Your Path to Calculus Mastery: Full Program Details

Welcome to the ultimate resource for mastering calculus. Below, you'll find a detailed breakdown of every module and topic covered in our comprehensive program, designed to take you from foundational concepts to advanced applications with confidence.

Reinforce your learning! Each video is paired with a carefully designed worksheet complete with answers, so you can practice and solidify the concepts you've just learned.

Go to Module 1: Geometry and Trigonometry
• Covers: This foundational module establishes a strong understanding of essential geometric principles, including Pythagoras' Theorem, calculating distances, and estimating volumes and surface areas of solids. It then delves into both right-angled and non-right-angled trigonometry, covering trigonometric ratios, sine and cosine rules, calculating areas of triangles, and real-world applications like angles of elevation and bearings.
• Outcomes:  Apply Pythagoras' Theorem to solve geometric problems.  Calculate distances, volumes, and surface areas of various solids.  Utilize trigonometric ratios (SOH CAH TOA), the Sine Rule, and the Cosine Rule to solve problems involving both right-angled and non-right-angled triangles.  Determine angles of elevation, angles between lines and planes, and find bearings in practical scenarios.

Go to Module 2: Algebra Fundamentals
• Covers: This foundational module provides a comprehensive introduction to algebraic concepts and techniques. It covers writing, manipulating, expanding, factorizing, and simplifying algebraic expressions, including operations with algebraic fractions. The module then progresses to solving various types of equations using methods like balancing, collecting like terms, and handling brackets and fractions. It concludes with an in-depth exploration of simultaneous equations, utilizing both substitution and elimination methods.  
• Outcomes:  Confidently write, manipulate, expand, factorize, and simplify algebraic expressions. Perform operations (multiplication, division, addition, subtraction) with algebraic fractions.  Solve linear equations, equations with brackets, and equations involving fractions using systematic methods.  Rearrange equations to change the subject.  Solve inequations and basic quadratic equations.  Master techniques for solving simultaneous equations using both substitution and elimination.

Go to Module 3: Quadratic Equations
• Covers: This module provides a thorough exploration of quadratic expressions and equations. It begins with techniques for manipulating quadratics, including expanding brackets, various methods of factorizing (simple, complex, and special quadratics), and completing the square. The module then focuses on solving quadratic equations using factorization, completing the square, and the quadratic formula. Finally, it delves into the applications of quadratics, examining the discriminant, the nature of roots, solving quadratic inequations, and applying quadratics to geometry and kinematics problems.
•  Outcomes:  Master the expansion and factorization of various types of quadratic expressions.  Fluently apply the method of completing the square.  Solve quadratic equations efficiently using factorization, completing the square, and the quadratic formula.  Understand and use the discriminant to determine the nature of quadratic roots.  Solve quadratic inequations.  Apply quadratic concepts to solve problems in geometry and kinematics.

Go to Module 4: Simultaneous Equations and Matrices
• Covers: This module provides an in-depth exploration of simultaneous equations, progressing from two-dimensional systems solved graphically, by elimination, and by substitution, to three-dimensional systems utilizing both calculator methods and advanced techniques like row reduction. It distinguishes between consistent and inconsistent systems. The module then introduces matrices, covering their formal definition, calculating determinants, finding inverse matrices using row operations, minors, cofactors, and adjoints, and applying determinants in geometric contexts.
• Outcomes:  Solve 2D systems of simultaneous equations using graphical, elimination, and substitution methods.  Solve 3D systems of simultaneous equations, including using a graphics calculator, elimination, and row reduction.  Identify and interpret consistent and inconsistent systems of equations.  Understand the formal introduction of matrices, including calculating determinants and inverse matrices.  Apply determinants to solve various mathematical problems, including geometric applications.

Go to Module 5: Sequences and Series
• Covers: This comprehensive module delves into the fundamental concepts of sequences and series. It begins with arithmetic sequences and series, including recursive definitions, sigma notation, and the arithmetic sum formula. It then progresses to geometric sequences and series, covering their classification, sum formulas for n terms, and the concept of infinite sum series. The module concludes with an in-depth exploration of infinite series, discussing methods like the differences method, various convergence tests (direct, limit comparison, integral), conditional and absolute convergence, convergence intervals of power series, and the approximation of infinite series, culminating in Taylor and Maclaurin series.
• Outcomes:  Identify, define, and calculate terms and sums of arithmetic sequences and series, including using sigma notation.  Recognize, classify, and apply formulas for geometric sequences and series, including infinite sums.  Determine the convergence or divergence of various infinite series using multiple test methods.  Understand and apply concepts of conditional and absolute convergence.  Work with convergence intervals of power series.  Approximate infinite series and construct Taylor and Maclaurin series.

Go to Module 6: Exponentials and Logarithmic Equations
• Covers: This module provides a comprehensive exploration of exponential and logarithmic functions and equations. It begins with exponentials, covering the manipulation of negative and fractional exponents, the application of exponential laws, and an introduction to exponential equations, including those with base 'e'. The module then delves into solving exponential equations by introducing and applying logarithms (both decimal and natural), combining logarithms, and linearizing exponential equations. Finally, it focuses on logarithmic equations, teaching how to solve them, reduce complex forms, and change bases, equipping students to tackle challenging problems.
• Outcomes:  Confidently manipulate and rewrite exponential expressions using various exponent laws.  Solve a wide range of exponential equations, including those involving the natural base 'e'.  Understand the relationship between exponentials and logarithms.  Apply decimal and natural logarithms to solve complex exponential and logarithmic equations.  Linearize exponential equations for easier analysis.  Change the base of logarithmic expressions and solve challenging logarithmic problems

Go to Module 7: Algebra and Complex Equations
• Covers: This module provides an in-depth exploration of solving various types of equations, distinguishing between real and complex roots. It begins by covering techniques for manipulating quadratics and polynomials, introducing the Factor and Remainder Theorems, and solving quadratic, surd, and power equations with real roots, along with examining the properties of polynomial roots. The module then introduces complex numbers, explaining powers of 'i', manipulation of complex numbers, modulus and argument, complex conjugates, and converting between rectangular and polar forms, including multiplication, division, and using De Moivre's Theorem for powers. Finally, it focuses on solving equations with complex roots, including quadratics, polynomials, and power equations, and introduces the concept of the locus of a point.
• Outcomes:  Manipulate and solve quadratic, polynomial, surd, and power equations to find real roots.  Apply the Factor and Remainder Theorems effectively.  Understand and perform operations with complex numbers in both rectangular and polar forms.  Calculate modulus, argument, and complex conjugates, and apply De Moivre's Theorem.  Solve quadratic, polynomial, and power equations that yield complex roots.  Comprehend and determine the locus of a point in the complex plane.

Go to Module 8: Trigonometric Equations and Functions
• Covers: This module provides a comprehensive exploration of trigonometry, starting with circular measurements, including the introduction of radians for sectors, calculating perimeters and areas of segments, and utilizing the unit circle. It then delves into trigonometric graphs and functions, focusing on modeling real-world scenarios, transforming graphs, finding parameters from given information, and understanding the properties of trigonometric functions. The module concludes with advanced techniques for solving trigonometric equations and identities, covering various methods for solving equations, introducing and applying trigonometric identities (compound angle, double angle), solving quadratic trigonometric equations, expressing products as sums, and an introduction to inverse trigonometric functions.
• Outcomes:  Work confidently with radians, and calculate perimeters and areas of sectors and segments.  Utilize the unit circle to solve trigonometric problems.  Model and interpret real-world phenomena using trigonometric graphs and functions.  Solve a wide range of trigonometric equations, including those requiring quadratic methods.  Apply fundamental, compound angle, and double angle trigonometric identities to simplify expressions and prove relationships.  Understand the basics of inverse trigonometric functions.
 
Go to Module 9: Binomial and Proof
• Covers: This module provides a comprehensive exploration of binomial expansion, arrangements, selections, and various proof techniques. It begins with binomial expansion, covering its introduction, using graphics calculators for coefficients, expanding with natural and integer powers, and finding specific terms. The module then delves into arrangements and selections, explaining factorials, permutations, and combinations, and linking these concepts to binomial coefficients. Finally, it provides an in-depth look at proof methods, including deductive proofs, disproving by counterexample, proof by contradiction, and extensive coverage of proof by induction (for factorials, divisibility, derivatives, and sums), culminating in the proof of the Binomial Theorem.
• Outcomes: Perform binomial expansions for expressions with both natural and integer powers. Efficiently find specific terms within a binomial expansion. Understand and apply factorials, permutations, and combinations to solve problems involving arrangements and selections. Formulate and execute various mathematical proofs, including deductive, contradiction, and counterexample methods. Master the technique of proof by induction for a range of mathematical statements, including those involving factorials, divisibility, derivatives, and sums. Understand and prove the Binomial Theorem.

Go to Module 10: Graphs and Functions
• Covers: This module provides a comprehensive guide to understanding, sketching, transforming, and manipulating various types of graphs and functions. It starts with sketching graphs, covering straight lines, parabolas (including finding vertices), polynomial graphs, identifying asymptotes in exponential graphs, and visualizing logarithmic graphs. It then moves into transforming graphs, detailing how to translate, stretch, and reflect general graphs, including specific applications to quadratic and absolute value functions. Finally, the module focuses on manipulating functions, teaching how to identify domain and range, compose functions, find inverse functions, and sketch rational and reciprocal functions.  
• Outcomes:  Confidently sketch and interpret graphs of straight lines, parabolas, polynomials, exponentials, and logarithms.  Identify key features of graphs such as vertices and asymptotes.  Apply various transformations (translation, stretching, reflection) to general and specific functions.  Determine the domain and range of given functions.  Compose functions and find their inverse functions.  Accurately sketch rational and reciprocal functions.

Go to Module 11: Differential Calculus 
• Covers: This comprehensive module provides an in-depth exploration of differential calculus. It starts with core differentiation techniques, covering power, composite, trigonometric, exponential, logarithmic, reciprocal, and inverse functions, along with the product, quotient, and chain rules. The module then delves into applications of derivatives, including finding stationary points, tangents and normals, rates of change (including related rates), optimization problems, implicit differentiation, and kinematics. Finally, it introduces advanced concepts such as continuity and differentiability, differentiation by first principles, limits and L'Hôpital's Rule, the Mean Value Theorem, gradient functions, and analyzing inflection points and concavity to understand graph features.  
• Outcomes:  Master differentiation techniques for a wide range of functions, including composite, trigonometric, exponential, logarithmic, reciprocal, and inverse functions.  Confidently apply the product, quotient, and chain rules.  Solve problems involving stationary points, tangents, normals, and various rates of change.  Utilize differentiation for optimization problems and in kinematics.  Apply implicit differentiation.  Understand foundational concepts like continuity, differentiability, and limits (including L'Hôpital's Rule).  Analyze graph features using derivatives, including inflection points and concavity.

Go to Module 12: Integral Calculus
• Covers: This comprehensive module provides a deep dive into integral calculus and its applications. It starts with antidifferentiation, covering the introduction to integration, integrating polynomials, special functions like 1/x, and advanced techniques such as integration by substitution and integration by parts. It also includes the integration of trigonometric functions, exponentials, and integrals leading to inverse functions. The module then progresses to definite integrals, exploring their introduction, calculating areas between curves, areas between curves and the y-axis, volumes of revolution, numerical integration, and applications in kinematics. Finally, it delves into differential equations, covering their introduction, finding general and particular solutions, separating variables, using substitution for homogeneous equations, the integrating factor method, and Euler's Method.  
• Outcomes:  Master various integration techniques, including integration by substitution and integration by parts.  Confidently integrate polynomials, trigonometric functions, exponentials, and functions leading to inverse trigonometric forms.  Understand and apply definite integrals to calculate areas between curves, volumes of revolution, and solve kinematics problems.  Utilize numerical integration techniques.  Solve various types of differential equations, including those requiring separation of variables, substitution for homogeneous equations, the integrating factor method, and Euler's Method.