Aesthetic digital concise summaries of all the theory and some examples you need to know for mathematics 114.
The whole semester's content for mathematics 114 is summarised.
Self-study: Revision of coordinate geometry & straight lines
These are topics that you should be familiar with from school. Some students forget some of this material by the time they start their first year, and therefore need to revise them. These topics are all required for Calculus topics later on.
Week 1: Numbers, inequalities, absolute values, trigonometry and radian measure
We spend some time revising numbers, how to solve basic inequalities and trigonometry. We introduce radian measure, the absolute value function and solve some absolute value inequalities.
Week 2: Sets and logic
Everything in mathematics builds on sets and logic. This week is spent introducing the concept of a set and understanding basic mathematical logic.
Week 3 Functions & inverse functions
Most students will have seen functions at high school, but for Calculus you will need an improved understanding of functions. We will discuss properties of functions that you may not have come across, relate them to sets and statements, and later revisit them from the point of view of Calculus. We introduce the concept of the inverse of a function and what it means to be injective (one-to-one).
Week 4: Limits & continuity
The concept of a limit is one of the most important concepts in the course and underlies other important concepts, including continuity, derivatives and definite integrals. The second topic is for this week is that of continuity which gives a formal definition of the intuitive idea that a graph has
no breaks in it.
Week 5: Derivatives & Induction I
We introduce the derivative. It is one of the most important concept sin this course, and most of the course revolves around it. First we define it and develop some properties, then we learn how to differentiate most elementary functions, and then we show how it can be applied. We prove the product rule and the quotient rule. We introduce induction and look at basic examples. Induction is a proof technique for proving a statement holds for all natural numbers.
Week 6: Induction II & the Binomial Theorem
We continue to look at it induction this week and use it to prove the Binomial Theorem. We also look at how the binomial theorem can be used to solve certain problems.
Week 7: Derivatives II: Trigonometric Derivatives and the Chain rule
Here we develop the techniques that allow us to differentiate most elementary functions.
Week 8: Implicit Differentiation, Rates of Change
The first section gives you some idea of how Calculus may be used in practice. We also begin develop the theory that allow us to use Calculus to find minimum and maximum values. This discussion continues into the next week.
Week 9: , Max & Min values, the Mean Value Theorem, How derivatives affect the shape of a graph
Towards the end of the week we begin studying how derivatives affect the shape of a graph, the start of learning how to sketch and read a graph.
The reason for learning how to sketch graphs of functions is to consolidate the understanding of the relation between a function and its derivative and to be able to read a graph. As exercises you will need to sketch the graphs of some functions to see this principle in action.
Week 11: Exponential & logarithmic functions & Related Rates
We show how to make sense of arbitrary exponential functions and define logarithmic functions as their inverses. We discuss an application of derivatives.
Week 12: Anti-derivatives, the definite integrals & the Fundamental Theorem of Calculus
If you want to know if you understand something, see if you can do it in reverse. But that’s not the only reason for introducing anti-derivatives. They will also provide a connection to integration. Definite integrals are a way of calculating areas. The Fundamental Theorem provides a connection between differentiation and integration and is the most important result in Calculus, since it allows the efficient calculation of integrals.
This week we study a technique of integration called the substitution rule which is based on the chain rule of differentiation. In addition we study optimisation which as mentioned before is probably the main application of differentiation and is how it is most often applied in practice. Our final topic for this week is Newton’s method which is a method (based on differentiation) for approximating the solution to equations.
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