Engineering Mathematics

Calculus answers two questions: how fast is something changing right now, and how much has built up in total? A derivative gives the rate of change, and an integral gives the running total. They are opposites of each other.

• Derivative: the slope of a curve at one exact point.
• Power rule: the derivative of xⁿ is n·xⁿ⁻¹, so x² becomes 2x.
• Integral: the total area under a curve.
• Where it is used: speed from position, current from charge, power from energy.

f(x)  = x²
f'(x) = 2x
f'(3) = 6      slope at x = 3

∫₀³ x² dx = [x³/3]₀³ = 9

Real systems have many unknowns tied together, such as the currents in a circuit. Writing them as a matrix lets you solve them with one reliable method, by hand or by computer.

• Matrix: a grid of numbers. Ax = b packs a whole system into one line.
• Gaussian elimination: subtract rows to remove unknowns, then work backwards.
• Determinant: if it is zero, there is no single unique solution.
• Where it is used: circuits, graphics, machine learning, and structural analysis.

[ 1  1 ] [ x ]   [ 5 ]
[ 2  1 ] [ y ] = [ 8 ]

A differential equation describes how something changes based on its current state. Cooling coffee, charging capacitors and swinging springs all follow one. Solving it tells you where the system will be later.

• Model: dT/dt = -k(T - 20): the hotter the coffee, the faster it cools.
• Numerical solving: Euler's method takes small steps forward using the current slope.
• Step size: smaller steps are more accurate but take more work.
• Laplace transform: turns these equations into algebra, which is a key tool in control systems.

dT/dt = -0.1 (T - 20)
T(next) = T + step × dT/dt