Numerical Method

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What is the condition for the equation f(x) = 0 to have at least one real root between a and b?

f(a) and f(b) have the same sign

f(a) and f(b) are both positive

f(a) and f(b) have opposite signs (correct)

f(a) and f(b) are both negative

What is the graphical method used to find the real roots of an equation?

Plotting the graphs of y = f(x) and y = x

Plotting the graphs of y = f1(x) and y = f2(x)

Plotting the graphs of y = f(x) and y = 0

Plotting the graphs of y = f1(x) and y = f2(x) and finding the points of intersection (correct)

What is the condition for all roots of the equation x^3 - 3x + k = 0 to be real?

k > 2

k belongs to the range (-2, 2) (correct)

k belongs to the range [2, 3]

k < 2

What is the purpose of the Location Theorem in finding the roots of an equation?

To determine the intervals in which the root lies Signup and view all the answers

What is the method of finding the real roots of an equation by plotting the graphs of y = f1(x) and y = f2(x) and finding the abscissae of the points of intersection?

Graphical Method Signup and view all the answers

When a number is rounded off to n decimal places, what is the maximum bound on the magnitude of relative error?

0.5 × 10^(−n+1) Signup and view all the answers

What type of equation is ax^2 + bx + c = 0?

Algebraic equation Signup and view all the answers

When a number is truncated to n decimal places, what is the maximum bound on the absolute error?

10^(−n+1) Signup and view all the answers

What is the number of significant digits in the number 0.0001?

1 Signup and view all the answers

What is the absolute error when the number 2.089 is rounded off to three significant digits?

-0.001 Signup and view all the answers

What is the maximum number of real roots of the equation x²ⁿ - 1 = 0?

2 Signup and view all the answers

What is the iterative method used for in solving algebraic and transcendental equations?

To approximate the root of the equation to any desired degree of accuracy Signup and view all the answers

What is the condition for all roots of the given equation to be real?

k + 2 > 0 and k - 2 < 0 Signup and view all the answers

In the iterative method, what is the value of x₁ if x₀ is the initial approximation to the root α?

g(x₀) Signup and view all the answers

What is the interval in which the smallest positive root of the transcendental equation x - e^x = 0 lies?

(0, 1) Signup and view all the answers

What is the purpose of the iterative method in solving algebraic and transcendental equations?

To successively approximate the root of an equation to any desired degree of accuracy Signup and view all the answers

What is the nature of the function g(x) in the iterative method?

Non-constant Signup and view all the answers

What is the relationship between the successive approximations x₀, x₁, and x₂ in the iterative method?

x₂ = g(x₁) = g(g(x₀)) Signup and view all the answers

What is the nature of the roots of the equation x²ⁿ - 1 = 0?

Exactly two real Signup and view all the answers

What is the purpose of locating the root of an equation in the successive bisection method?

To determine the interval in which the root of an equation lies Signup and view all the answers

What determines the number of significant digits in a number?

The position of the non-zero digits Signup and view all the answers

When rounding off a number to n significant digits, what happens to the (n+1)th digit if it is greater than 5?

It is discarded Signup and view all the answers

What is the purpose of rounding off numbers in numerical methods?

To decrease the complexity of the calculation Signup and view all the answers

How many significant digits are in the number 35.9200?

6 Signup and view all the answers

What is the rule for rounding off a number to n significant digits if the (n+1)th digit is 5 and is followed by zero or zeros?

Increase the nth digit by 1 if it is odd and remain unchanged if it is even Signup and view all the answers

What is the characteristic of an algebraic equation?

It has a polynomial function Signup and view all the answers

What type of equation is ae^x + b sin x = 0?

Transcendental equation Signup and view all the answers

What is the purpose of the Location Theorem in finding the roots of an equation?

To find the interval where the root lies Signup and view all the answers

What is the iterative method used for in solving algebraic and transcendental equations?

To find the approximate root of the equation Signup and view all the answers

What is the condition for an equation to have at least one real root between a and b?

f(a) and f(b) have opposite signs Signup and view all the answers

Numerical Method

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Questions and Answers

What is the condition for the equation f(x) = 0 to have at least one real root between a and b?

What is the graphical method used to find the real roots of an equation?

What is the condition for all roots of the equation x^3 - 3x + k = 0 to be real?

What is the purpose of the Location Theorem in finding the roots of an equation?

What is the method of finding the real roots of an equation by plotting the graphs of y = f1(x) and y = f2(x) and finding the abscissae of the points of intersection?

When a number is rounded off to n decimal places, what is the maximum bound on the magnitude of relative error?

What type of equation is ax^2 + bx + c = 0?

When a number is truncated to n decimal places, what is the maximum bound on the absolute error?

What is the number of significant digits in the number 0.0001?

What is the absolute error when the number 2.089 is rounded off to three significant digits?

What is the maximum number of real roots of the equation x²ⁿ - 1 = 0?

What is the iterative method used for in solving algebraic and transcendental equations?

What is the condition for all roots of the given equation to be real?

In the iterative method, what is the value of x₁ if x₀ is the initial approximation to the root α?

What is the interval in which the smallest positive root of the transcendental equation x - e^x = 0 lies?

What is the purpose of the iterative method in solving algebraic and transcendental equations?

What is the nature of the function g(x) in the iterative method?

What is the relationship between the successive approximations x₀, x₁, and x₂ in the iterative method?

What is the nature of the roots of the equation x²ⁿ - 1 = 0?

What is the purpose of locating the root of an equation in the successive bisection method?

What determines the number of significant digits in a number?

When rounding off a number to n significant digits, what happens to the (n+1)th digit if it is greater than 5?

What is the purpose of rounding off numbers in numerical methods?

How many significant digits are in the number 35.9200?

What is the rule for rounding off a number to n significant digits if the (n+1)th digit is 5 and is followed by zero or zeros?

What is the characteristic of an algebraic equation?

What type of equation is ae^x + b sin x = 0?

What is the purpose of the Location Theorem in finding the roots of an equation?

What is the iterative method used for in solving algebraic and transcendental equations?

What is the condition for an equation to have at least one real root between a and b?

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