Applied Calculus
Applied Mathematics, Unit III. Covers Derivatives in applications; Marginal cost and marginal revenue; Maxima, minima and monotonicity; Integration and area; Consumer and producer surplus; Differential equations.
About this unit
Applied Calculus takes the derivatives and integrals you already know and puts them to work, especially in economics and business. On the differentiation side you revise second-order, parametric and implicit derivatives and rates of change, then use derivatives as marginal cost and marginal revenue, and to find where a function rises, falls and reaches its highest or lowest value. On the integration side you find areas under curves and use them to measure consumer surplus and producer surplus. The unit ends with differential equations: forming them from situations such as growth and decay, and solving simple first-order equations. The aim throughout is to read a real situation, model it with a function and interpret the answer.
How to prepare
- Revise the standard derivatives and integrals until they are automatic.
- Learn the economic meanings: MC = C′(x), MR = R′(x), profit is maximum where MR = MC and the second-derivative test confirms it.
- Learn the consumer and producer surplus formulas together with a sketch of the shaded area.
- Practise word problems on optimisation, growth and decay, and always state units in the final answer.
Topics in this chapter
Derivatives in applications
Second-order, parametric and implicit derivatives; rates of change.
5 questionsMarginal cost and marginal revenue
Using derivatives in economics.
5 questionsMaxima, minima and monotonicity
Local and absolute extrema and applied problems.
5 questionsIntegration and area
Indefinite and definite integrals and area under a curve.
5 questionsConsumer and producer surplus
Applications of integration in economics.
5 questionsDifferential equations
Order, degree, formulating and solving simple differential equations.
5 questions