# Using calculus to solve engineering problems

These resources support the use of calculus to solve engineering problems with particular reference to:

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using graphs to find the solution to engineering problems

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use graphs to represent variables in engineering systems

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using differentiation and integration to determine the rate of change in engineering systems and to identify turning points, maximum, minimum and optimum values

### Wind Turbine Power Calculations

When planning a wind farm it is important to know the expected power and energy output of each wind turbine to be able to calculate its economic viability.

In this resource, students are required to calculate the rotational kinetic power produced in a wind turbine at its rated wind speed, which is the minimum wind speed at which a wind turbine produces its rated power.

Detailed notes and examples are provided and there are extension activities for students to complete, together with learning outcomes and assessment criteria.

In completing these tasks students will:

- understand the idea of mathematical modelling
- be familiar with a range of models of change, and growth and decay
- know how 2-D and 3-D coordinate geometry is used to describe lines, planes and conic sections within engineering design and analysis
- know how to use differentiation and integration in the context of engineering analysis and problem solving
- construct rigorous mathematical arguments and proofs in engineering context
- comprehend translations of common realistic engineering contexts into mathematics

### Draining Hazardous Liquids During Chemical Processing

In this resource, students explore how to build a mathematical model of liquid draining through a tank and how to use the model to determine the time required for a tank to completely drain.

Bernoulli’s conservation of energy equation, which requires integration and differentiation, is applied to the draining of hazardous liquids.

Detailed notes and examples are provided and there are extension activities for students to complete, together with learning outcomes and assessment criteria.

In completing these activities students will:

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understand the idea of mathematical modelling

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be familiar with a range of models of change, and growth and decay

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know how to use differentiation and integration

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construct rigorous mathematical arguments and proofs

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comprehend translations of common realistic contexts into mathematics

### Study of Dynamic Systems in JCB Construction Machinery

In this resource, students explore how calculations for displacement, velocity and acceleration, caused during loading, are used to ensure that they are not so large as to adversely affect the performance of a dump truck.

Detailed notes and examples are provided and there are extension activities for students to complete, together with learning outcomes and assessment criteria.

In completing these tasks students will:

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understand the idea of mathematical modelling

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be familiar with a range of models of change, and growth and decay

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understand the use of trigonometry to model situations involving oscillations

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understand the mathematical structure of a range of functions and be familiar with their graphs

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know how to use differentiation and integration in the context of engineering analysis and problem solving

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construct rigorous mathematical arguments and proofs in engineering context

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comprehend translations of common realistic engineering contexts into mathematics

### Gas Compression and Expansion

Many machines compress or expand gas or fluid as part of their working design. Examples include a simple bicycle pump, a refrigerator and an internal combustion engine. To compress gas energy needs to be expended to reduce its volume. When gas is allowed to expand energy is released.

In this engineering resource students are asked the question "How can you calculate the energy used, or made available, when the volume of a gas is changed?"

Boyle’s law is used and students need to be able to integrate to complete the activities. Isothermal change and adiabatic change are considered. An interactive file graphs the motion of a piston in a cylinder.

The mathematics students will be required to use in this activity is to:

• be able to interpret data

• distinguish between definite and indefinite integrals and interpret a definite integral as an area.

### Differentiation

These resources cover aspects of differentiation, often used in the field of engineering.

Topics covered are:

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introduction to differentiation

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table of derivatives

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linearity rules

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product and quotient rules

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the chain rule

Comprehensive notes, with clear descriptions, for each resource are provided, together with relevant diagrams and examples. Students wishing to review, and consolidate, their knowledge and understanding of differentiation will find them useful, as each topic includes a selection of questions to be completed, for which answers are provided.

### Integration

These resources cover aspects of integration, often used in the field of engineering.

Topics covered are:

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integration as the reverse of differentiation

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table of integrals

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linearity rules of integration

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evaluating definitive integrals

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integration by parts

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integration by substitution

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integration as summation

Comprehensive notes, with clear descriptions, for each resource are provided, together with relevant diagrams and examples. Students wishing to review, and consolidate, their knowledge and understanding of Integration will find them useful, as each topic includes a selection of questions to be completed, for which answers are provided.

### Calculus Refresher

This calculus refresher resource contains a comprehensive review of derivatives, the product, quotient and chain rules, differentiation of functions, integration by parts and substitution, as well as partial fractions.

The topics covered are:

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derivatives of basic functions

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linearity in differentiation

· the product rule, quotient rule and chain rule for differentiation

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differentiation of functions defined implicitly and of functions defined parametrically

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integrals of basic functions

· linearity in integration

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evaluating definite integrals, integration by parts and integration by substitution

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integration using partial fractions

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integration using trigonometrical identities