IGCSE Additional Mathematics (0606)

Connected rates of change

What this topic tests, the mistakes that cost marks, and one exam-style question worked through step by step.

Key ideas and common mistakes

For small changes, the approximate change in y is found by multiplying the derivative at the starting x-value by the small increment in x.
Common mistake: Students often forget to use the derivative at the correct point or omit this multiplication.
To relate the height, radius, and slant height of a cone, use the Pythagorean theorem with the correct assignment of each length.
Common mistake: A common mistake is to confuse which side corresponds to which part of the triangle.
To relate the radius and height in a cone with a sloping edge at a given angle to the vertical, use tan(angle) = radius/height.
Common mistake: A common error is to use sine or cosine, which swaps or misassigns the sides.
To approximate the change in x for a small change in y, use δx ≈ δy/(dy/dx) evaluated at the correct x-value found by solving y = f(x).
Common mistake: The common mistake is to use the wrong x or not substitute it into dy/dx before dividing.
To approximate the small change in y for y = x³ sin(2x) as x increases by h, use the derivative at x = pi/4, correctly applying the product rule and evaluating trigonometric functions at 2x, not x.
When using the small-increment (differential) approximation for related variables, you must use the correct sign for dy/dx at the relevant x-value.

The value of x now increases by a small amount p, starting at x = 0. Write down, in terms of p, the approximate change in y.

Try it first, then show the working

For a small increase, change in y is approximately (dy/dx) multiplied by the change in x.
Here dy/dx = 3/4 at x = 0 and the change in x is p.
So the approximate change in y is (3/4)p = 3p/4.

Answer: 3p/4 (that is, 0.75p)

13 exam-style questions on Connected rates of change, each marked to the scheme.