Newton–Raphson Worked Solutions — A-Level Maths

Fully worked, step-by-step solutions to A-Level Newton–Raphson questions. See exactly how to solve problems on newton-raphson, differentiation, iterative-formula, iteration.

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A-Level70 questionsStep-by-step solutions
Question 1
2 markseasy
Given f(x)=x3−2x−5f(x)=x^{3} - 2 x - 5, find f′(x)f'(x) (the derivative needed for the Newton-Raphson method).

Worked solution

  1. Write down the function

    f(x)=x3−2x−5f(x)=x^{3} - 2 x - 5

    Identify the function that is to be differentiated.

  2. Differentiate with respect to x

    3x2−23 x^{2} - 2

    Apply the standard rules of differentiation term by term.

  3. State the derivative

    f′(x)=3x2−2f'(x)=3 x^{2} - 2

    This derivative is used in the Newton-Raphson formula.

Answer
3x2−23 x^{2} - 2
Question 2
2 markseasy
Given f(x)=x2−5f(x)=x^{2} - 5, find f′(x)f'(x) (the derivative needed for the Newton-Raphson method).

Worked solution

  1. Write down the function

    f(x)=x2−5f(x)=x^{2} - 5

    Identify the function that is to be differentiated.

  2. Differentiate with respect to x

    2x2 x

    Apply the standard rules of differentiation term by term.

  3. State the derivative

    f′(x)=2xf'(x)=2 x

    This derivative is used in the Newton-Raphson formula.

Answer
2x2 x
Question 3
2 markseasy
Given f(x)=x3+x−3f(x)=x^{3} + x - 3, find f′(x)f'(x) (the derivative needed for the Newton-Raphson method).

Worked solution

  1. Write down the function

    f(x)=x3+x−3f(x)=x^{3} + x - 3

    Identify the function that is to be differentiated.

  2. Differentiate with respect to x

    3x2+13 x^{2} + 1

    Apply the standard rules of differentiation term by term.

  3. State the derivative

    f′(x)=3x2+1f'(x)=3 x^{2} + 1

    This derivative is used in the Newton-Raphson formula.

Answer
3x2+13 x^{2} + 1
Question 4
2 markseasy
Given f(x)=x+exf(x)=x + e^{x}, find f′(x)f'(x) (the derivative needed for the Newton-Raphson method).

Worked solution

  1. Write down the function

    f(x)=x+exf(x)=x + e^{x}

    Identify the function that is to be differentiated.

  2. Differentiate with respect to x

    ex+1e^{x} + 1

    Apply the standard rules of differentiation term by term.

  3. State the derivative

    f′(x)=ex+1f'(x)=e^{x} + 1

    This derivative is used in the Newton-Raphson formula.

Answer
ex+1e^{x} + 1
Question 5
2 markseasy
Given f(x)=x−cos⁡(x)f(x)=x - \cos{\left(x \right)}, find f′(x)f'(x) (the derivative needed for the Newton-Raphson method).

Worked solution

  1. Write down the function

    f(x)=x−cos⁡(x)f(x)=x - \cos{\left(x \right)}

    Identify the function that is to be differentiated.

  2. Differentiate with respect to x

    sin⁡(x)+1\sin{\left(x \right)} + 1

    Apply the standard rules of differentiation term by term.

  3. State the derivative

    f′(x)=sin⁡(x)+1f'(x)=\sin{\left(x \right)} + 1

    This derivative is used in the Newton-Raphson formula.

Answer
sin⁡(x)+1\sin{\left(x \right)} + 1

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