Numerical Methods and Calculus

Question 81

n-th derivative of xⁿ is

• n x^n-1
• nⁿ . n!
• nxⁿ !
• n!

Question 82

The arithmetic mean of attendance of 49 students of class A is 40% and that of 53 students of class B is 35%. Then the percentage of arithmetic mean of attendance of class A and B is

• 27.2%
• 50.25%
• 51.13%
• 37.4%

Question 83

A root of equation f(x) = 0 can be computed to any degree of accuracy if a \'good\' initial approximation x₀ is chosen for which

• f (x₀) > 0
• f (x₀) f (x₀)" > 0
• f (x₀) f (x₀)" < 0
• f (x₀)" > 0

Question 84

The value of x at which y is minimum for y = x² − 3x + 1 is

• -3/2
• 3/2
• 0
• -5/4

Question 85

The shift operator E is defined as E[f(x_i)] = f(x_i + h) and E\'[f(x_i)] = f(x_i - h) then △ (forward difference) in terms of E is

• E^-1
• E
• 1 - E^-1
• 1 - E

Question 86

The formula:

• Simpson rule
• Trapezoidal rule
• Romberg\'s rule
• Gregory\'s formula

Question 87

The image is called?

• Newton\'s backward formula
• Gauss forward formula
• Gauss backward formula
• Stirling\'s formula

Question 88

If we define the functions f, g and h that map R into R by : f(x) = x⁴ , g(x) = √(x² + 1), h(x) = x² + 72, then the value of the composite functions ho(gof) and (hog)of are given as

• x⁸ – 71 and x⁸ – 71
• x⁸– 73 and x⁸ – 73
• x⁸ + 71 and x⁸ + 71
• x⁸ + 73 and x⁸ + 73

Question 89

The domain of the function log( log sin(x) ) is

• 0 < x < π
• 2nπ < x < (2n + 1) π , for n in N
• Empty set
• None of the above

Question 90

Consider the functions I. [Tex]e^{-x} [/Tex] II. [Tex]x^{2} - \\sin x [/Tex] III. [Tex]\\sqrt{x^3+1} [/Tex] Which of the above functions is/are increasing everywhere in [0, 1] ?

• Ⅲ only
• Ⅱ only
• Ⅱ and Ⅲ only
• Ⅰ and Ⅲ only