Newton’s Method for Finding Solutions to Equations (VCE SSCE Mathematical Methods): Quizzes

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Newton's Method for Finding Solutions to Equations
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What does Newton's method find for equations of the form f(x)=0f(x) = 0?

Approximate solutions

What is the iterative formula for Newton's method?

xn+1=xn−f(xn)f′(xn)x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)}

In Newton's formula, what does f′(xn)f'(x_n) represent?

The derivative (gradient) at xnx_n

Why must f′(xn)f'(x_n) never equal zero when applying Newton's method?

Division by zero is undefined

For f(x)=x2−2f(x) = x^2 - 2 starting at x0=4x_0 = 4, what is x1x_1?

2.252.25

Solving the cubic −x3+5x2−3x+4=0-x^3 + 5x^2 - 3x + 4 = 0, which x0x_0 needs the fewest iterations?

3.83.8 (about 6 iterations)

What happens with f(x)=x3−5xf(x) = x^3 - 5x starting at x0=1x_0 = 1?

Oscillating sequence: 1,−1,1,−1,…1, -1, 1, -1, \ldots

What happens at a stationary point where f′(x)=0f'(x) = 0?

Tangent is horizontal, never crosses x-axis

What is x1x_1 in the general procedure?

x-axis intercept of tangent at P0P_0

For f(x)=x3−5xf(x) = x^3 - 5x, starting in (−1,1)(-1, 1) converges to which value?

x=0x = 0

Explore VCE SSCE Mathematical Methods Revision Notes by Topics

Tangents and Normals

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Newton’s Method for Finding Solutions to Equations

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Tangents and Normals

Rates of Change

Stationary Points

Absolute Maximum and Minimum Values

Maximum and Minimum Problems

Families of Functions

Applications to Motion in a Straight Line

Newton’s Method for Finding Solutions to Equations

Explore VCE SSCE Mathematical Methods Flashcards by Topics

Tangents and Normals

Rates of Change

Stationary Points

Absolute Maximum and Minimum Values

Maximum and Minimum Problems

Families of Functions

Applications to Motion in a Straight Line

Newton’s Method for Finding Solutions to Equations

Explore VCE SSCE Mathematical Methods Exam Questions by Topics

Tangents and Normals

Rates of Change

Stationary Points

Absolute Maximum and Minimum Values

Maximum and Minimum Problems

Families of Functions

Applications to Motion in a Straight Line

Newton’s Method for Finding Solutions to Equations

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