Numerical Methods and Calculus

Question 61

The trapezoidal method to numerically obtain ^b∫_a f(x) dx  has an error E bounded by ((b-a)/12)h² max  f \'\'(x), x ∈ [a,b] where h is the width of the trapezoids. The minimum number of trapezoids guaranteed to ensure E ≤ 10^-4 in computing ln 7 using f= 1/x is

• 60
• 100
• 600
• 10,000

Question 62

The formula used to compute an approximation for the second derivative of a function f at a point X₀ is   

[caption width="800"] [/caption]

• A

• B

• C

• D

Question 63

Newton-Raphson iteration formula for finding ³√c, where c > 0 is               .

• a
• b
• c
• d

Question 64

Let f be a function defined by           Find the values for the constants a, b, c and d so that f is continuous and differentiable every where on the real line.

Question 65

What is the value of the limit - [Tex]\\lim_{x\\to 0} \\frac{a^{mx}-b^{mx}}{\\sin {(kx)}}[/Tex]

• A
• B
• C
• D

Question 66

Evaluate [Tex]\\lim_{y\\rightarrow 7}\\frac{y^2-4y-21}{3y^2-17y-28}[/Tex]

• 2/7
• 3/5
• 3/7
• 2/5

Question 67

What is the number of closed regions in the Hypercube [Tex]Q_3 [/Tex]?

• 6

• 5

• 3

• 4

Question 68

The value of [Tex]\\int_{0}^{\\pi / 4} x \\cos(x^2) dx[/Tex] correct to three decimal places (assuming that [Tex]\\pi = 3.14[/Tex]) is _______ . Note -This was Numerical Type question.

• 0.289
• 0.389
• 0.829
• 0.428

Question 69

If x = -1 and x = 2 are extreme points of f(x) = α log |x| + β x² + x then

• α = -6, β = -1/2
• α = 2, β = -1/2
• α = 2, β = 1/2
• α = -6, β =1/2

Question 70

Let f(x) = log|x| and g(x) = sin x . If A is the range of f(g(x)) and B is the range of g(f(x)) then A ∩ B is

• [-1, 0]
• [-1, 0)
• [-∞, 0]
• [-∞,1]