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Functions
Abstract
The function is one of the basic concepts in mathematics. The description starts with the definition of a function and introductory properties of functions: a sign of a function, intervals of monotonicity, odd and even functions, and limits of a function. Elementary functions, like linear, quadratic, exponential, logarithmic and trigonometric functions will be introduced.
This chapter provides examples of equations with terms involving these functions, and illustrates the algebraic techniques necessary to solve them and the importance of maritime issues. In short, it is essential to be familiar and comfortable with the fundaments of functions before proceeding to the formal introduction of calculus in the next chapter.
Aim
To learn basic concepts in functions, like sign of a function, increasing or decreasing function and limit of a function and to solve maritime problems using functions.
Learning outcomes:
- Understanding the definition of function and recognize whether the mapping is a function
- Determine whether the function is injection, surjection and bijection
- Find the intervals of monotonicity of a function
- Solve the limits of a function
- Use the appropriate function in solving maritime problems.
Previous knowledge of mathematics
algebraic expressions, algebraic identities, linear equations and inequalities
Relatedness with solving problems in the maritime field
Functions are widely used in solving many engineering problems. Practical application of functions is in navigation theory, calculating the set and drift, velocity of a vehicle, the traveling time, and distance.
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Elementary Functions
Abstract
This chapter describes elementary functions, their properties and applications. Elementary functions are the functions which we can meet in most of the calculus and basic math applications. They include polynomial functions – linear, quadratic, etc., rational functions, trigonometric functions, exponential functions and their inverse functions, including the inverse trigonometry functions and logarithms, and, also functions defined as a sum, product and/or composition of finitely many elementary functions. Lots of functionswhich appear in technical or economic applications are elementary functions.
In this chapter, we will focus on understanding these functions and transforming those using basic algebraic operations. Examples (and diagrams/graphs) are given throughout the text to provide guidance on how to approach and solve various problems. Exercises are also included with solutions at the end of each task. Activities for individual learning and assessing your knowledge are available.
Aim
To acquire skills in plotting graphs of functions, solving equations, systems of equations, to understand the concepts behind making computations. It turns out that elementary functions are important not only for solving mathematical problems but also in many real applications.
Learning outcomes
To acquire skills in plotting graphs of functions, solving equations, systems of equations, to understand the concepts behind making computations. It turns out that elementary functions are important not only for solving mathematical problems but also in many real applications.
Previous knowledge of mathematics
basic knowledge from primary and high school, acquaintance of basic rules and formulae.
Relatedness with solving problems in the maritime field
We apply linear functions to solve problems about rates of change, curve fitting, linear regression formula. There are many real-world situations that deal with quadratics and parabolas. Throwing a ball, shooting a cannon, diving from a platform and hitting i.e., a golf ball are examples of situations that can be modeled by quadratic functions. In many of these situations you will want to know the highest or lowest point of the parabola. Applications of exponential decay can be found with the following examples: medications/caffeine leaving the body, radioactive decay, half-life, carbon dating, depreciation of material objects, etc. Applications of exponential growth can be found with the following examples: Population growth, bacteria growth, appreciation of material objects, compound interest. Scientists agree that the greenhouse effect is approximately logarithmic . Square roots are used in many jobs. They are used by engineers, carpenters, architects, designers, construction managers, technicians, statisticians, drafters, economists, lawyers, social scientists, timekeepers, agricultural workers, floor installers, medical assistants and many others jobs.
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Trigonometry
Abstract
In this chapter basic trigonometric concepts will be introduced, including the link between degrees and radians, conversion of angles from one measure to another, and calculating co-terminal angles. The trigonometric functions will firstly be introduced as ratios in a right triangle, and their definitions will then be expanded to onto the number line. Techniques for algebraic and graphical solving of equations and inequalities will be provided in the final part of this chapter. Finally, specific usage of trigonometry in navigation will be provided in the end of this chapter.
Aim
To successfully convert between radian and degree measure, to calculate the trigonometric functions in a right triangle, to solve trigonometric equations and inequalities, and to use trigonometric functions and concepts in everyday tasks.
Previous knowledge of mathematics
functions, coordinate geometry, basic geometry, Pythagorean theorem, ratios
Relatedness with solving problems in the maritime field
polar coordinates, navigation using polar coordinates, constraints problems (limited docking space), …
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Limits
Abstract
The concept of a limit is a fundamental concept of calculus and mathematical analysis. This chapter begins by describing how to find the limit of a function at a given point. Not all functions have limits at all points, and it is discussed how we can tell if a function does or does not have a limit at a particular value. This chapter has been created in an informal, intuitive fashion, but this is not always enough if we need to prove a mathematical statement involving limits. The last section of this chapter presents the more precise definition of a limit and shows how to prove whether a function has a limit.
Learning outcomes
- Describe the limit of a function.
- Use a table of values to estimate the limit of a function or to identify when the limit does not exist.
- Apply a graph to estimate the limit of a function or to identify when the limit does not exist.
- Determine one-sided limits and provide examples.
- Identify the importance of limits.
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Curves
Abstract
Mathematics is present in the movements of planets, bridge and tunnel construction, navigational systems used to keep track of a ship’s location, manufacture of lenses for telescopes, and even in a procedure for disintegrating kidney stones. The mathematics behind these applications involves conic sections. Conic sections are curves that result from the intersection of a right circular cone and a plane. Figure below illustrates the four conic sections: the circle, the ellipse, the parabola, and the hyperbola…
