Precalculus 101

Unlock the Power of Mathematical Foundations

Precalculus 101

Unlock the Power of Mathematical Foundations

Unlock your potential and explore the entire landscape of opportunity with "Precalculus 101: Your Gateway to Advanced Mathematics." Imagine confidently navigating the perplexing world of calculus, empowered by a robust foundation. This course is your transformative journey from basic algebra to the elegance of calculus; it's not just educational--it's life-changing. Learn how to decode complexities easily and gain the skills that will unlock advanced problem-solving across diverse fields. With engaging, accessible content, master the building blocks of future success and break through any limitations, fortified by clarity and confidence. Your calculus journey begins here--let us guide you with precision, ensuring you stride fearlessly into new mathematical realms. Enroll now and redefine what's possible for your future.

Self Paced • Online Class • 35 Hours

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OVERVIEW

CURRICULUM

Precalculus 101

Name: Precalculus 101
Price: 95.00 USD

CRN: 9770456

Lesson 1. Real and Complex Numbers: A Pre-Calculus Introduction

Complex numbers extend the real number system by incorporating imaginary numbers, represented by the symbol i, where i squared equals -1. The lesson explores their representation in the form a + bi and describes basic arithmetic rules and properties like the complex conjugate.

Lesson 2. Exploring the Foundations of Functions: A Pre-Calculus Journey

The lesson highlights the critical role of functions in mathematics by distinguishing unique output-producing relations and illustrates the plotting of these functions on coordinate axes to track continuity and changes. A practical exercise on graph plotting solidifies these theoretical underpinnings with hands-on problem-solving.

Lesson 3. Understanding Domain and Range in Functions: Key Insights and Applications

The lesson delves into domain and range, illustrating limitations within real-valued functions and utilizing different tests to identify one-to-one functions. Students learn to construct composite functions and compute inverses, fundamental for understanding function interactions.

Lesson 4. Understanding Mathematical Relationships: Equations and Inequalities

This lesson explores how equations and inequalities relate algebraic expressions, with solutions verified either as distinct points or continuous intervals. Mastery involves both algebraic manipulation and the graphical portrayal of solution sets.

Lesson 5. The Intricacies of Polynomials and Rational Functions

Polynomials are fundamental in calculus, representing expressions with constant coefficients and variable powers, whereas rational functions focus on polynomial-based fractions acknowledging domain limitations. The domain, roots, and intercepts of these functions help in graphing and solving them, with tools like the quadratic formula applied for quadratic functions.

Lesson 6. Roots and Asymptotes: Uncovering Mathematical Patterns

Synthetic division offers a streamlined method to determine the roots of higher-degree polynomials, shedding light on their underlying structure and factors. Rational functions' asymptotic behavior, whether approaching infinity or shifting diagonally, is critical to understanding their limits and domain.

Lesson 7. Understanding Logarithmic Functions

Exponential and logarithmic functions are the lesson's protagonists, with their rules showcased through applications in continuous compounding and decay modeling. The session underscores the functionality of the constant e and natural logarithms as tools for solving complex equations in these contexts.

Lesson 8. Sine, Cosine, and Tangent: A Pre-Calculus Journey

Trigonometric functions stem from right triangle geometry, assisting in determining unknown measurements in triangles via sine, cosine, and tangent. Calculators serve as valuable tools for generating function values required in both theoretical and practical problem-solving.

Lesson 9. Angles in Radians: A New Perspective

By examining the unit circle, this lesson interrelates trigonometry with circular motion, transitioning from traditional angle measurements in degrees to radians. Students are equipped to tackle complex problems involving trigonometric functions and circular geometry.

Lesson 10. Mastering Trig Identities and Inverse Functions

A focus on graphical analysis and the properties of trigonometric functions emphasizes their symmetries and inverse relationships, illustrating the impact on their algebraic and geometric applications. Key exercises provided train students in identifying the correct function manipulations and solutions specific to the characteristics of triangles.

Lesson 11. Mastering Vectors in Pre-Calculus

While scalars embody pure magnitude, vectors integrate direction, offering a richer toolkit for problem-solving. This lesson guides through representing vectors in a coordinate system, adding, and subtracting them, with practical exercises to reinforce learning.

Lesson 12. Vectors Unleashed: Exploring Unit Vectors and Scalar Products

Lesson Summary 4: This lesson extends basic vector theory by introducing unit vectors in algebraic form and discusses the influence of scalar multiplication on vectors. Learners also explore the scalar product computation as a means to determine vector magnitude, enhancing their mathematical toolkit.

Lesson 13. Understanding Polar Coordinates through Trigonometry

Polar coordinates offer an alternative to the traditional Euclidean system, simplifying various mathematical problems by expressing a point's location as a distance and an angle from the origin. This lesson covers the relationship between polar and rectangular coordinates, conversion techniques, and the benefits of using polar coordinates in certain algebraic expressions.

Lesson 14. Circles and Parabolas: A First Look

Conic sections are derived from slicing a cone at various angles, resulting in shapes like circles and parabolas, which are explored in this lesson. The algebraic and graphical properties of these shapes demonstrate their underlying mathematical principles while highlighting their unique characteristics.

Lesson 15. Exploring the Intricacies of Ellipses and Hyperbolas

Ellipses, described as flattened circles, accommodate the Sun in their focus, portraying celestial orbits, while hyperbolas, akin to parabolas, demonstrate more open loops. Conic sections dictate planetary paths, emphasizing varying speeds and distances that result in hyperbolic or elliptical motions.

Lesson 16. Limits: The Gateway to Calculus

Limits enable us to predict how functions behave near specific points, playing a critical role in calculus concepts like differentiation and integration. By examining both one-sided and two-sided limits, we delve into the nuances of function behavior and their continuity.

Lesson 17. Diving into Sequences and Series

Exploring sequences and series, the lesson offers a holistic overview of these comprehensive topics, detailing how sequences are ordered number sets, while series unite sequence terms into sums. Essential concepts like convergence and divergence provide a sneak peek into the world of limits, paving the way for calculus applications.

Lesson 18. Derivatives Simplified

Lesson Summary 1: This pre-calculus lesson introduces the concept of derivatives, connecting it to algebra and limits by exploring how they quantify the rate of change in a function. Through examining the slope of both linear and nonlinear functions, the lesson demonstrates how differentiation offers insights into varying rates of change.

Lesson 19. Integrals and Antiderivatives: A Prelude

This lesson delves into integrals, complementing derivatives, and uses rectangles through Riemann sums to approximate curve-under areas. The resultant calculated areas reflect a fundamental calculus concept foundational for physics and advanced math studies.

In This Course

35 Hours average completion time
3.5 CEUs
19 Lessons
26 Exams & Assignments
55 Discussions
20 Videos
40 Reference Files
Mobile Friendly
Last Updated April 2024

Description

For many, calculus looms as a daunting peak, a challenging summit seemingly out of reach. Such apprehensions can inadvertently curtail the vast array of opportunities our modern, technology-centric world offers.

Enter "Precalculus 101," your bridge from foundational algebra to the captivating realm of calculus. Designed meticulously for learners, this course illuminates the path, breaking down complex concepts into digestible, relatable pieces, thereby transitioning you smoothly from algebraic foundations to the doors of calculus.

Course Takeaways:

Reconnecting with the Basics: Dive deep into the foundational realms of real and complex numbers, cementing the core upon which advanced topics are built.
The Symphony of Functions: Navigate through the nuanced world of functions -- from the rhythmic patterns of polynomials to the exponential crescendos and logarithmic decrescendos.
Trigonometry Demystified: Rediscover trigonometry through the lens of simple geometries like right triangles and circles, making it relatable and intuitive.
Venturing into Pre-Calculus: Engage with foundational pre-calculus topics like vectors, polar coordinates, and the elegance of conic sections, setting the tone for advanced calculus principles.
The Prelude to Calculus: Delve into the crux of calculus preparation by unraveling the mysteries of limits, sequences, and series, offering a sneak peek into what lies ahead.
Unleashing Calculus: By utilizing algebraic fundamentals, embark on an enlightening journey as you're introduced to the cornerstone of calculus: differentiation and integration. These tools, once mastered, open the gates to a myriad of problem-solving techniques across diverse fields.

On completion of "Precalculus 101," students won't just be equipped with knowledge; they'll be imbued with confidence. A newfound understanding will empower learners to delve deeper into calculus, unearthing its potential to decode challenges across mathematics, physics, and beyond.

Join us on this transformative journey, and let "Precalculus 101" be your compass, guiding you from familiar shores to uncharted territories, all while making the journey exhilarating and enlightening.

Skills You'll Develop

Interpret limits and continuity intuitively
Solve polynomial and rational equations effectively
Visualize vectors and their applications
Apply logarithmic rules in practical problems
Transition smoothly to calculus concepts
Evaluate sequences and series systematically
Master conic sections geometrically
Analyze trigonometric identities seamlessly
Graph complex functions confidently

Skills You'll Develop

Interpret limits and continuity intuitively
Solve polynomial and rational equations effectively
Visualize vectors and their applications
Apply logarithmic rules in practical problems
Transition smoothly to calculus concepts
Evaluate sequences and series systematically
Master conic sections geometrically
Analyze trigonometric identities seamlessly
Graph complex functions confidently

More About This Course

Reconnecting with the Basics: Strengthen your core mathematical knowledge.
Understanding Polar Coordinates: Transition smoothly between polar and rectangular systems.
The Prelude to Calculus: Grasp limits, sequences, and series introduction.
Unleashing Calculus: Get introduced to differentiation and integration essentials.
Vectors Unleashed: Explore vectors, unit vectors, and scalar products.
Graph Plotting Skills: Develop and refine skills in plotting various functions.
Complex Numbers Simplified: Explore arithmetic with complex numbers, enhance problem-solving.
Venturing into Pre-Calculus: Dive into vectors and conic sections, simplify problems.
Symphony of Functions: Master polynomials, exponential, and logarithmic functions.
Trigonometry Demystified: Understand trigonometric principles and solve practical problems.
Trig Identities Mastery: Comprehend and use trigonometric identities effectively.

What You'll Achieve

Identify and illustrate real numbers and complex numbers on the number line and describe their basic properties and differences using examples.
Use interval notation to accurately express a given set of real numbers and apply the concept of absolute value to evaluate numerical expressions.
Define functions as algebraic relations where each independent variable corresponds to one dependent variable
Plot coordinate graphs of simple functions given a set range for the independent variable
Determine the domain and range of given functions using examples and graphical analysis.
Identify and construct composite functions by correctly substituting expressions according to the function composition rules.
Apply the horizontal line test to determine if functions are one-to-one, and calculate their inverses if applicable.
Demonstrate understanding of function characteristics, including domain, range, and inverses, through solving exercises and practice problems.
Define and solve equations to accurately determine the solution sets, demonstrating understanding of equations' characteristics and graphical interpretations.
Identify and solve inequalities, effectively communicating the range of their solution sets through algebraic manipulation and graphical representation.
Identify the degree and number of roots for a given polynomial function.
Determine the domain and range of a simple rational function.
Apply synthetic division to factor polynomials and verify results using roots.

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