Mathematics - Calculus and Geometry
Mathematics - Calculus and Geometry combines differentiation, integration, tangents and normals, areas, coordinate geometry, and basic vectors. Expect a mix of exact algebra and interpretation of graphs.
Core rules
- Power rule: d/dx x^n = nx^{n−1}; chain rule for composite functions.
- Gradient of tangent = dy/dx at the point; normal gradient = −1/m.
- Definite integral gives signed area; take absolute values or split intervals for total area.
- ∫ x^n dx = x^{n+1}/(n+1) + C for n ≠ −1.
- Circle: (x − h)² + (y − k)² = r²; distance formula and midpoint are standard tools.
Worked example
Find the equation of the tangent to y = x³ − 3x at x = 2. First find the point: y = 8 − 6 = 2, so the point is (2, 2). Differentiate: dy/dx = 3x² − 3. At x = 2, gradient m = 12 − 3 = 9. Use point–slope form: y − 2 = 9(x − 2), so y = 9x − 16. The normal at the same point would have gradient −1/9, not 9. Always evaluate the derivative at the given x before forming the line.
Traps
- Using the derivative as the y-coordinate rather than the gradient.
- Forgetting the constant of integration in indefinite integrals.
- Finding a normal when the question asks for a tangent.
- Ignoring regions below the x-axis when asked for total area.
- Applying the power rule to a constant term incorrectly; d/dx of a constant is zero.