# Methods in Calculus

## Scheme of work: Year 13 A-Level Further Maths: Core Pure 2: Methods in Calculus

#### Prerequisite Knowledge

1. Differentiation:
• Standard differentiation rules (product, quotient, chain).
• Differentiation of standard functions (polynomials, trigonometric functions, exponential functions, and logarithms).
2. 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.
3. Sequences and Series:
• Convergence and divergence of series, as they relate to improper integrals.
• Familiarity with arithmetic and geometric progressions.
4. Basic Trigonometry:
• Core trigonometric identities, such as Pythagorean identities.
• Basic understanding of the graphs, properties, and values of sine, cosine, and tangent functions.
5. Algebraic Fractions:
• Decomposing fractions into partial fractions, especially those with linear factors.
• Fundamental knowledge about polynomial division and factor theorem.
6. Function Transformations and Graphs:
• The behaviour of functions, including asymptotic behaviour, relates to improper integrals.
• Understanding transformations of basic functions.
7. Logarithms:
• Properties and differentiation of natural logarithms, which is foundational for certain integration techniques.

#### Success Criteria

1. 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.
2. Mean Value of a Function:
• Understand and explain the geometric interpretation of the Mean Value Theorem.
• Apply the Mean Value Theorem to solve problems.
3. 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.
4. 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.
5. 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.
6. 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.

#### Teaching Points

1. 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.
2. 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.
3. 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.
4. 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.
5. 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.
6. 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.

#### Common Misconceptions

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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