Numerical Methods and Calculus

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Question 1

Which one of the following functions is continuous at x = 3?

  • [Tex]f(x)=\left\{\begin{array}{lll}2, & \text { if } & x=3 \\ x-1, & \text { if } & x>3 \\ \frac{x+3}{3}, & \text { if } & x<3\end{array}\right.[/Tex]

  • [Tex]f(x)=\left\{\begin{array}{lll}4, & \text { if } & x=3 \\ 8-x & \text { if } & x \neq 3\end{array}\right.[/Tex]

  • [Tex]f(x)=\left\{\begin{array}{lll}x+3, & \text { if } & x \leq 3 \\ x-4, & \text { if } & x>3\end{array}\right.[/Tex]

  • [Tex]f(x)=\frac{1}{x^3-27}, [/Tex]if x ≠ 3

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Question 2

Function f is known at the following points: 

 

gatecs201310



 

  • 9.003 
     

  • 9.017 
     

  • 8.983 
     

  • 9.045
     

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Question 3

Consider the function f(x) = sin(x) in the interval [π/4, 7π/4]. The number and location(s) of the local minima of this function are

  • One, at π/2

  • One, at 3π/2

  • Two, at π/2 and 3π/2

  • Two, at π/4 and 3π/2

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Question 4

The bisection method is applied to compute a zero of the function f(x) = x4 – x3 – x2 – 4 in the 
interval [1,9]. The method converges to a solution after ––––– iterations
 

  • 7
     

  • 5
     

  • 3
     

  • 1
     

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Question 5

Given i=√-1, what will be the evaluation of the integral [Tex]\int_{0}^{\pi/2} \frac{\cos x + i\sin x}{\cos x - i\sin x} dx[/Tex]?
 

  • -i
     

  • 2
     

  • 0
     

  • i
     

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Question 6

Newton-Raphson method is used to compute a root of the equation x2-13=0 with 3.5 as the initial value. The approximation after one iteration is
 

  • 3.607
     

  • 3.667
     

  • 3.676
     

  • 3.575
     

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Question 7

What is the value of Limn→∞(1-1/n)2n ?
 

  • 1
     

  • e-2
     

  • e-1/2
     

  • 0
     

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Question 8

Two alternative packages A and B are available for processing a database having 10k records. Package A requires 0.0001n2 time units and package B requires 10nlog10n time units to process n records. What is the smallest value of k for which package B will be preferred over A?

  • 12

  • 10

  • 6

  • 5

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Question 9

[Tex]\int_{0}^{\pi/4} \frac{1 - \tan x}{1 + \tan x} dx[/Tex] is equivalent to

  • 0

  • 1

  • ln 2

  • 1/2 ln 2

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Question 10

[Tex]\lim_{x \to \infty } \frac{x - \sin(x)}{x + \cos(x)}[/Tex] equals

  • 1

  • -1

  • ∞

  • -∞

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