Download PTU B.Sc (Non-Medical) 2020 March 1st Sem 75746 Differential Calculas Question Paper

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Roll No. | Total No. of Pages : 02
Total No. of Questions : 09
B.Sc. (Non Medical) (2018 & Onwards) (Sem.-1)
DIFFERENTIAL CALCULAS
Subject Code : BSNM-105-18

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INSTRUCTIONS TO CANDIDATES :

1. SECTION-A is COMPULSORY consisting of TEN questions carrying ONE mark each.
2. SECTION-B contains FIVE questions carrying FIVE marks each and students have to attempt any FOUR questions.

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3. SECTION-C contains THREE questions carrying TEN marks each and students have to attempt any TWO questions.

SECTION-A

1. Answer briefly :
   1. Define sequence.
   2. Define limit inferior with example.
   3. Define Left Hand Limit.

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   4. Define uniform continuity.
   5. Define Right hand derivatives.
   6. Find
      if f=x²-xsiny and g=x²y²+x+y
   7. Show that the function f(x, y) = | x | + | y | is continuous at the origin.

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   8. State Euler’s Theorem on homogeneous function.
   9. Prove that a real polynomial function is continuous everywhere.
   10. Give an example of a decreasing sequence which diverges to —8.

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SECTION-B

1. State and prove Cauchy’s first theorem on limits.

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2. State and prove Bolzano’s Intermediate Value Theorem.
3. Prove that the function f (x, y) = v|xy| is not differentiable at the origin but it is continuous at the origin, both fx, fy exist at the origin & have the value 0.
4. Apply Taylor’s Theorem with Lagrange’s form of remainder to the function f (x) = log x i [1, x].
5. If v1=x1+x2+x3+x4
   v1v2 =x2+ x3+ x4
   

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   v1v2v3 = x3+ x4
   v1v2v3v4 = x4, show that ?(x1,x2,x3,x4)/?(v1,v2,v3,v4)=v1

SECTION-C

1. a) Show that the alternating series S(-1)ⁿ [n/(n+1) + 2/3 + 4/5 + .........] oscillates finitely.
   b) Use definition of limit to prove that lim x?3 (1 — 3x) =— 8.

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2. a) Show that the function f (x) = 1/vx is differentiable at x > 0.
   b) If z= X tan^-1(y/x), prove that x² ?²z/?x?y = (x²-y²)/(x²+y²)
3. a) Prove that sequence {1/(n+1)} is Cauchy sequence.
   b) Prove that if f is continuous at x = a, the | f| is also continuous at x = a.

NOTE : Disclosure of Identity by writing Mobile No. or Making of passing request on any page of Answer Sheet will lead to UMC against the Student.

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