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Are You Ready for Calculus?
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Calculus is not as difficult as you may have heard, but if you want to succeed in your study of the subject, then you have to have the proper foundations.
One of the foundations that you will need is a familiarity with certain basic concepts.
The most important precalculus concept is the notion of a functional relationship between two variable quantities. This relationship may take many forms: linear functions, power functions, exponential functions, logarithmic functions, trig functions, polynomial functions, rational functions, etc. Functions from these basic families may be combined, transformed, and inverted to produce still more functional forms. Functions also appear in various representations: formulas, graphs, data sets. You will have to be familiar with the basic families of functions, and all of their representations, in order to succeed in your study of calculus. The concept of a function underlies everything that calculus considers.
You will also need to be able to carry out certain computational tasks with efficiency and accuracy if you are going to succeed in calculus. These include manipulations of functional symbolism, solving algebraic equations involving the functions mentioned above, interpreting numerical values given by formulas, graphs, and tables, using and manipulating data, and knowing how, and when, to use your computer.
Finally, it is helpful to be acquainted with the main ideas of calculus, even before you get there. These are the ideas of rate of change and accumulation. If you have an intuitive understanding of how these ideas appear in discussions of functions, and you have some familiarity with the sorts of applications that lead to these ideas, then you are in good shape. You will define these ideas carefully in calculus, and then practice them repeatedly.
But there is more to being prepared for calculus than memorizing definitions, acquainting yourself with basic concepts, and mastering computational skills. Being prepared for calculus means more than making it through the prerequisite courses, having "seen" functions, or knowing how to solve certain equations. Calculus requires a new kind of mathematical maturity from you.
The strategies that may have worked for you in previous math courses may not have worked as well for you in this course. Precalculus and calculus are more conceptual than any of the mathematics you have studied before. To succeed in calculus you must know more than "how" to do things, you must also understand "why". You must also see how the different things that you do fit into a bigger picture: understanding functions and how they behave. If you have worked through the materials in this course, you should have a good foundation in both the "how" and the "why" of solving precalculus and calculus problems.
Don't hurry to study calculus. Calculus is the lingua franca of mathematics, engineering, and all of the sciences. You want to speak it well, with genuine understanding. You want to carry out calculations involving realistic problems with confidence. You want to savor calculus, and appreciate all of the truly beautiful things that it reveals about the world around you. Prepare yourself to enjoy it.
Don't be afraid to take the next step, however. Calculus is not brand new, it only involves a deeper understanding, and a more efficient use, of the concepts in this course. Want to learn more? Then it's time to move on...
One of the foundations that you will need is a familiarity with certain basic concepts.
The most important precalculus concept is the notion of a functional relationship between two variable quantities. This relationship may take many forms: linear functions, power functions, exponential functions, logarithmic functions, trig functions, polynomial functions, rational functions, etc. Functions from these basic families may be combined, transformed, and inverted to produce still more functional forms. Functions also appear in various representations: formulas, graphs, data sets. You will have to be familiar with the basic families of functions, and all of their representations, in order to succeed in your study of calculus. The concept of a function underlies everything that calculus considers.
You will also need to be able to carry out certain computational tasks with efficiency and accuracy if you are going to succeed in calculus. These include manipulations of functional symbolism, solving algebraic equations involving the functions mentioned above, interpreting numerical values given by formulas, graphs, and tables, using and manipulating data, and knowing how, and when, to use your computer.
Finally, it is helpful to be acquainted with the main ideas of calculus, even before you get there. These are the ideas of rate of change and accumulation. If you have an intuitive understanding of how these ideas appear in discussions of functions, and you have some familiarity with the sorts of applications that lead to these ideas, then you are in good shape. You will define these ideas carefully in calculus, and then practice them repeatedly.
But there is more to being prepared for calculus than memorizing definitions, acquainting yourself with basic concepts, and mastering computational skills. Being prepared for calculus means more than making it through the prerequisite courses, having "seen" functions, or knowing how to solve certain equations. Calculus requires a new kind of mathematical maturity from you.
The strategies that may have worked for you in previous math courses may not have worked as well for you in this course. Precalculus and calculus are more conceptual than any of the mathematics you have studied before. To succeed in calculus you must know more than "how" to do things, you must also understand "why". You must also see how the different things that you do fit into a bigger picture: understanding functions and how they behave. If you have worked through the materials in this course, you should have a good foundation in both the "how" and the "why" of solving precalculus and calculus problems.
Don't hurry to study calculus. Calculus is the lingua franca of mathematics, engineering, and all of the sciences. You want to speak it well, with genuine understanding. You want to carry out calculations involving realistic problems with confidence. You want to savor calculus, and appreciate all of the truly beautiful things that it reveals about the world around you. Prepare yourself to enjoy it.
Don't be afraid to take the next step, however. Calculus is not brand new, it only involves a deeper understanding, and a more efficient use, of the concepts in this course. Want to learn more? Then it's time to move on...
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