Numerical Methods and Calculus

• Last Updated : 21 Jan, 2014

 Question 1

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

 C D B A

Question 1-Explanation:

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:

 9.003 9.017 8.983 9.045

Question 2-Explanation:

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

 One, at π/2 One, at 3π/2 Two, at π/2 and 3π/2 Two, at π/4 and 3π/2

Question 3-Explanation:

Sine function increases till π/2 and so for the considered interval π/4 would be a local minimum. From π/2, value of sine keeps on decreasing till 3π/2 and hence 3π/2 would be another local minima.

 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

Question 4-Explanation:

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 ?

 -i 2 0 i

 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

Question 6-Explanation:

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 Limn->∞(1-1/n)2n ?

 1 e-2 e-1/2 0

Question 7-Explanation:

$\lim_{n \to \infty}\left(1 - \frac{1}{n}\right)^{2n}\\ = e^{(-1)*\frac{1}{n}*2n}\\ =e^{-2}$

 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

Question 8-Explanation:

Since, 10nlog10n ≤ 0.0001n2

Given n = 10k records.

Therefore,

⟹10×(10k)log1010k ≤ 0.0001(10k)2

⟹10k+1k ≤ 0.0001 × 102k

⟹k ≤ 102k−k−1−4

⟹k ≤ 10k−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
 0 1 ln 2 1/2 ln 2

Question 9-Explanation:
(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
 1 -1 inf -inf

Question 10-Explanation:
There are 93 questions to complete.