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Longhand

Before the midterm

Combining rates the way the quantities combine

The habit: Rates follow the derivative of the relation, not the relation itself: differentiate first, then substitute.

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

Practise against it: 4 questions

What it looks like

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

Introductory2 marks · 3 min

Three quantities depend on time and are related by R=PQR = \dfrac{P}{Q}. At a certain moment PP is increasing at an instantaneous rate of 6 % per year and QQ is increasing at 2 % per year; that is, P′P=0.06\dfrac{P'}{P} = 0.06 and Q′Q=0.02\dfrac{Q'}{Q} = 0.02. Find the percentage rate of change of RR at that moment.

Give the answer as a percentage per year.

With the slip
3\displaystyle 3
Worked through
R′R=0.04, i.e. 4% per year\displaystyle \frac{R'}{R} = 0.04, \text{ i.e. } 4\% \text{ per year}

6/2 = 3 divides the percentages because R = P/Q. Logs turn the quotient into log P − log Q: see what that does to the rates.

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
    At noon, ship A is 30 km due north of a port and sailing north at 20 km/h, while ship B is 40 km due east of the same port and sailing east at 15 km/h. How fast is the distance between the ships increasing at noon?

    Related ratesStandard

  2. 02
    A bead slides along a wire bent into the curve y=2xy = 2\sqrt{x}, with xx and yy in centimetres. Its xx-coordinate increases at 2 cm/s. At what rate is the bead’s distance from the origin changing as it passes the point (4,4)(4, 4)?

    Related ratesStandard

  3. 03
    A crate on a warehouse floor is dragged toward a wall by a rope tied to it at floor level. The rope runs up to a pulley fixed on the wall 3 m above the floor, and a winch hauls the rope in at 1 m/s. At what rate is the crate’s distance from the wall changing at the moment 5 m of rope lies between the pulley and the crate?

    Related ratesStandard

  4. 04
    Doctors set the doses of some medicines by body surface area. A common estimate of a person’s body surface area, in square metres, is S=hw60S = \dfrac{\sqrt{hw}}{60}, where hh is height in centimetres and ww is mass in kilograms. A child is 144 cm tall with a mass of 36 kg, and is growing at 6 cm per year while gaining 4 kg per year. How fast is the child’s body surface area increasing?

    Related ratesStandard

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