There are 366 different Starters of The Day, many to choose from. You will find in the left column below some starters on the topic of Calculus. In the right column below are links to related online activities, videos and teacher resources.

A lesson starter does not have to be on the same topic as the main part of the lesson or the topic of the previous lesson. It is often very useful to revise or explore other concepts by using a starter based on a totally different area of Mathematics.

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### Calculus Starters:

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.

#### Trapezium Rule

Practise using the trapezium rule to find an approximate value for the area under a curve.

Transum.org/go/?to=trapeziumrule

### 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.
Integrate ekx, 1/x, sin kx , cos kx and related sums, differences and constant multiples more...

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...

#### International Baccalaureate

See the Calculus sub-topics, syllabus statements, exam-style questions and learning resources for the IB AA course here.

### Exam-Style Questions:

There are almost a thousand exam-style questions unique to the Transum website.

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### Notes:

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.

### Calculus Teacher Resources:

Calculus Grapher: Given a function, predict the integral, first derivative, or second derivative curves.

Refreshing Revision: Every time you refresh the page you get different revision questions for whole class retrieval practice.

Integration Flowchart: This flowchart will only be of use to those who have already learnt the techniques mentioned in the IB AA Syllabus.

### Calculus Activities:

Trapezium Rule: Practise using the trapezium rule to find an approximate value for the area under a curve.

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.

Partial Fractions: Exercises on mastering the art of partial fraction decomposition.

Parametric Equations: Develop the skills required to manipulate a set of equations involving a paramater.

Alternatively, for the more advanced student, there is an ever-growing collection of Exam-Style Questions with worked solutions on the topic of Calculus.

### Calculus Investigations:

Maxvoltray: Find the maximum volume of a tray made from an A4 sheet of paper. A practical mathematical investigation.

### Calculus Videos:

Transum's Calculus Video

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.

Differentiation Video: A reminder of how to differentiate different types of functions and how to find the equations of tangents and normals.

Integration Video: You can't possibly learn all about integration from a 28 minute video so all that this resource can do is provide a quick revision to help you do the online exercise.

The Birth Of Calculus: A fascinating BBC Two television programme from 1986. A documentary on Leibniz and the calculus.

Links to other websites containing resources for Calculus are provided for those logged into 'Transum Mathematics'. Subscribing also opens up the opportunity for you to add your own links to this panel. You can sign up using one of the buttons below:

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### Other

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#### Parametric Equations

Develop the skills required to manipulate a set of equations involving a paramater.

Transum.org/go/?to=parametric

### Teaching Notes:

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