Numerical Methods and Calculus

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

Function f is known at the following points: 

 

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

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
 

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]?
 

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
 

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

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?

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

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