The Certificate Of Numerical Methods Exam Be Carefully

Explanation
The value of y'/x' in terms of the angle 0 is given by tanθ because the tangent of an angle is equal to the ratio of the length of the side opposite the angle to the length of the side adjacent to the angle in a right triangle.

Explanation
The Newton Raphson method is also known as the Tangent method because it involves finding the tangent line to the curve of a function at a given point. This tangent line is then used to approximate the root of the function by iteratively improving the estimate. The method is based on the idea that the tangent line provides a good approximation of the function near the root, allowing for efficient convergence to the actual root. The other options, Secant method, Chord method, and Diameter method, are different numerical methods used for root finding but are not synonymous with the Newton Raphson method.

Explanation
To decrease the number of iterations in the Newton-Raphson method, the value of f'(x) must be increased. This is because f'(x) represents the derivative of the function f(x), which provides information about the slope of the function at a given point. By increasing the value of f'(x), the slope becomes steeper, allowing the Newton-Raphson method to converge more quickly towards the root of the function.

Explanation
The correct answer is y’/x’ because in the Newton Raphson method, f’(x) represents the derivative of the function f(x) with respect to x. The derivative of a function measures the rate at which the function is changing at a given point. In this case, y’/x’ represents the derivative of y with respect to x, which is the appropriate formula for f’(x) in the Newton Raphson method.

Explanation
The iterative formula for the Newton Raphson Method is given by x(n+1)=0.5 [x(n)+166x(n)]. This formula is derived from the Newton Raphson Method, which is an iterative method used to find the roots of a given equation. In this case, the equation is f(x) = x^2 - 117 = 0. By applying the Newton Raphson Method, the formula x(n+1)=0.5 [x(n)+166x(n)] is obtained, which represents the next approximation of the root (x(n+1)) based on the current approximation (x(n)).

Explanation
The Newton-Raphson method fails at stationary points. In mathematical terms, a stationary point is a point where the derivative of a function is equal to zero. At these points, the Newton-Raphson method is unable to converge and find the root of the function accurately. This is because the method relies on the derivative of the function to update the estimation of the root, and if the derivative is zero, the method cannot progress further. Therefore, the Newton-Raphson method fails at stationary points.

Explanation
The Newton Raphson method is an iterative numerical method used to find the root of a function. It relies on an initial assumed value and iteratively improves the approximation until it converges to the actual root. The convergence of the method depends on the initial assumed value chosen.

Explanation
To find the value of x1, we can use the Newton-Raphson method. The formula for finding the next approximation is x1 = x0 - f(x0)/f'(x0), where x0 is the initial approximation.

Given that the initial approximation is x=1, we can substitute this value into the equation f(x) = x^3 + 4x + 1 and calculate f(1) and f'(1).

f(1) = 1^3 + 4(1) + 1 = 6
f'(1) = 3(1)^2 + 4 = 7

Substituting these values into the formula, we get:
x1 = 1 - 6/7 = 1 - 0.857 = 0.143

Therefore, the value of x1 is 0.143. However, none of the given answer choices match this value, so it seems that the question may be incomplete or not readable.

Explanation
The modification of Gauss elimination method that is called Gauss Jordan.