Skip to content
Longhand

Before the final

Dividing the wrong way round

The habit: Write the equation out in full before clearing fractions, and substitute the answer back to check.

23 practice questions are built to catch it, in 12 topics.

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.

Introductory1 mark · 2 min

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}?

With the slip
xn+1=xn−f′(xn)f(xn)\displaystyle x_{n+1} = x_{n} - \dfrac{f'(x_{n})}{f(x_{n})}
Worked through
xn+1=xn−f(xn)f′(xn)\displaystyle x_{n+1} = x_{n} - \dfrac{f(x_{n})}{f'(x_{n})}

The ratio is upside down. The step is the run that takes the tangent line down to zero: height over slope, not slope over height.

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
    Solve the equation below for xx. log⁡(x+1)−log⁡x=1\displaystyle \log(x + 1) - \log x = 1

    Functions, graphs and inversesIntroductory

  2. 02
    Find the value of the constant cc that makes ff continuous at x=0x = 0.

    ContinuityIntroductory

  3. 03
    Biologists counted 2340 salmon spawning in a creek in 2016 and 1620 in 2024. Find the average rate of change of the count over that period: the slope of the line through the two data points.

    The definition of the derivativeIntroductory

  4. 04
    A horse’s daily ration is xx kilograms of oats and yy kilograms of hay. Oats supply 12 megajoules (MJ) of energy per kilogram and hay supplies 8 MJ per kilogram, and the ration must supply exactly 72 MJ, so 12x+8y=7212x + 8y = 72. Plotted with xx on the horizontal axis and yy on the vertical axis, these rations lie on a line. Find its slope.

    The definition of the derivativeIntroductory

  5. 05
    The angle θ\theta satisfies sin⁡θ=27\sin\theta = \dfrac{2}{7} and π2<θ<π\dfrac{\pi}{2} < \theta < \pi. Find the exact value of tan⁡θ\tan\theta.

    Functions, graphs and inversesStandard

  6. 06
    Find the value of the constant aa for which the limit below holds. lim⁡x→0ax+25−5x=32\displaystyle \lim_{x \to 0} \frac{\sqrt{ax + 25} - 5}{x} = \frac{3}{2}

    LimitsStandard

Start the set