Fence Optimisation: Find the length of a rectangle enclosing the largest possible area.
Maximum Product: Two numbers add up to 10. What's the largest possible product they could have?
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.
Curriculum for Calculus:
Years 12 and 13
Pupils should be taught to understand and use the derivative of f (x) as the gradient of the tangent to the graph of y = f ( x) at a general point (x, y); the gradient of the tangent as a limit; interpretation as a rate of change, sketching the gradient function for a given curve, second derivatives, differentiation from first principles for small positive integer powers of x and for sin x and cos x. Understand and use the second derivative as the rate of change of gradient; connection to convex and concave sections of curves and points of inflection more...
Pupils should be taught to know and use the Fundamental Theorem of Calculus more...
Pupils should be taught to differentiate xn, for rational values of n, and related constant multiples, sums and differences. Differentiate ekx and akx, sin kx, cos kx, tan kx and related sums, differences and constant multiples. Understand and use the derivative of ln x more...
Pupils should be taught to integrate xn (excluding n = -1) and related sums, differences and constant multiples.
Pupils should be taught to apply differentiation to find gradients, tangents, normals, maxima, minima and points of inflection. Identify where functions are increasing or decreasing more...
Pupils should be taught to evaluate definite integrals; use a definite integral to find the area under a curve and the area between two curves more...
Pupils should be taught to differentiate using the product rule, the quotient rule and the chain rule, including problems involving connected rates of change and inverse functions more...
Pupils should be taught to Understand and use integration as the limit of a sum. more...
Pupils should be taught to understand and use numerical integration of functions, including the use of the trapezium rule and estimating the approximate area under a curve and limits that it must lie between more...
Pupils should be taught to differentiate simple functions and relations defined implicitly or parametrically, for first derivative only more...
Pupils should be taught to carry out simple cases of integration by substitution and integration by parts; understand these methods as the inverse processes of the chain and product rules respectively (Integration by substitution includes finding a suitable substitution and is limited to cases where one substitution will lead to a function which can be integrated; integration by parts includes more than one application of the method but excludes reduction formulae) more...
Pupils should be taught to construct simple differential equations in pure mathematics and in context, (contexts may include kinematics, population growth and modelling the relationship between price and demand) more...
Pupils should be taught to integrate using partial fractions that are linear in the denominator more...
Pupils should be taught to evaluate the analytical solution of simple first order differential equations with separable variables, including finding particular solutions (Separation of variables may require factorisation involving a common factor.) more...
Pupils should be taught to interpret the solution of a differential equation in the context of solving a problem, including identifying limitations of the solution; includes links to kinematics more...
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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.
Differentiation: Practise the technique of differentiating polynomials with this self marking exercise.
Integration: Exercises on indefinite and definite integration of basic algebraic and trigonometric functions.
Alternatively, for the more advanced student, there is an ever-growing collection of Exam-Style Questions with worked solutions on the topic of Calculus.
Introduction to Calculus: This video will give you a brief introduction to calculus. It does this by explaining that calculus is the mathematics of change. A couple of examples are presented, and then limits, derivatives, and integrals are introduced.
The Birth Of Calculus: A fascinating BBC Two television programme from 1986. A documentary on Leibniz and the calculus.
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