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Longhand

Before the final

Adding the Newton correction instead of subtracting it

The habit: The next estimate is x0 − f(x0)/f′(x0): subtract the ratio, and let the signs of f(x0) and f′(x0) decide which way the estimate moves.

9 practice questions are built to catch it, in 1 topic.

See it fail in a figure: Which way to step, in the Newton’s method lesson

Practise against it: 6 questions

What it looks like

A question from Newton’s method, with the answer this slip gives beside the right one.

Introductory2 marks · 3 min

Let f(x)=x3+x−3f(x) = x^{3} + x - 3. Starting from x0=1x_{0} = 1, use one step of Newton’s method to find x1x_{1}.

With the slip
0.75\displaystyle 0.75
Worked through
x1=54\displaystyle x_{1} = \frac{5}{4}

f(1)/f′(1) = −1/4, and subtracting a negative step moves right. f is negative and rising at 1, so the root lies to the right of 1, not at 3/4.

See the worked solution

Where it comes up

Practise against it

Each of these has an answer this slip gives. Work them on paper: if you fall into it, the verdict names it and the question comes back in review.

  1. 01
    Newton’s method improves an estimate xnx_{n} of a root of f(x)=0f(x) = 0 by following the tangent line at xnx_{n} to the point where it crosses the xx-axis. Which formula gives the next estimate xn+1x_{n+1}?

    Newton’s methodIntroductory

  2. 02
    The equation cos⁡x=x\cos x = x has exactly one solution, the zero of f(x)=cos⁡x−xf(x) = \cos x - x. Use one step of Newton’s method on ff from x0=π4x_{0} = \dfrac{\pi}{4} to estimate the solution. Give x1x_{1} exactly.

    Newton’s methodStandard

  3. 03
    After a dose, the concentration of a drug in the blood is C(t)=5t e−0.4tC(t) = 5t\,e^{-0.4t} mg/L, where tt is in hours. The drug takes effect once the concentration reaches 3 mg/L. Starting from t0=1t_{0} = 1, use one step of Newton’s method on f(t)=C(t)−3f(t) = C(t) - 3 to estimate the time at which this first happens.

    Newton’s methodStandard

  4. 04
    Let x∗x^{*} be the xx-coordinate of the point on the curve y=exy = e^{x} closest to the point (2,0)(2, 0). Write the equation D′(x)=0D'(x) = 0, where D(x)D(x) is the squared distance from (x,ex)(x, e^{x}) to (2,0)(2, 0), and estimate x∗x^{*} with one step of Newton’s method on the function D′D' itself, from x0=0x_{0} = 0.

    Newton’s methodStandard

  5. 05
    The function f(x)=x3−xf(x) = x^{3} - x has roots −1-1, 00 and 11. Newton’s method is started at x0=12x_{0} = \tfrac{1}{2}. Which root do the estimates approach, and why?

    Newton’s methodChallenging

  6. 06
    Newton’s method is applied to f(x)=x3−3x2+x+3f(x) = x^{3} - 3x^{2} + x + 3, whose only real root is near x=−0.77x = -0.77, starting from x0=1x_{0} = 1. What happens to the estimates?

    Newton’s methodChallenging

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