Lesson 6 - More about Limits
As the study of calculus and applied mathematics grows deeper, the types of limits we will need to evaluate grows much more complicated. In many cases, the evaluation of a limit is "indeterminate", meaning that the basic techniques we have for understanding limits don't give us enough information to actually evaluate a limit. Luckily, a wide range of more powerful techniques have been developed by mathematicians, including the Squeeze Theorem introduced in this lesson.
Lesson 4: Introduction to The Calculus
Calculus deals with the infinite and the infinitisemal, and finding ways to deal with their non-finite extent in a finite way. At the heart of this adventure into infinity, and the of the great value of The Calculus, is the notion of a limit.
Traditionally, the great value of limits is emphasized in the context of two very graphical interpretations - finding the tangent to and area under a curve that isn't a line. Both of these examples are motivated by important physical correlates, such as finding the speed of a moving object, the rate of change of a variable of interest, such as population of a species, value of a stock, or price of gas or oil.
Traditionally, the great value of limits is emphasized in the context of two very graphical interpretations - finding the tangent to and area under a curve that isn't a line. Both of these examples are motivated by important physical correlates, such as finding the speed of a moving object, the rate of change of a variable of interest, such as population of a species, value of a stock, or price of gas or oil.
Lesson 3: Modeling Data with Functions
This lesson helps to bring students to an understanding of how using theoretical approximations can often help us to understand real world phenomena that are less accessible to pure mathematical calculations. This is done by introducing students to the concept of "regression", where scattered data has a line or curve fit to it that, in some sense, best approximates the data. In this lesson, we explore how many very important phenomena in the physical world can be approximated excellently by the range of mathematical functions we have so far reviewed.
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Lesson 2: Relations and Functions
In Lesson 2, the topics of Domain and Range, and how they apply to the interpretation of functions are reviewed. A number of important graphs of functions are presented, including the the absolute value of x, sinusoidal functions, and more advanced polynomial functions. The purpose of this lesson is to continue the student's review of important topics in functions and analytical geometry that will be utilized in introducing the topics of calculus in subsequent chapters. An important skill for the continued learning of calculus is how to identify a function from its graph, and to be able to graph a function based on its algebraic formula.
The calculation of the Domain and Range is extremely helpful to begin to graph a function, and also to identify a function from its graph. For many students, the calculation of the Domain and Range can become very formulaic, which is often exacerbated by exercises which only ask students to identify the Domain and Range of a potpourri of functions without providing any physical or visual intuition as to why the
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The calculation of the Domain and Range is extremely helpful to begin to graph a function, and also to identify a function from its graph. For many students, the calculation of the Domain and Range can become very formulaic, which is often exacerbated by exercises which only ask students to identify the Domain and Range of a potpourri of functions without providing any physical or visual intuition as to why the
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Lession 1: Equations, Functions and Graphs
To begin the study of calculus, we begin by having the student review some important properties of equations and functions, and how to graph different kinds of functions. A solid understanding of analytic geometry is essential to developing the techniques of differentiation and integration presented in this book.
In this lesson, the idea of a function being a table of numbers, where one column is the input variable (x) and the other comumn is the output variable (y). An important technique students will need to understand throughout their study of calculus is how to evaluate functions by either plugging in values for the input variable, or in more complicated cases like in developing the equations for derivatives and integrals, substituting algebraic expressions in for the input variable of a function.
For instance, one definition of the derivative of a function will have students evaluating functions for the value x + h, i.e. f(x+h), and then finding the value of the evaluated function in the limit as h approaches zero. Making certain that students understand how to do this, particularly the technique of substituting the entire expression in the function's argument for the input variable in the function expression, is essential for them to develop facility with understanding many topics throughout this course.
Although much of the notation that is being introduced in this and subsequent lessons is very formal, it is important to stress that functions are so important because they enable us to effectively model a number of real world phenomena. In the exercises for this chapter, the relationship between the independent variable x, and the example of modeling costs is provided through both linear and nonlinear examples.
In this lesson, the idea of a function being a table of numbers, where one column is the input variable (x) and the other comumn is the output variable (y). An important technique students will need to understand throughout their study of calculus is how to evaluate functions by either plugging in values for the input variable, or in more complicated cases like in developing the equations for derivatives and integrals, substituting algebraic expressions in for the input variable of a function.
For instance, one definition of the derivative of a function will have students evaluating functions for the value x + h, i.e. f(x+h), and then finding the value of the evaluated function in the limit as h approaches zero. Making certain that students understand how to do this, particularly the technique of substituting the entire expression in the function's argument for the input variable in the function expression, is essential for them to develop facility with understanding many topics throughout this course.
Although much of the notation that is being introduced in this and subsequent lessons is very formal, it is important to stress that functions are so important because they enable us to effectively model a number of real world phenomena. In the exercises for this chapter, the relationship between the independent variable x, and the example of modeling costs is provided through both linear and nonlinear examples.
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