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Longhand

Before the midterm

Reporting a different quantity from the one asked for

The habit: Reread what the question asks, and check that the number you give is the thing it names, not one step before or after it.

70 practice questions are built to catch it, in 18 topics.

Practise against it: 6 questions

What it looks like

A question from the definition of the derivative, with the answer this slip gives beside the right one.

Introductory1 mark · 2 min

Let T(d)T(d) be the temperature of a lake, in degrees Celsius, at a depth of dd metres below the surface on a summer afternoon. What are the units of T′(d)T'(d), and what sign would you expect it to have?

With the slip
Degrees Celsius; positive, because the water is above 0 °C at every depth.
Worked through
Degrees Celsius per metre; negative, because the water gets colder as you go deeper.

That describes T itself: its unit and the sign of its values. T′ is a rate, in degrees Celsius per metre, and its sign says whether T rises or falls as you go deeper.

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
    A person’s core body temperature, in degrees Celsius, is modelled by the function below, where tt is the time in hours after midnight. At what time tt, with 0≤t<240 \leq t < 24, is the temperature highest? T(t)=36.9+0.4sin⁡ ⁣(π12(t−11))\displaystyle T(t) = 36.9 + 0.4\sin\!\left(\frac{\pi}{12}(t - 11)\right)

    Power functions and first sketchesStandard

  2. 02
    Find the sum of all real solutions of the equation below. 22x−6⋅2x+8=0\displaystyle 2^{2x} - 6\cdot 2^{x} + 8 = 0

    Functions, graphs and inversesStandard

  3. 03
    Two lamps stand 8 m apart on a straight path. At a distance of dd metres from a lamp, the brightness it gives is a power function of dd: 9d2\dfrac{9}{d^{2}} for the stronger lamp and 1d2\dfrac{1}{d^{2}} for the weaker one. At what distance from the stronger lamp, on the path between the two lamps, is the brightness from each lamp the same?

    Power functions and first sketchesChallenging

  4. 04
    There is exactly one value of the constant aa for which the limit below exists. Find that value of aa, and then enter the value of the limit. lim⁡x→3x2+ax−12x2−x−6\displaystyle \lim_{x \to 3} \frac{x^{2} + ax - 12}{x^{2} - x - 6}

    LimitsChallenging

  5. 05
    The function below has a horizontal asymptote as x→∞x \to \infty and another as x→−∞x \to -\infty. Find the xx-coordinate of the point where its graph crosses one of these asymptotes, or type none if it never does. f(x)=4x+3x2+1\displaystyle f(x) = \frac{4x + 3}{\sqrt{x^{2} + 1}}

    LimitsChallenging

  6. 06
    A function ff satisfies the identity below for every x≠2x \neq 2. Find f−1(3)f^{-1}(3). f(3x−1)=2x+5x−2\displaystyle f(3x - 1) = \frac{2x + 5}{x - 2}

    Functions, graphs and inversesCapstone

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