
,?? 2
) such that ?? (?? )=0 by Rolle's theorem.
Page 2
Edurev123
1. Function of a Real Variable
1.1 Suppose that ?? ''
is continuous on [?? ,?? ] and that ?? has three zeros in the
interval (?? ,?? ) . Shov that ?? ''
has at least one zero in the interval (?? ,?? ) .
(2009 : 12 . Marks)
Solution:
Insight: This question uses the fact that continuity of any derivative of a function ensures
continuity and differentiability of lower order derivatives and the Rolle's theorem.
?? ''
is continuous on [1,2]
??? is continuous and differentiable on [1,2]
??? is continuous and differentiable on [1,2]
?? has three zeros in (1,2) . Let them be ?? 1
,?? 2
,?? 3
with ?? 1
<?? 2
<?? 3
.
Inflection Point
In the interval [?? 1
,?? 2
], applying Rolle's theorem.
?? is continuous on [?? 1
,?? 2
].
?? is differentiable on (?? 1
,?? 2
) .
?? (?? 1
)=?? (?? 2
)=0
???? ?(?? 1
,?? 2
) such that ?? (?? )=0 by Rolle's theorem.
Similarly, applying Rolle's theorem in interval [?? 2
,?? 3
],??? 2
?(?? 2
,?? 3
) such that ?? (?? 2
)=0.
As ?? 1
<?? 2
and ?? 2
>?? 2
??? 1
<?? 2
.
Applying Rolle's theorem on ?? in (?? 1
,?? 2
?
) .
?? is continuous on [?? 1
,?? 2
].
?? is differentiable on (?? 1
,?? 2
) as ?? '
is continuous on that interval.
?? (?? 1
)=?? (?? 2
)=0
???? ?(?? 1
,?? 2
) so that ?? '
(?? )=0 by Rolle's theorem.
Also, (?? 1
,?? 2
)?(1,2)??? ?(1,2)
1.2 If ?? is the derivative of some function defined on [?? ,?? ] prove that there exists a
number ?? ?[?? ,?? ] such that
? ?
?? ?? =?? ?? (?? )???? =?? (?? )(?? -?? )
(2009 : 12 Marks)
Solution:
Insight: This is the mean value theorem of integral calculus with the difference that ?? is
not given as continuous but as derivative of some function. We use 2nd Fundamental
Theorem of Calculus.
?? is derivative on some function
? ?? (?? )=?? (?? ) on [?? ,?? ]
i.e., ?? (?? ) has an anti-derivative ?? (?? ) defined on [?? ,?? ].
By 2nd Fundamental Theorem of Algebra, for any ?? 1
,?? 2
?[?? ,?? ]
? ?
?? 2
?? 1
?? (?? )???? =?? (?? 2
)-?? (?? 1
)
Proof of 2nd Fundamental Theorem: ?? (?? ) is an anti-derivative of ?? (?? ) .
Let
?? (?? )=? ?
?? ?? 1
?? (?? )???? where ?? 1
?(?? ,?? )
Then ?? (?? ) is an anti-derivative.
As two anti-derivatives differ by a constant
Page 3
Edurev123
1. Function of a Real Variable
1.1 Suppose that ?? ''
is continuous on [?? ,?? ] and that ?? has three zeros in the
interval (?? ,?? ) . Shov that ?? ''
has at least one zero in the interval (?? ,?? ) .
(2009 : 12 . Marks)
Solution:
Insight: This question uses the fact that continuity of any derivative of a function ensures
continuity and differentiability of lower order derivatives and the Rolle's theorem.
?? ''
is continuous on [1,2]
??? is continuous and differentiable on [1,2]
??? is continuous and differentiable on [1,2]
?? has three zeros in (1,2) . Let them be ?? 1
,?? 2
,?? 3
with ?? 1
<?? 2
<?? 3
.
Inflection Point
In the interval [?? 1
,?? 2
], applying Rolle's theorem.
?? is continuous on [?? 1
,?? 2
].
?? is differentiable on (?? 1
,?? 2
) .
?? (?? 1
)=?? (?? 2
)=0
???? ?(?? 1
,?? 2
) such that ?? (?? )=0 by Rolle's theorem.
Similarly, applying Rolle's theorem in interval [?? 2
,?? 3
],??? 2
?(?? 2
,?? 3
) such that ?? (?? 2
)=0.
As ?? 1
<?? 2
and ?? 2
>?? 2
??? 1
<?? 2
.
Applying Rolle's theorem on ?? in (?? 1
,?? 2
?
) .
?? is continuous on [?? 1
,?? 2
].
?? is differentiable on (?? 1
,?? 2
) as ?? '
is continuous on that interval.
?? (?? 1
)=?? (?? 2
)=0
???? ?(?? 1
,?? 2
) so that ?? '
(?? )=0 by Rolle's theorem.
Also, (?? 1
,?? 2
)?(1,2)??? ?(1,2)
1.2 If ?? is the derivative of some function defined on [?? ,?? ] prove that there exists a
number ?? ?[?? ,?? ] such that
? ?
?? ?? =?? ?? (?? )???? =?? (?? )(?? -?? )
(2009 : 12 Marks)
Solution:
Insight: This is the mean value theorem of integral calculus with the difference that ?? is
not given as continuous but as derivative of some function. We use 2nd Fundamental
Theorem of Calculus.
?? is derivative on some function
? ?? (?? )=?? (?? ) on [?? ,?? ]
i.e., ?? (?? ) has an anti-derivative ?? (?? ) defined on [?? ,?? ].
By 2nd Fundamental Theorem of Algebra, for any ?? 1
,?? 2
?[?? ,?? ]
? ?
?? 2
?? 1
?? (?? )???? =?? (?? 2
)-?? (?? 1
)
Proof of 2nd Fundamental Theorem: ?? (?? ) is an anti-derivative of ?? (?? ) .
Let
?? (?? )=? ?
?? ?? 1
?? (?? )???? where ?? 1
?(?? ,?? )
Then ?? (?? ) is an anti-derivative.
As two anti-derivatives differ by a constant
?? (?? )=?? (?? )+??
? ?? (?? 1
) =?? (?? 1
)+??
and ? ?
?? 1
?? 1
??? (?? )=?? (?? 1
)=0
? ?? (?? 1
)+?? =0? ?? =-?? (?? 1
)
? ?? (?? )=?? (?? )-?? (?? 1
)
? ? ?
?? ?? 1
??? (?? )???? =?? (?? )-?? (?? 1
)
? ? ?
?? ?? 2
??? (?? )???? =?? (?? 2
)-?? (?? 1
)
Now, ?? '
(?? )=?? (?? ) on [?? ,?? ] so ?? (?? ) is continuous on [?? ,?? ] as every differentiable
function is continuous and ?? (?? ) is differentiable on (?? ,?? ) .
So by Mean Value Theorem, ??? ?(?? ,?? ) such that
?? (?? )=
?? (?? )-?? (?? )
?? -?? ? ?? (?? )-?? (?? )=(?? -?? )?? (?? )
? ? ?
?? ?? ??? (?? )???? =(?? -?? )?? (?? )
1.2 A twice differentiable function ?? (?? ) is such that ?? (?? )=?? =?? (?? ) and ?? (?? )>?? for
?? <?? <?? . Prove that there is at least one point ?? ,?? <?? <?? , for which ?? ''
(?? )<?? .
(2010 : 12 Marks)
Solution:
Given ?? (?? )=?? (?? )=0 and ?? ?(?? ,?? ) such that ?? (?? )>0.
By Lagrange's Mean Value Theorem (LMVT), ??? ?(?? ,?? ) and ?? ?(?? ,?? ) :
?? (?? )=
?? (?? )-?? (?? )
?? -?? and ?? (?? )=
?? (?? )-?? (?? )
?? -?? (1)
Now, let ?? ?(?? ,?? )
? ?? ?(?? ,?? ) or ?? <?? <??
By Lagrange's Mean Value Theorem (LMVT)
?? '
(?? )=
?? '
(?? )-?? '
(?? )
?? -??
Page 4
Edurev123
1. Function of a Real Variable
1.1 Suppose that ?? ''
is continuous on [?? ,?? ] and that ?? has three zeros in the
interval (?? ,?? ) . Shov that ?? ''
has at least one zero in the interval (?? ,?? ) .
(2009 : 12 . Marks)
Solution:
Insight: This question uses the fact that continuity of any derivative of a function ensures
continuity and differentiability of lower order derivatives and the Rolle's theorem.
?? ''
is continuous on [1,2]
??? is continuous and differentiable on [1,2]
??? is continuous and differentiable on [1,2]
?? has three zeros in (1,2) . Let them be ?? 1
,?? 2
,?? 3
with ?? 1
<?? 2
<?? 3
.
Inflection Point
In the interval [?? 1
,?? 2
], applying Rolle's theorem.
?? is continuous on [?? 1
,?? 2
].
?? is differentiable on (?? 1
,?? 2
) .
?? (?? 1
)=?? (?? 2
)=0
???? ?(?? 1
,?? 2
) such that ?? (?? )=0 by Rolle's theorem.
Similarly, applying Rolle's theorem in interval [?? 2
,?? 3
],??? 2
?(?? 2
,?? 3
) such that ?? (?? 2
)=0.
As ?? 1
<?? 2
and ?? 2
>?? 2
??? 1
<?? 2
.
Applying Rolle's theorem on ?? in (?? 1
,?? 2
?
) .
?? is continuous on [?? 1
,?? 2
].
?? is differentiable on (?? 1
,?? 2
) as ?? '
is continuous on that interval.
?? (?? 1
)=?? (?? 2
)=0
???? ?(?? 1
,?? 2
) so that ?? '
(?? )=0 by Rolle's theorem.
Also, (?? 1
,?? 2
)?(1,2)??? ?(1,2)
1.2 If ?? is the derivative of some function defined on [?? ,?? ] prove that there exists a
number ?? ?[?? ,?? ] such that
? ?
?? ?? =?? ?? (?? )???? =?? (?? )(?? -?? )
(2009 : 12 Marks)
Solution:
Insight: This is the mean value theorem of integral calculus with the difference that ?? is
not given as continuous but as derivative of some function. We use 2nd Fundamental
Theorem of Calculus.
?? is derivative on some function
? ?? (?? )=?? (?? ) on [?? ,?? ]
i.e., ?? (?? ) has an anti-derivative ?? (?? ) defined on [?? ,?? ].
By 2nd Fundamental Theorem of Algebra, for any ?? 1
,?? 2
?[?? ,?? ]
? ?
?? 2
?? 1
?? (?? )???? =?? (?? 2
)-?? (?? 1
)
Proof of 2nd Fundamental Theorem: ?? (?? ) is an anti-derivative of ?? (?? ) .
Let
?? (?? )=? ?
?? ?? 1
?? (?? )???? where ?? 1
?(?? ,?? )
Then ?? (?? ) is an anti-derivative.
As two anti-derivatives differ by a constant
?? (?? )=?? (?? )+??
? ?? (?? 1
) =?? (?? 1
)+??
and ? ?
?? 1
?? 1
??? (?? )=?? (?? 1
)=0
? ?? (?? 1
)+?? =0? ?? =-?? (?? 1
)
? ?? (?? )=?? (?? )-?? (?? 1
)
? ? ?
?? ?? 1
??? (?? )???? =?? (?? )-?? (?? 1
)
? ? ?
?? ?? 2
??? (?? )???? =?? (?? 2
)-?? (?? 1
)
Now, ?? '
(?? )=?? (?? ) on [?? ,?? ] so ?? (?? ) is continuous on [?? ,?? ] as every differentiable
function is continuous and ?? (?? ) is differentiable on (?? ,?? ) .
So by Mean Value Theorem, ??? ?(?? ,?? ) such that
?? (?? )=
?? (?? )-?? (?? )
?? -?? ? ?? (?? )-?? (?? )=(?? -?? )?? (?? )
? ? ?
?? ?? ??? (?? )???? =(?? -?? )?? (?? )
1.2 A twice differentiable function ?? (?? ) is such that ?? (?? )=?? =?? (?? ) and ?? (?? )>?? for
?? <?? <?? . Prove that there is at least one point ?? ,?? <?? <?? , for which ?? ''
(?? )<?? .
(2010 : 12 Marks)
Solution:
Given ?? (?? )=?? (?? )=0 and ?? ?(?? ,?? ) such that ?? (?? )>0.
By Lagrange's Mean Value Theorem (LMVT), ??? ?(?? ,?? ) and ?? ?(?? ,?? ) :
?? (?? )=
?? (?? )-?? (?? )
?? -?? and ?? (?? )=
?? (?? )-?? (?? )
?? -?? (1)
Now, let ?? ?(?? ,?? )
? ?? ?(?? ,?? ) or ?? <?? <??
By Lagrange's Mean Value Theorem (LMVT)
?? '
(?? )=
?? '
(?? )-?? '
(?? )
?? -??
Using values in eqn. (1), we get
?? '
(?? )=
?? (?? )-?? (?? )
?? -?? -
?? (?? )-?? (?? )
?? -?? ?? -?? ? ?? ''
(?? )=
-
?? (?? )
?? -?? -
?? (?? )
?? -?? ?? -?? [?? (?? )=?? (?? )=0]
?? '
(?? )=-[
?? (?? )
?? -?? +
?? '
(?? )
?? -?? ?? -?? ]<0 as ?? >?? ,?? >?? ,?? >??
1.4 Show that a box (rectangular parallelopiped) of maximum volume ?? with
prescribed surface area is a cube.
(2010 : 20 Marks)
Solution:
Let ?? ,?? ,?? be length of edges of given rectangular parallelopiped. Its surface area be ?? ,
volume be ?? .(?? ,?? , ?? ?0) .
? ?? =2???? +2???? +2???? ,?? =??????
Let ?? be the Lagrange's multiplier (?? ?0)
So,
?? =?????? +?? (2???? +2???? +2???? -?? )
? At extremum,
?? ?? =0????? +?? (2?? +2?? )=0 (1)
?? ?? =0????? +?? (2?? +2?? )=0 (2)
?? ?? =0????? +?? (2?? +???? )=0 (3)
Multiplying (1) by ?? and (2) by ?? and subtracting then, we get
?????? +?? (2???? +2???? )-?????? -?? (2???? +2???? )=0
Page 5
Edurev123
1. Function of a Real Variable
1.1 Suppose that ?? ''
is continuous on [?? ,?? ] and that ?? has three zeros in the
interval (?? ,?? ) . Shov that ?? ''
has at least one zero in the interval (?? ,?? ) .
(2009 : 12 . Marks)
Solution:
Insight: This question uses the fact that continuity of any derivative of a function ensures
continuity and differentiability of lower order derivatives and the Rolle's theorem.
?? ''
is continuous on [1,2]
??? is continuous and differentiable on [1,2]
??? is continuous and differentiable on [1,2]
?? has three zeros in (1,2) . Let them be ?? 1
,?? 2
,?? 3
with ?? 1
<?? 2
<?? 3
.
Inflection Point
In the interval [?? 1
,?? 2
], applying Rolle's theorem.
?? is continuous on [?? 1
,?? 2
].
?? is differentiable on (?? 1
,?? 2
) .
?? (?? 1
)=?? (?? 2
)=0
???? ?(?? 1
,?? 2
) such that ?? (?? )=0 by Rolle's theorem.
Similarly, applying Rolle's theorem in interval [?? 2
,?? 3
],??? 2
?(?? 2
,?? 3
) such that ?? (?? 2
)=0.
As ?? 1
<?? 2
and ?? 2
>?? 2
??? 1
<?? 2
.
Applying Rolle's theorem on ?? in (?? 1
,?? 2
?
) .
?? is continuous on [?? 1
,?? 2
].
?? is differentiable on (?? 1
,?? 2
) as ?? '
is continuous on that interval.
?? (?? 1
)=?? (?? 2
)=0
???? ?(?? 1
,?? 2
) so that ?? '
(?? )=0 by Rolle's theorem.
Also, (?? 1
,?? 2
)?(1,2)??? ?(1,2)
1.2 If ?? is the derivative of some function defined on [?? ,?? ] prove that there exists a
number ?? ?[?? ,?? ] such that
? ?
?? ?? =?? ?? (?? )???? =?? (?? )(?? -?? )
(2009 : 12 Marks)
Solution:
Insight: This is the mean value theorem of integral calculus with the difference that ?? is
not given as continuous but as derivative of some function. We use 2nd Fundamental
Theorem of Calculus.
?? is derivative on some function
? ?? (?? )=?? (?? ) on [?? ,?? ]
i.e., ?? (?? ) has an anti-derivative ?? (?? ) defined on [?? ,?? ].
By 2nd Fundamental Theorem of Algebra, for any ?? 1
,?? 2
?[?? ,?? ]
? ?
?? 2
?? 1
?? (?? )???? =?? (?? 2
)-?? (?? 1
)
Proof of 2nd Fundamental Theorem: ?? (?? ) is an anti-derivative of ?? (?? ) .
Let
?? (?? )=? ?
?? ?? 1
?? (?? )???? where ?? 1
?(?? ,?? )
Then ?? (?? ) is an anti-derivative.
As two anti-derivatives differ by a constant
?? (?? )=?? (?? )+??
? ?? (?? 1
) =?? (?? 1
)+??
and ? ?
?? 1
?? 1
??? (?? )=?? (?? 1
)=0
? ?? (?? 1
)+?? =0? ?? =-?? (?? 1
)
? ?? (?? )=?? (?? )-?? (?? 1
)
? ? ?
?? ?? 1
??? (?? )???? =?? (?? )-?? (?? 1
)
? ? ?
?? ?? 2
??? (?? )???? =?? (?? 2
)-?? (?? 1
)
Now, ?? '
(?? )=?? (?? ) on [?? ,?? ] so ?? (?? ) is continuous on [?? ,?? ] as every differentiable
function is continuous and ?? (?? ) is differentiable on (?? ,?? ) .
So by Mean Value Theorem, ??? ?(?? ,?? ) such that
?? (?? )=
?? (?? )-?? (?? )
?? -?? ? ?? (?? )-?? (?? )=(?? -?? )?? (?? )
? ? ?
?? ?? ??? (?? )???? =(?? -?? )?? (?? )
1.2 A twice differentiable function ?? (?? ) is such that ?? (?? )=?? =?? (?? ) and ?? (?? )>?? for
?? <?? <?? . Prove that there is at least one point ?? ,?? <?? <?? , for which ?? ''
(?? )<?? .
(2010 : 12 Marks)
Solution:
Given ?? (?? )=?? (?? )=0 and ?? ?(?? ,?? ) such that ?? (?? )>0.
By Lagrange's Mean Value Theorem (LMVT), ??? ?(?? ,?? ) and ?? ?(?? ,?? ) :
?? (?? )=
?? (?? )-?? (?? )
?? -?? and ?? (?? )=
?? (?? )-?? (?? )
?? -?? (1)
Now, let ?? ?(?? ,?? )
? ?? ?(?? ,?? ) or ?? <?? <??
By Lagrange's Mean Value Theorem (LMVT)
?? '
(?? )=
?? '
(?? )-?? '
(?? )
?? -??
Using values in eqn. (1), we get
?? '
(?? )=
?? (?? )-?? (?? )
?? -?? -
?? (?? )-?? (?? )
?? -?? ?? -?? ? ?? ''
(?? )=
-
?? (?? )
?? -?? -
?? (?? )
?? -?? ?? -?? [?? (?? )=?? (?? )=0]
?? '
(?? )=-[
?? (?? )
?? -?? +
?? '
(?? )
?? -?? ?? -?? ]<0 as ?? >?? ,?? >?? ,?? >??
1.4 Show that a box (rectangular parallelopiped) of maximum volume ?? with
prescribed surface area is a cube.
(2010 : 20 Marks)
Solution:
Let ?? ,?? ,?? be length of edges of given rectangular parallelopiped. Its surface area be ?? ,
volume be ?? .(?? ,?? , ?? ?0) .
? ?? =2???? +2???? +2???? ,?? =??????
Let ?? be the Lagrange's multiplier (?? ?0)
So,
?? =?????? +?? (2???? +2???? +2???? -?? )
? At extremum,
?? ?? =0????? +?? (2?? +2?? )=0 (1)
?? ?? =0????? +?? (2?? +2?? )=0 (2)
?? ?? =0????? +?? (2?? +???? )=0 (3)
Multiplying (1) by ?? and (2) by ?? and subtracting then, we get
?????? +?? (2???? +2???? )-?????? -?? (2???? +2???? )=0
? ?? (2???? -2???? )=0
? 2???? -2???? =0 (?? ?0)
? ?? =?? as ?? ?0 (4)
Similarly, multiplying ( 2 ) by ?? and (3) by ?? and subtracting, we get
?????? +?? (2???? +2?? ?? )-?????? -?? (2???? +2???? )=0
? ?? (2???? -2???? )=0
? ?? =?? (?? ?0) (5)
? from (4) and (5), it can be concluded ?? =?? =?? which is a cube.
1.5 Let ?? be a function defined on R such that ?? (?? )=-?? and ?? '
(?? )=?? for all
values of ?? in R. How long can ?? (?? ) possibly be?
(2011: 10 marks)
Solution :
?? (?? ) =?? ??? ???
Let ?? (?? )
=5+ where
?? (?? )=0
? ?? (?? )
=5?? +?? (?? )+?? 1
,h(?? )=??? (?? )???? is a
decreasing function
? ?? (0) =?? (0)+?? 1
=-3
? ?? 1
=-3-?? (0)
? ?? (?? ) =5?? +?? (?? )-3-?? (0)
? ?? (2) =10+?? (2)-3-?? (0)
=7+(?? (2)-?? (0))
=7 (??? is a decreasing function, ?? (2)-?? (0)<
0)
? (2) =7
?? '
(?? )=5??? ???
Read More
|
387 videos|203 docs
|
Related Exams
About this Document
|
4.85/5 Rating |
|
Dec 22, 2024 Last updated |
Document Description: Function of a Real Variable for UPSC 2024 is part of Mathematics Optional Notes for UPSC preparation.
The notes and questions for Function of a Real Variable have been prepared according to the UPSC exam syllabus. Information about Function of a Real Variable covers topics
like and Function of a Real Variable Example, for UPSC 2024 Exam. Find important definitions, questions, notes, meanings, examples, exercises and tests below for Function of a Real Variable.
Introduction of Function of a Real Variable in English is available as part of our Mathematics Optional Notes for UPSC
for UPSC & Function of a Real Variable in Hindi for Mathematics Optional Notes for UPSC course.
Download more important topics related with notes, lectures and mock test series for UPSC
Exam by signing up for free. UPSC: Function of a Real Variable | Mathematics Optional Notes for UPSC
Description
Full syllabus notes, lecture & questions for Function of a Real Variable | Mathematics Optional Notes for UPSC - UPSC | Plus excerises question with solution to help you revise complete syllabus for Mathematics Optional Notes for UPSC | Best notes, free PDF download
Information about Function of a Real Variable
In this doc you can find the meaning of Function of a Real Variable defined & explained in the simplest way possible. Besides explaining types of
Function of a Real Variable theory, EduRev gives you an ample number of questions to practice Function of a Real Variable tests, examples and also practice UPSC
tests
|
Explore Courses for UPSC exam
|
|
Signup for Free!
Signup to see your scores go up within 7 days! Learn & Practice with 1000+ FREE Notes, Videos & Tests.
Related Searches
Viva Questions
,ppt
,Semester Notes
,Function of a Real Variable | Mathematics Optional Notes for UPSC
,Previous Year Questions with Solutions
,Sample Paper
,MCQs
,shortcuts and tricks
,study material
,practice quizzes
,past year papers
,Important questions
,video lectures
,Exam
,Extra Questions
,mock tests for examination
,Function of a Real Variable | Mathematics Optional Notes for UPSC
,Objective type Questions
,Free
,Summary
,Function of a Real Variable | Mathematics Optional Notes for UPSC
;
Additional Information about Function of a Real Variable for UPSC Preparation
Function of a Real Variable Free PDF Download
The Function of a Real Variable is an invaluable resource that delves deep into the core of the UPSC 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 Function of a Real Variable now and kickstart your journey towards success in the UPSC exam.
Importance of Function of a Real Variable
The importance of Function of a Real Variable cannot be overstated, especially for UPSC aspirants.
This document holds the key to success in the UPSC 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.
Function of a Real Variable Notes
Function of a Real Variable Notes offer in-depth insights into the specific topic to help you master it with ease.
This comprehensive document covers all aspects related to Function of a Real Variable.
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.
Additionally, the paper analysis provides valuable tips for tackling the exam strategically.
Access to Toppers' notes gives you an edge in understanding complex concepts.
Whether you're a beginner or aiming for advanced proficiency, Function of a Real Variable Notes on EduRev are your ultimate resource for success.
Function of a Real Variable UPSC Questions
The "Function of a Real Variable UPSC Questions" guide is a valuable resource for all aspiring students preparing for the
UPSC 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 Function of a Real Variable on the App
Students of UPSC can study Function of a Real Variable alongwith tests & analysis from the EduRev app,
which will help them while preparing for their exam. Apart from the Function of a Real Variable,
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 Function of a Real Variable is prepared as per the latest UPSC syllabus.
|
© EduRev
|
Education Revolution
|
|
Signup to see your scores
go up within 7 days!
Access 1000+ FREE Docs, Videos and Tests
Takes less than 10 seconds to signup