Precalculus, Calculus and Infinitesimals

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See my insights article for those interested in an unconventional approach to doing Precalculus at an accelerated pace and beginning Calculus.

It is different from the usual way that a precalculus is done text in that it covers in the US what is called Algebra 1, Geometry, Algebra 2, and Precalculus in one text. Instead of taking 4 years to cover the material, IMHO, an average student can do it in 2 years; a good student could do it in a year. The text has some 'issues' in that it uses things like 2^x without first defining 2^x, where x is a real number. There is only one way to fix this - with calculus. So when it reaches that point in the text, I introduce calculus and define it properly. This also includes an introduction to set theory and infinitesimals but at a level appropriate for the audience.

Thanks
Bill
 
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1. What is the difference between precalculus and calculus?

Precalculus is a branch of mathematics that builds upon algebra and geometry and prepares students for the study of calculus. It covers topics such as functions, equations, and graphs. Calculus, on the other hand, is a branch of mathematics that deals with the study of change and motion. It involves the calculation of derivatives and integrals to solve real-world problems.

2. What are infinitesimals in calculus?

Infinitesimals are infinitely small numbers that are used in calculus to represent the change in a variable. They are often denoted as dx or dy and are used in the calculation of derivatives and integrals. Infinitesimals are essential in the study of calculus as they allow for the precise measurement of continuously changing quantities.

3. How is calculus used in real life?

Calculus has numerous real-life applications, such as in physics, engineering, economics, and medicine. It is used to model and analyze the motion of objects, calculate the rate of change, optimize functions, and solve complex problems involving continuous change. Some examples of real-life applications of calculus include predicting the trajectory of a projectile, designing bridges and buildings, and determining the optimal dosage of medication.

4. What is the fundamental theorem of calculus?

The fundamental theorem of calculus is a fundamental concept in calculus that connects the two main branches of calculus, differentiation, and integration. It states that the derivative of a function can be calculated by evaluating the corresponding indefinite integral at the upper and lower limits. In other words, it provides a method for finding the area under a curve by evaluating the antiderivative of the function.

5. How can I improve my understanding of precalculus and calculus?

To improve your understanding of precalculus and calculus, it is essential to practice regularly and seek help from a tutor or teacher when needed. You can also use online resources such as videos, practice problems, and interactive tutorials to supplement your learning. Additionally, it is crucial to have a solid foundation in algebra and geometry before studying precalculus and calculus.

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