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Calculus Quest: Exploring the World of Change

Categories: Math & Physics
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About Course

Get introduced to the basics of calculus, understanding how things change and move. Ever wondered how things change and why? Join us on a quest to explore motion, growth, and the magic of calculus. Let’s unravel the stories behind the curves and slopes!

What Will You Learn?

  • Introduction to the fundamental concepts of calculus.
  • Understanding of derivatives and how they measure change.
  • Exploration of integrals and the concept of accumulation.
  • Real-world applications of calculus in motion, growth, and other phenomena.
  • Techniques to solve calculus problems using mathematical tools and logic.
  • Insights into the history and development of calculus.
  • Hands-on experience with plotting curves and understanding their significance.
  • Deep dive into the relationship between geometry and calculus.

Course Content

Understand Basic Functions
Before diving into calculus, ensure you have a strong grasp of basic algebra, including linear and quadratic functions. You should also be comfortable with algebraic operations like addition, subtraction, multiplication, and division of polynomials.

  • Why Study Calculus? Overcoming the Fear
  • The Paradox of the derivative
  • Derivative formulas through geometry

Learn About Rational Expressions
Start by understanding what rational expressions are. A rational expression is a fraction where both the numerator and denominator are polynomial expressions. For example, (3x^2 - 2x + 1) / (2x - 1) is a rational expression.

Simplify Rational Expressions
Learn how to simplify rational expressions by factoring both the numerator and denominator and canceling common factors. This is crucial for working with rational functions.

Identify Vertical Asymptotes
Understand how to find vertical asymptotes by setting the denominator equal to zero and solving for the values of x that make the denominator zero. Vertical asymptotes represent points where the function approaches infinity or negative infinity.

Determine Horizontal Asymptotes
Learn how to determine horizontal asymptotes by comparing the degrees of the highest power terms in the numerator and denominator. Depending on the degree relationships, you can determine whether there are horizontal asymptotes and their values.

Study Limits
Calculus involves the concept of limits. Understand how to find the limit of a rational expression as it approaches a particular value. Limits are essential for understanding continuity and the behavior of functions.

Differentiate Rational Functions
Learn how to find the derivative of a rational function. The quotient rule is commonly used for this purpose. It allows you to find the derivative of the numerator and denominator separately and then simplify.

Use the Chain Rule
If your rational expression contains composite functions, practice applying the chain rule when taking derivatives.

Solve for Critical Points
To analyze the behavior of a rational function, solve for critical points by finding where the derivative is equal to zero or undefined. Critical points can help you identify local maxima, minima, and inflection points.

Integration of Rational Functions
Learn how to integrate rational functions. Depending on the complexity of the function, you may need to use techniques like partial fraction decomposition.

Practice, Practice, Practice
To truly grasp calculus and rational expressions, practice solving various problems, including finding limits, derivatives, integrals, and graphing rational functions.

Explore Real-Life Applications
Understand how calculus and rational expressions are applied in real-world scenarios, such as physics, engineering, economics, and biology. This can provide motivation and context for your learning.

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