Chapter 5: Further calculus Notes

Sections

1. 5A Parametric differentiation Mastered

   Curves defined parametrically, finding dy/dx as dy/dt divided by dx/dt, gradients and tangents at a point given by a parameter, and checking with the Cartesian equation.

   Open notes
2. 5B Inverse functions and their derivatives Mastered

   One-to-one functions and inverse functions, reflection in y = x, swapping domain and range, restricting a domain to create an inverse, and the derivative of an inverse function using dy/dx = 1 ÷ dx/dy.

   Open notes
3. 5C Differentiating inverse trigonometric functions Mastered

   The derivatives of inverse sine, inverse cosine and inverse tangent and their proofs, the standard forms with x/a, using the chain and product rules, and tangents to inverse trigonometric graphs.

   Open notes
4. 5D Integrals giving inverse trigonometric functions Mastered

   The standard integrals that give inverse sine and inverse tangent, evaluating definite integrals exactly in terms of π, adjusting coefficients such as 1/(4 + 9x²), and areas under these curves.

   Open notes
5. 5E The reverse chain rule Mastered

   Integrating functions of the form f'(x) times g(f(x)) by recognising a derivative, including powers, exponentials, logarithms, trigonometric functions and inverse trigonometric functions of a function.

   Open notes
6. 5F Integration by substitution Mastered

   Integration by substitution for indefinite and definite integrals, finding du, changing the limits, substitutions of the form u = x − a that need x written in terms of u, and substituting x as a function of u.

   Open notes
7. 5G Integrating sin² x and cos² x Mastered

   Using the double angle formulas to integrate sin²x and cos²x, integrals such as sin²(ax), the area of a circle by the substitution x = sin θ, and a summary of the standard integrals.

   Open notes