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Longhand

Before the midterm

Dropping the sign of a decreasing rate

The habit: A quantity that shrinks has a negative derivative, so write the minus in before substituting.

13 practice questions are built to catch it, in 2 topics.

Practise against it: 6 questions

What it looks like

A question from related rates, with the answer this slip gives beside the right one.

Introductory2 marks · 3 min

An online shop sells qq copies of a board game a week when the price is pp dollars, where q=3000pq = \dfrac{3000}{\sqrt{p}}. The price is now $25 and is being raised by $0.50 a week. At what rate is the weekly demand changing?

Give a signed rate in games per week.

With the slip
6\displaystyle 6
Worked through
dqdt=−6 games per week\displaystyle \frac{dq}{dt} = -6 \text{ games per week}

That is how fast demand is falling. A falling quantity has a negative rate.

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
    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?

    The definition of the derivativeIntroductory

  2. 02
    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

  3. 03
    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

  4. 04
    The function ff is piecewise linear: its graph is made of straight segments joining (−2,1)(-2, 1), (1,7)(1, 7), (3,7)(3, 7) and (5,−1)(5, -1), in that order, as shown. Which statement about its derivative is correct?

    The definition of the derivativeIntroductory

  5. 05
    A 5 m ladder leans against a vertical wall. The foot of the ladder slides away from the wall at 0.6 m/s. At what rate is the height of the top of the ladder changing when the foot is 3 m from the wall?

    Related ratesStandard

  6. 06
    A snowball melts so that its radius decreases at 0.5 cm per minute. A student finds how fast its volume is changing when the radius is 6 cm, as shown. Find the first line that is wrong.

    Related ratesStandard

Start the set