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**Differentiation:**- Standard differentiation rules (product, quotient, chain).
- Differentiation of standard functions (polynomials, trigonometric functions, exponential functions, and logarithms).

**Integration:**- Basic integration techniques, such as integration by substitution and by parts.
- Integration of standard functions (as mentioned under differentiation).
- Understanding of definite and indefinite integrals.

**Sequences and Series:**- Convergence and divergence of series, as they relate to improper integrals.
- Familiarity with arithmetic and geometric progressions.

**Basic Trigonometry:**- Core trigonometric identities, such as Pythagorean identities.
- Basic understanding of the graphs, properties, and values of sine, cosine, and tangent functions.

**Algebraic Fractions:**- Decomposing fractions into partial fractions, especially those with linear factors.
- Fundamental knowledge about polynomial division and factor theorem.

**Function Transformations and Graphs:**- The behaviour of functions, including asymptotic behaviour, relates to improper integrals.
- Understanding transformations of basic functions.

**Logarithms:**- Properties and differentiation of natural logarithms, which is foundational for certain integration techniques.

**Improper Integrals:**- Demonstrate an understanding of what makes an integral “improper”.
- Successfully evaluate integrals with infinite limits and integrals of unbounded functions.
- Determine the convergence or divergence of an improper integral.

**Mean Value of a Function:**- Understand and explain the geometric interpretation of the Mean Value Theorem.
- Apply the Mean Value Theorem to solve problems.

**Differentiating Inverse Trigonometric Functions:**- Recall and apply the derivatives of arcsin, arccos, and arctan.
- Use the chain rule in combination with the derivatives of inverse trigonometric functions.

**Integrating Inverse Trigonometric Functions:**- Recall and apply the antiderivatives of arcsin, arccos, and arctan.
- Use the reverse chain rule, substitution and integration techniques in combination with inverse trigonometric functions.
- Evaluate definite integrals involving inverse trigonometric functions.

**Integrating Partial Fractions with Quadratic Factors:**- Decompose rational expressions with quadratic factors into appropriate partial fractions.
- Integrate expressions derived from partial fraction decomposition.
- Recognize when a quadratic factor cannot be further factorized and apply the correct decomposition.

**Integration with Partial Fractions and Inverse Trigonometric Functions:**- Decompose rational functions into partial fractions suitable for integration using inverse trigonometric functions.
- Combine knowledge of partial fraction decomposition and inverse trigonometric antiderivatives to evaluate integrals.
- Solve problems that require multiple integration techniques.

**Improper Integrals:**- Definition and classification of improper integrals: those with infinite intervals and those with unbounded functions.
- Techniques for evaluating improper integrals, including substitution and comparison tests.
- Convergence and divergence of improper integrals.

**Mean Value of a Function:**- Introduction and statement of the Mean Value Theorem.
- Geometric interpretation and visualization of the theorem using graphs.
- Applications and implications of the Mean Value Theorem in mathematical analysis and real-world contexts.

**Differentiating Inverse Trigonometric Functions:**- Introduction to inverse trigonometric functions: arcsin, arccos, and arctan.
- Derivatives of these functions with proofs.
- Practical applications and problems involving differentiation of inverse trigonometric functions.

**Integrating Inverse Trigonometric Functions:**- Antiderivatives of arcsin, arccos, and arctan with derivations.
- Integration techniques involving substitution and parts for these functions.
- Real-world applications requiring the integration of inverse trigonometric functions.

**Integrating Partial Fractions with Quadratic Factors:**- Review of partial fraction decomposition with emphasis on quadratic factors.
- Techniques for breaking down rational expressions with irreducible quadratic factors.
- Integration of resulting partial fractions.

**Integration with Partial Fractions and Inverse Trigonometric Functions:**- Decomposition of rational functions that result in integrands involving inverse trigonometric functions.
- Combined integration techniques: using partial fractions first, then integrating resulting terms that involve inverse trigonometric functions.
- Complex problems that test proficiency in multiple integration techniques.

**Improper Integrals:**

**Finite Appearance:**Some students think that if an integral looks finite (i.e., doesn’t have an infinity symbol), it cannot be improper. They might overlook functions that become unbounded.**Always Divergent:**A misconception is that all improper integrals are divergent or don’t have a finite area.

**Mean Value of a Function:**

**Existence Everywhere:**Students might believe the Mean Value Theorem applies to all functions, not realizing the conditions that must be met (e.g., the function must be continuous and differentiable on the interval).**Single Point Guarantee:**The notion that there’s always only one “c” in the interval [a, b] where f'(c) equals the average rate of change, when in fact there can be more than one such point.

**Differentiating Inverse Trigonometric Functions:**

**General Trig Confusion:**Mixing up the derivatives of regular trigonometric functions with their inverse counterparts.**Sign Mistakes:**Often students misremember the signs for the derivatives, particularly with arcsin and arccos.

**Integrating Inverse Trigonometric Functions:**

**Inverse Confusion:**Students sometimes try to integrate using the rules for regular trigonometric functions instead of the inverse functions.**Integration Constants:**Forgetting to add the integration constant after integrating.

**Integrating Partial Fractions with Quadratic Factors:**

**Full Factorization:**Assuming that all quadratic denominators can be factorized into linear factors.**Wrong Decomposition:**Misunderstanding the correct form for partial fractions when dealing with irreducible quadratics, leading to incorrect integrands.

**Integration with Partial Fractions and Inverse Trigonometric Functions:**

**Order of Operations:**Attempting to integrate before completing the partial fraction decomposition.**Mixing Techniques:**Confusing the integration techniques needed for inverse trigonometric functions with those for other function types.

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