International Baccalaureate Mathematics

Syllabus Content

Implicit differentiation.
Related rates of change.
Optimisation problems.

Here are some exam-style questions on this statement:

"A circle with equation $x^2+y^2=25 $ has centre $(0,0)$ and radius 5." ... more

"The edge lengths, $x$ cm, of a cube are increasing at a rate of 6 cm s^-1." ... more

"The north corridor at Addwell High School has a width of $2 \, \text{m}$. There is a ninety-degree corner at point $C$. Points $A$ and $B$ are variable points on the base of the walls such that $A$, $C$, and $B$ lie on a straight line." ... more

See all these questions

Here are some Advanced Starters on this statement:

Fence Optimisation
Find the length of a rectangle enclosing the largest possible area. more

Maximum Product
Two numbers add up to 10. What's the largest possible product they could have? more

Road Connections
Design roads to connect four houses that are on the corners of a square, side of length one mile, to minimise the total length of the roads. more

Click on a topic below for suggested lesson Starters, resources and activities from Transum.

Calculus It is said that the word calculus comes from the Latin word for the small pebble used for counting and calculations. The two major branches, differentiation and integration, are studied by pupils only towards the end of their school days but does then form a major part of their studies. A course in calculus is a prerequisite for other, more advanced courses in mathematical analysis. Isaac Newton is best known for developing calculus. He was born in England in 1642 and is considered one of the most influential scientists and mathematicians in history.

Furthermore

Official Guidance, clarification and syllabus links:

Appropriate use of the chain rule or implicit differentiation, including cases where the optimum solution is at the end point.

Implicit differentiation is a technique used to find the derivative of a function that is not written in the form of $y = f(x)$, but rather as an equation involving $x$ and $y$. To use implicit differentiation, we differentiate both sides of the equation with respect to $x$ and then solve for the derivative of $y$ in terms of $x$ and $y$.

The key formula for implicit differentiation is the chain rule:

$$\dfrac{dy}{dx} = \dfrac{dy}{dt} \times \dfrac{dt}{dx}$$

where $t$ is a parameter that defines the curve.

Here is an example of how to use implicit differentiation to find the derivative of the equation $x^2 + y^2 = 25$:

$$\begin{aligned} x^2 + y^2 &= 25 \\ \frac{d}{dx}(x^2 + y^2) &= \frac{d}{dx}(25) \\ \frac{d}{dx}(x^2) + \frac{d}{dx}(y^2) &= 0 \\ 2x + 2y\frac{dy}{dx} &= 0 \\ \frac{dy}{dx} &= -\frac{x}{y} \end{aligned} $$

Therefore, the derivative of the equation $x^2 + y^2 = 25$ with respect to $x$ is given by $y' = -x/y$.

Related rates of change. Here is a suggested step-by-step process for solving many problems:

Label a Diagram of the situation and notice which values are variables and which are constants;
Write an Equation relating the variables;
Differentiate the equation with respect to time (t);
Use the values at a Particular time into the differentiated equation;
Make the required rate the Subject.

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