calculus

Integration by Substitution

Current UK exam textbooks pass over proofs and mathematical discussions in a hurry to show the ‘how to’ of exam questions. Integration by substitution is a little more than just backwards chain-rule and deserves a fuller treatment. Try this, Let, then, Suppose that there exists a function g, of another variable , such that and …

Differentiation From First Principles

The gradient of a smooth curve, , at a point is the gradient of the tangent to the curve at the point . Point is on the curve and is a neighbouring point whose value is displaced a small quantity, . The idea behind differentiation is that as becomes very small, the gradient of tends …

The Fundamental Theorem of Calculus

Integration is introduced as the reversal of differentiation i.e. in solving a differential equation, . The link between integration and area is often passed over and is the subject of the Fundamental Theorem of Calculus. [The following discussion can be adapted for a decreasing function or, piece-wise, a function which successively increases or decreases.] Consider …

The Ghosts of Departed Quantities and the difficulties of teaching and learning caculus

Bishop Berkeley writes this attack on the apparent supernatural reasoning involved in calculus.  The infidel was probably Halley (of comet fame) or Newton. If pupils find the subject difficult to understand at school, and teachers find it difficult to teach, then the reason may be articulated in this book by the great man. Quotes include: …

Euler and the Properties of the Second Derivative

The summer examination season sees pupils searching for maximums and minimums on their text papers. This long-standing pursuit was initiated by the likes of Newton and Leibniz in their calculus. It can be all too difficult to think about for some: “our modern Analysts are not content to consider only the Differences of finite Quantities: …

Exdexcel Formula Sheet lays Pooh Traps for Further Mathematics Pupils

FP3 integration standard forms could lead unwary pupils and teachers into a Pooh Trap. The following issue arises each year as the finer points of this course get sharpened. When consulting the Edexcel formula sheet to find the integral, two anti-derivatives are given, Some might be tempted to infer that these functions are equal, this …