Numerical Methods and Calculus

Question 1

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

• A. C
  
• B. D
  
• C. B
  
• D. A
  

Answer: (D)

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A function is continuous at some point c, 

Value of f(x) defined for x > c = Value of f(x) defined for x < c = Value of f(x) defined for x = c 

All values are 2 in option A

Question 2

Function f is known at the following points: 

• A. 9.003 
  
• B. 9.017 
  
• C. 8.983 
  
• D. 9.045
  

Answer: (D)

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Since the intervals are uniform, apply the uniform grid formula of trapezoidal rule.

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

• A. One, at π/2
• B. One, at 3π/2
• C. Two, at π/2 and 3π/2
• D. Two, at π/4 and 3π/2

Answer: (D)

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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

• A. 5
  
• B. 7
  
• C. 3
  
• D. 1
  

Answer: (C)

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In bisection method, we calculate the values at extreme points of given interval, if signs of values are opposite, then we find the middle point. Whatever sign we get at middle point, we take the corner point of opposite sign and repeat the process till we get 0. 

f(1) < 0 and f(9) > 0 
mid = (1 + 9)/2 = 5 

f(5) > 0, so zero value lies in [1, 5] 
mid = (1+5)/2 = 3 

f(3) > 0, so zero value lies in [1, 3] 
mid = (1+3)/2 = 2 

f(2) = 0

Question 5

Given i=√-1, what will be the evaluation of the integral ?

• A. 2
  
• B. -i
  
• C. 0
  
• D. i
  

Answer: (D)

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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

• A. 3.667
  
• B. 3.607
  
• C. 3.676
  
• D. 3.575
  

Answer: (B)

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In Newton-Raphson's method, We use the following formula to get the next value of f(x). f'(x) is derivative of f(x). 

f(x)  =  x2-13
f'(x) =  2x

Applying the above formula, we get
Next x = 3.5 - (3.5*3.5 - 13)/2*3.5
Next x = 3.607

Question 7

What is the value of Lim_n->∞(1-1/n)²ⁿ ?

• A. 1
  
• B. e^-2
  
• C. e^-1/2
  
• D. 0
  

Answer: (B)

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Question 8

Two alternative packages A and B are available for processing a database having 10^k 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?

• A. 12
• B. 10
• C. 6
• D. 5

Answer: (C)

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Since, 10nlog₁₀n ≤ 0.0001n²

Given n = 10^k records. 

Therefore, 

⟹10×(10^k)log₁₀10^k ≤ 0.0001(10^k)²

 ⟹10^k+1k ≤ 0.0001 × 10^2k

 ⟹k ≤ 10^{2k−k−1−4}

⟹k ≤ 10^k−5

 Hence, value 5 does not satisfy but value 6 satisfies. 6 is the smallest value of k for which package B will be preferred over A. Option (C) is correct.

Question 9

is equivalent to

• A. 0
• B. 1
• C. ln 2
• D. 1/2 ln 2

Answer: (D)

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(1-tanx)/(1+tanx) = (cosx - sinx)/(cosx + sinx) Let cosx + sinx = t (-sinx + cosx)dx = dt (1/t)dt = ln t => ln(sinx + cosx) => ln(sin Π/4 + cos Π/4) => ln(1/√2 + 1/√2) => 1/2 ln 2

Question 10

• A. 1
• B. -1
• C. INF
• D. -INF

Answer: (A)

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