Calculus (Differentiation and Integration)
Calculus spans differentiation and integration of algebraic, trigonometric, exponential and logarithmic functions, together with applications: tangents and normals, stationary points, rates of change, areas and volumes of revolution. Integration technique - substitution, by parts, partial fractions - is examined, and the choice of technique should be justified.
Differentiation
- Chain rule dy/dx = dy/du × du/dx; product (uv)′ = u′v + uv′; quotient (u/v)′ = (u′v − uv′)/v².
- d/dx(sin x) = cos x, d/dx(cos x) = −sin x, d/dx(tan x) = sec²x, d/dx(eˣ) = eˣ, d/dx(ln x) = 1/x.
- Implicit differentiation: differentiate term by term, remembering that y terms bring a factor dy/dx.
- Parametric curves: dy/dx = (dy/dt)/(dx/dt).
Stationary points and rates
- Solve dy/dx = 0, then classify with d²y/dx² (negative for a maximum, positive for a minimum) or a sign table.
- Tangent gradient m, normal gradient −1/m; write each with the point–slope form.
- Connected rates use dV/dt = dV/dr × dr/dt.
Integration
- ∫xⁿ dx = x^(n+1)/(n+1) + c for n ≠ −1; ∫x⁻¹ dx = ln|x| + c; ∫e^(kx) dx = e^(kx)/k + c.
- By parts: ∫u dv = uv − ∫v du, choosing u as the factor that simplifies when differentiated.
- Substitution: change dx and the limits together for a definite integral.
- Area between a curve and the x-axis is ∫y dx; volume about the x-axis is π∫y² dx.
Worked example
Find ∫x e^(2x) dx. Take u = x and dv = e^(2x) dx, so du = dx and v = e^(2x)/2. Then the integral is x e^(2x)/2 − ∫(e^(2x)/2) dx = x e^(2x)/2 − e^(2x)/4 + c.
Traps
- Omitting + c from an indefinite integral.
- Forgetting the inner-derivative factor when integrating e^(kx) or (ax + b)ⁿ.
- Adding signed areas instead of splitting the integral where the curve crosses the axis.