When calculus is applied to problems of subjects like physics and economics, it usually leads to equations involving the first and second derivatives of functions, and the task is to recover the original function from these equations. If the derivative is completely known and is continuous, we can use the second fundamental theorem

However, we usually have only a relation between the function and its derivatives rather than a full knowledge of the derivative. For example, we may know that f (x)= f (x)2 for every x. So, we need to find more ways of relating information about f with information about f .We already have two important instances: Fermat's theorem and the monotonicity theorem. In this chapter we will explore several consequences of these results. The payoffs will be new techniques of calculating limits (§6.2), approximation of functions by polynomials (§6.3), use of integration to measure arc length, surface area, and volume (§6.4), and error estimates for numerical calculations of integrals (§6.5). Sections 6.2 and 6.3 are required for the final two chapters on sequences and series, while sections 6.4 and 6.5 are important for the applications of calculus.

Darboux's theorem says that if a function f is differentiable on an interval then f will have the intermediate value property on that interval. Thus, f cannot have a jump discontinuity and it behaves like a continuous function in some ways. Nevertheless, it need not be continuous or even bounded.

As an example, consider the function defined by f (0) = 0 and f (x) = x2 sin(1/x) when x ≠ 0. This function is differentiable at non-zero points by the chain rule. It is also differentiable at zero by a direct calculation:

Thus, f is differentiable at every point. However, f is not continuous at zero: which does not exist.

We can modify the above example slightly to get a function that is differentiable but whose derivative is not bounded. Define g(0) = 0 and g(x) = x3/2 sin(1/x) when x ≠ 0. We have

We say f is continuously differentiable on an interval I if it is differentiable on I and f is continuous on I.

Appendix: Solutions to Odd-Numbered Exercises

Significance of Transport Processes

The conversion of raw materials into useful products in a predictable, efficient, economical and environment-friendly manner is an essential part of many branches of engineering. There are two types of transformations: chemical transformations (involving chemical reactions) and physical transformations (melting, evaporation, filtering, mixing, etc.). Both of these transformations involve the motion of constituents relative to each other, and they often involve the transfer of energy in the form of heat. In operations involving fluid flow and mixing, there are forces exerted on the fluid due to pumps, impellers, etc. (input of mechanical energy), in order to overcome the frictional resistance generated by the flow. The subject of this text is the transport of the components in materials relative to each other, the transport of heat energy and the transport of momentum due to applied forces.

This text is limited to operations carried out in the fluid phase. Although solids transport and mixing does form an important part of material transformation processes, fluid-phase operations are the preferred mode for conversion because the transport is enabled by the two fundamental processes: convection and diffusion. Convection is the transport of mass, momentum and energy along with the flowing fluid. Diffusion is transport due to the fluctuating motion of the molecules in a fluid, which takes place even in the absence of fluid flow. Convection does not take place in solids since they do not flow, and diffusion in solids due to vacancy or interstitial migration is a very slow process, which makes it infeasible to effect material transformations over industrial timescales.

Fluids are of two types: liquids and gases. In liquids, the molecules are closely packed, and the distance between molecules is comparable to the molecular diameter. In contrast, in gases, the distance between molecules is about 10 times larger than the molecular diameter under conditions of standard temperature and pressure (STP). Due to this, the density of a liquid is about 103 times that of a gas. In a gas, the molecules interact through discrete collisions, and the period of a collision is much smaller than the average time between collisions.

The two transport mechanism considered in this text are convection and diffusion. Convection is transport due to the flow. It is directional, and takes place only along the flow streamlines. Transport across streamlines, and transport across surfaces (where there is no fluid velocity perpendicular to the surface) necessarily takes place due to diffusion.

Diffusion is the process by which material is transported by the random thermal motion of the molecules within the fluid, even in the absence of fluid flow. The random velocity fluctuations of the molecules are isotropic, and they have no preferred direction. The characteristic velocity and length for the thermal motion are the molecular velocity and the microscopic length scale, which is the molecular size in a liquid or the mean free path (distance between intermolecular collisions) in a gas. While random molecular motion is always present in fluids, when the concentration/temperature/velocity fields are uniform, there is no net transport due to the random motion. Diffusion takes place only when there is a spatial variation, and transport is along direction of variation.

The molecular mechanisms of mass, momentum and thermal diffusion, are discussed in this chapter. Constitutive relations for the fluxes are derived from a molecular description, and the diffusion coefficients are estimated.

The gas diffusivities are estimated using kinetic theory for an ideal gas made of hard spheres, which undergo instantaneous collisions when the surfaces are in contact, but which do not exert any intermolecular force when not in contact. Real gas molecules do not interact like hard spheres—the interaction force between molecules is repulsive at small separations and attractive at larger separations. Diatomic and polyatomic molecules are also not spherically symmetric, and their interaction depends on the relative orientation of the molecules. The diffusion coefficients in the hard sphere model are proportional to √T, where T is the absolute temperature. For molecules with continuous intermolecular potential, the diffusion coefficients are proportional to a power of the temperature which higher than ½. The pressure-density relationship for real gases is also more complicated than that for an ideal gas, and the virial corrections need to be included for dense gases.

In calculus, we mainly study continuous change. However, there are situations where discrete changes have to be considered. For example, when we try to describe a number such as _ by its decimal representation, we actually create an iterative process of successively better approximations: 3, 3.1, 3.14, 3.141, 3.1415, 3.14159, and so on. A similar situation arises when we work with the Taylor polynomials of a function— we successively approximate a function by polynomials of increasing degree. What is common to the two examples is that there is a first stage, a second stage, a third stage, and we are interested in what happens as we keep going. Clearly, we need to develop a theory of limits for this context.We shall do so in this chapter. Further, we shall work out in detail the situation when discrete changes accumulate and we are interested in the total. This will have many similarities as well as a direct relation with integration.

As an example, let us consider a geometry problem that leads to an iterative method for approximating square roots by fractions. It is named Heron's method after a Greek mathematician, but the evidence is strong that this kind of reasoning was carried out earlier in ancient Iraq and India, three to four thousand years ago. The statement of the problem is: “Given a rectangle, construct a square with the same area.” Now, if the rectangle has sides a and b, the square needs to have side √ab. To us, this may be a triviality, but what if the only numbers you know are the fractions? Then the problem will, in general, have only approximate solutions. How do we find good fractional approximations to √ab? Consider the following steps.

The final square is obviously a bit too big. Nevertheless, its side of (a + b)/2 is visibly better than the initial sides of a and b. If we have ab = N, we can repeat the process with a rectangle whose sides are a = (a + b)/2 and b0 = N/a. This will lead to a new and further improved square with side (a + b)/2.

This chapter explores two key concepts in contract law. First, it identifies the parties to a contract and the rules that help in that process. In particular, the doctrines of privity and agency, which assist with determining who incurs rights and obligations under a contract, are discussed. Second, the chapter considers the terms of a contract, including how to identify, incorporate and interpret them. Specific attention is paid to the various types of contract terms and how they should be interpreted.

This chapter considers how intangible assets that businesses develop, such as inventions, designs and brands, as well as business ideas and information, can be protected pursuant to Australian intellectual property (IP) laws. It identifies that various IP rights are protected by statute, while others are protected pursuant to common law. It explains that some forms of intellectual property require businesses to apply and register for protection before an IP right can be claimed, such as designs, patents and trade marks, while others, such as copyright, dont require any application or registration. The chapter highlights that an array of IP rights can be used to protect different aspects of a business’s goods and/or services. For example, a business’s product might have a patent that attaches to it regarding how it works, a registered trade mark protecting its brand name, and a design for its appearance. The more IP rights that can be used to protect a particular good or service, the more defiant the good or service will be to imitation and competition.

There are many taxes in Australia that operate at the federal, state and local levels. Taxpayers must understand their tax obligations by identifying which taxes apply to them and understanding the requirements they must meet in order to comply with the tax laws. Federal tax laws are administered by Australia’s federal revenue authority, the Australian Taxation Office (ATO), and state taxes are administered by state-based revenue authorities. This chapter will cover two types of federal taxes: income tax and the goods and services tax (GST).

In an ideal world, people who enter into contracts would choose to enter the contract, and agree to its terms, because they have accurate and comprehensive information on which to base their decision. This chapter will first explain when a contract may be invalid, because one or both parties entered into the contract under some sort of misapprehension, or on the basis of misinformation. We will look at mistake (mutual mistake, unilateral mistake and common mistake) and misrepresentation. We will also briefly explain the old action ‘non est factum’ (‘not my deed’) and the remedy of rectification. Second, the chapter will explore when a contract may be invalid because of ‘unfair’ conduct by one of the parties – for example, where one party (Party A) enters an agreement because another party (Party B) subjected A to undue pressure (duress or undue influence); or because the conduct of B is so ‘unconscientious’ or the contract is able to be set aside in equity for unconscionable conduct.

This chapter examines the law of sale of goods. The statutory regime across the states and territories is explained before the specific concept of contracts for the sale of goods is discussed. The chapter then considers the various implied terms that become a part of such contracts and the consequences for violation of these terms. A brief discussion of the various rules pertaining to delivery follows, before the chapter concludes with an outline of the various remedies available to aggrieved parties when sale of goods contracts are breached.

This chapter will introduce the idea of ‘ethics’, and then the subset of ‘business ethics’. You will read real-world examples of (often poor) behaviour from companies, and understand how that behaviour can be considered through the lenses of business ethics, the ethical director, corporate governance with a focus on corporate social responsibility, and ethical marketing and advertising. By the end of this chapter, you will have a broad understanding of how these concepts fit together, and how they interact with the legal regulations around companies and businesses discussed in other parts of this text.

This chapter focuses on what occurs when a contract is performed and brought to an end. It discusses the rules regarding performance, termination, and the resulting remedies available in the event of a breach of contract. The concepts covered in this chapter are of vital importance to contracting parties. For instance, parties may wish to know how to fulfil their contractual obligations in order to bring a contract to a natural end. Conversely, they may want to know how to get out of a contract altogether, potentially ending it early, resulting in a breach. Assessing the various remedies that are available in the event of a breach of contract will therefore assist parties in shaping their interactions with one another, as well as protect them where they are the subject of a breach.

Work is a universal human experience, making it highly applicable and interesting to study in a legal context. This chapter considers Australia’s workplace relations system, providing an overview of the complex framework of regulation, which differs between occupations and industries and, in some instances, between the federal and state/territory levels. It begins by considering the main sources of Australia’s employment law; operation of the Fair Work Act 2009 (Cth); the Fair Work Commission; the Fair Work Ombudsman; the National Employment Standards; Modern Awards and enterprise agreements; employment contracts; and the distinction between employees and independent contractors. Finally, it assesses how the various duties imposed by Australia’s workplace relations system can be enforced through an application for unfair dismissal.

This chapter covers personal property, which is a broad category and a developing one. It is the most important type of property today in the commercial world, partly because of its breadth. The chapter starts by placing personal property in the wider area of property, distinguishing it from land or interests in land. Whether something is land or personal property can have important consequences for its ownership, or security interests over it. We will also examine the test applied to decide whether something that was goods has become a ‘fixture’, and thus part of the land. Second, we will look at the usual classifications within personal property, which have legal consequences. Possession, and its acquisition or loss, plays a crucial role when considering ownership of personal property. Lastly, what can be done when a holder’s rights in personal property are interfered with? We will look at the main remedies available to enforce those rights.