## Numerical Methods and Calculus

 Question 1
Which one of the following functions is continuous at x = 3?
 A A B B C C D D
GATE CS 2013    Numerical Methods and Calculus
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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:
 A 8.983 B 9.003 C 9.017 D 9.045
GATE CS 2013    Numerical Methods and Calculus
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Question 2 Explanation:
Since the intervals are uniform, apply the uniform grid formula of trapezoidal rule. This solution is contributed by Anil Saikrishna Devarasetty
 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
GATE CS 2012    Numerical Methods and Calculus
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Question 3 Explanation:
 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 1 B 3 C 5 D 7
GATE CS 2012    Numerical Methods and Calculus
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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 ?
 A 0 B 2 C -i D i
GATE CS 2011    Numerical Methods and Calculus
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Question 5 Explanation:
 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 3.575 B 3.676 C 3.667 D 3.607
GATE CS 2010    Numerical Methods and Calculus
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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 ?
 A 0 B e-2 C e-1/2 D 1
GATE CS 2010    Numerical Methods and Calculus
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Question 7 Explanation:
The value of e (mathematical constant) can be written as following   And the value of 1/e can be written as following. Limn-> ∞( 1-1/n)2n  =  (Limn-> ∞(1-1/n)n)2  =  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?
 A 12 B 10 C 6 D 5
GATE CS 2010    Numerical Methods and Calculus
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Question 8 Explanation:
B must be preferred on A when the time taken taken by B is more than A, i.e.,
```0.0001 n2 < 10 n log10n
10-5n  < log10n```
 Question 9
is equivalent to
 A 0 B 1 C ln 2 D 1/2 ln 2
GATE-CS-2009    Numerical Methods and Calculus
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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
```

```
 A 1 B -1 C inf D -inf
Numerical Methods and Calculus    GATE CS 2008
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Question 10 Explanation:
There are 52 questions to complete.

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