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01. Limits and Functions
| Description | Exercise Links |
|---|
| Explores fundamental concepts of functions, including types, domain, range, limits, and continuity. | 01_Ex 1.1 |
| Introduces function composition, where two functions are combined to form a new one. | 02_Ex 1.2 |
| Evaluating limits using fundamental theorems: sum, product, quotient, and power of limits. | 03_Ex 1.3 |
| Focuses on limits and continuity, evaluating left-hand and right-hand limits, and determining the continuity of piecewise functions. | 04_Ex 1.4 |
| Graphing equations, exploring parametric curves, piecewise functions, symmetry, domain, range, and continuity. | 05_Ex 1.5 |
02. Differentiation
| Description | Exercise Links |
|---|
| Differentiation by definition, focusing on finding derivatives using first principles. | 01_Ex 2.1 |
| Application of binomial expansion in simplifying algebraic expressions for differentiation. | 02_Ex 2.2 |
| Applying basic differentiation rules like the Power, Product, and Quotient Rules. | 03_Ex 2.3 |
| Advanced techniques: substitution, chain rule, and implicit differentiation. | 04_Ex 2.4 |
| Differentiating trigonometric and inverse trigonometric functions, emphasizing standard derivatives. | 05_Ex 2.5 |
| Derivatives of exponential, hyperbolic, and logarithmic functions, essential in growth and oscillations. | 06_Ex 2.6 |
| Higher-order derivatives and the application of advanced rules to various functions. | 07_Ex 2.7 |
| Maclaurin series expansion and Taylor series for deriving and proving key functions. | 08_Ex 2.8 |
| Analyzing function behavior: finding intervals of increase/decrease, and determining extreme values. | 09_Ex 2.9 |
| Optimization problems: maximizing or minimizing specific quantities under constraints. | 10_Ex 2.10 |
03. Integration
| Description | Exercise Links |
|---|
| Using differential calculus to approximate changes in a function’s dependent variable. | 01_Ex 3.1 |
| Evaluating indefinite integrals using fundamental techniques such as substitution. | 02_Ex 3.2 |
| Various integration techniques, including substitution, trigonometric identities, and integration by parts. | 03_Ex 3.3 |
| Integration by Parts (IBP): crucial for integrating products of functions. | 04_Ex 3.4 |
| Partial fraction decomposition for integrating rational functions. | 05_Ex 3.5 |
| Evaluating definite integrals using substitution, algebraic manipulation, and standard formulas. | 06_Ex 3.6 |
| Finding areas under curves using definite integrals, a fundamental concept in calculus. | 07_Ex 3.7 |
| Solving differential equations using methods like separation of variables and integration. | 08_Ex 3.8 |
04. Analytical Geometry
| Description | Exercise Links |
|---|
| Analytic geometry: working with distance formula, midpoint formula, and slope calculations. | 01_Ex 4.1 |
| Coordinate transformations: specifically rotation and translation of points in the Cartesian plane. | 02_Ex 4.2 |
| Slopes and angles of inclination in geometry: calculating and interpreting relationships between lines. | 03_Ex 4.3 |
| Equations of lines, intersections, and the concept of concurrent lines. | 04_Ex 4.4 |
| Homogeneous quadratic equations, representing a pair of straight lines passing through the origin. | 05_Ex 4.5 |
05. Linear Inequalities
| Description | Exercise Links |
|---|
| Graphing linear inequalities: understanding solution sets and applying these skills to solve systems of inequalities. | 01_Ex 5.1 |
| Graphing systems of linear inequalities: finding the feasible region where all constraints are satisfied. | 02_Ex 5.2 |
| Maximizing a linear function under specified constraints: identifying feasible regions and evaluating the objective function at corner points. | 03_Ex 5.3 |
06. Conic Sections
| Description | Exercise Links |
|---|
| Deriving the equation of a circle: calculating the center, radius, and using the midpoint formula. | 01_Ex 6.1 |
| Tangents and normals to circles: calculating tangent lengths, chords, and determining point positions relative to a circle. | 02_Ex 6.2 |
| Normal lines, tangents, and perpendiculars: understanding their relationships and key geometric properties. | 03_Ex 6.3 |
| Properties of parabolas: finding the focus, vertex, and writing equations, including tangent properties. | 04_Ex 6.4 |
| Geometric properties of ellipses, including standard equations, foci, eccentricity, and axes. | 05_Ex 6.5 |
| Geometry of hyperbolas: covering equations, vertices, foci, and relationships between axes and eccentricity. | 06_Ex 6.6 |
| Computing tangent and normal equations for conics: leveraging differentiation to calculate slopes. | 07_Ex 6.7 |
| Shifting the origin and rotating axes: simplifying conic section equations by removing linear and cross terms. | 08_Ex 6.8 |
07. Vectors
| Description | Exercise Links |
|---|
| Fundamental vector concepts: unit vectors, position vectors, and operations on vectors. | 01_Ex 7.1 |
| Key vector analysis concepts: calculating magnitudes, unit vectors, direction cosines, and practical applications. | 02_Ex 7.2 |
| Dot product and its applications: calculating the cosine of angles, projections, and determining direction cosines. | 03_Ex 7.3 |
| Cross product: understanding how to compute it using the determinant formula and the right-hand rule. | 04_Ex 7.4 |
| Scalar triple product: its use in calculating volumes, and determining co-planarity of vectors. | 05_Ex 7.5 |
