Mathematical Tools
Physics numericals in Code 151 fail more often on mathematics than on physics. This unit supplies the specific tools the syllabus assumes: differentiation for velocity, acceleration and maxima; integration for work, charge and area under a graph; trigonometric identities for AC and wave problems; logarithms and exponentials for radioactive decay and capacitor discharge; and the binomial approximation for small corrections.
Core rules
- Standard derivatives: d/dx(xⁿ) = nxⁿ⁻¹, d/dx(sin x) = cos x, d/dx(cos x) = −sin x, d/dx(eˣ) = eˣ, d/dx(ln x) = 1/x. Use the chain rule for sin(ωt) type arguments.
- Standard integrals: ∫xⁿdx = xⁿ⁺¹/(n+1), ∫sin x dx = −cos x, ∫dx/x = ln x, ∫eˣdx = eˣ, with limits applied for definite integrals.
- Turning points: set dy/dx = 0; the second derivative decides maximum (negative) or minimum (positive).
- Useful identities: sin²θ + cos²θ = 1; sin 2θ = 2 sinθ cosθ; cos 2θ = 1 − 2sin²θ; 1 − cos θ ≈ θ²/2 for small θ.
- Binomial approximation: (1 + x)ⁿ ≈ 1 + nx when |x| ≪ 1 - used in error analysis and in approximations such as g variation with height.
- Exponential decay forms N = N₀e^(−λt) and q = q₀e^(−t/RC) are solved by taking logs: ln(N/N₀) = −λt.
Worked example
A particle's displacement is y = A sin(ωt). Find its velocity and the maximum speed. Differentiating with the chain rule: dy/dt = Aω cos(ωt). Since cos(ωt) ranges between −1 and +1, the maximum speed is Aω and the maximum acceleration (from differentiating again, −Aω² sin ωt) is Aω². This is exactly the SHM result a = −ω²y, obtained without memorising anything extra.
Traps that cost marks
- Forgetting the chain rule and writing d/dt sin(ωt) = cos(ωt) without the ω factor.
- Omitting the constant of integration or the limits in a definite integral.
- Using the binomial approximation when x is not small compared with 1.
- Losing a minus sign when integrating cos x or differentiating cos x.