MTH211 Calculus of Several Variables B.Sc Question Bank : nmu.ac.in
Name of the University : North Maharashtra University
Degree : B.Sc
Department :Mathematics
Year : II
Name Of The Subject : MTH211 Calculus of Several Variables
Document type : Question Bank
Website : nmu.ac.in
Download Model/Sample Question Papers : https://www.pdfquestion.in/uploads/nmu.ac.in/5275-S.Y.B.Sc.%20Mathematics%20(%20MTH%20-%20211%20)%20Question%20Bank.pdf
Calculus of Several Variables Model Paper :
North Maharashtra University , Jalgaon
Question Bank (New Syllabus w.e.f. June 2008)
Class- S. Y.B.Sc. Subject : Mathematics
Paper MTH-211 (Calculus of Several Variables)
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Unit -I (Functions of Two & Three Variables) :
I) Objective Type Questions (2 Marks each)
1) Define neighbourhood of a point in a plane .
2) Define simultaneous limit of a function f(x,y) as (x,y) – (a,b)
3) Define Continuity of a function f(x,y) at a point (a,b)
4) Define x f (a,b) and y f (a,b)
5) Define xx f (a,b) and yy f (a,b)
6) Define xy f (a,b) and yx f (a,b)
7) Define differentiability of a function f(x,y) at a point (a,b) of its domain.
8) State the necessary condition for differentiability of a function f(x,y) at a point (a,b).
9) State sufficient condition for differentiability of a function f(x,y) at a point (a,b)
10) State Young’s theorem for the equality of xy f and yx f
11) State Schwarz’s theorem for the equality of xy f and yx
II) Multipal Choice Questions (1 Marks each) :
Choose the correct option .
1) If a function f(x,y) is discontinuous at a point (a,b) then
a) ( x,y ) ( a,b ) lim f(x,y) exist and equal to f(a,b)
b) f(x,y) is not differentiable at (a,b)
c) f(x,y) is differentiable at (a,b)
d) None of these.
2) If the simultaneous limit exists and has the same value along any three different paths then
a) ( x,y ) ( a,b ) lim f(x,y) exists.
b) ( x,y ) ( a,b ) lim f(x,y) may or may not exist
c) ( x,y ) ( a,b ) lim f(x,y) does not exist
d) None of these.
3) If a function f(x,y) is differentiable at a point (a,b) then,
a) x f (a,b)& y f (a,b) may or maynot exist.
b) x f (a,b) & y f (a,b) both exist .
c) only one of x f (a,b) and y f (a,b) exists.
d) x f (a,b) and y f (a,b) both does not exist.
4) If f(x,y) = x3 + y3 – 2×2 y2 then f(x,x) (1,1) = ——–
a) 1
b) -1
c) 0
d) None of these.
III) Theory and Examples (4- Marks each) :
1) If a function f(x,y) is differentiable at a point (a,b) of its domain then show that
i) f(x,y) is continuous at (a,b)
ii) x f and y f exist at (a,b)
2) State and prove sufficient condition for differentiability of the function f(x,y).
3) State and prove Schwarz’s theorem for equality of xy f and yx f at a point (a,b)
4) State and prove Young’s theorem for equality of xy f and yx f at a point (a,b)
5) If f(x,y) = x3 – xy2 show that used in the mean value theorem applied to the points (2,1) and (4,1) satisfies the quadratic equation 3? 2 + 6? – 4 = 0
6) If f(x,y) = x2 y + 2xy2 show that the value of in the expression of the mean value theorem applied to the line segment joining the point (1,2) to (3,3) satisfies the equation
Unit – III ( Taylor’s Theorem and Extreme values ) :
I) Objective questions ( 2 marks each)
1) Write the simplified mathematical Taylor’s expansion for f(x,y)
2) Write the simplified mathematical Maclaurin’s expansion for f(x,y).
3) Define absolute maximum of the function f(x,y) at (a,b)
4) Define absolute minimum of the function f(x,y) at (a,b)
5) State the necessary condition for the extreme value.
6) Write the condition for critical point (a,b) to become a function f(x,y) maximum.
7)Write the condition for critical point (a,b) to become function f(x,y) minimum.
8) Under what condition the critical point (a,b) will be saddle point ?
9) Find the Stationary points for f(x,y) = x3 y2 (1- x – y)
10) Find the Stationary points for f(x,y) = x3 + y3 – 3axy
11) Find the minimum value of f(x,y) = 1+ x2 + y2