Engineering Mathematics I: Calculus

Limits, derivatives and integrals. Single-variable calculus for engineers: limits and continuity, differentiation and its rules, applications to optimisation, related rates and Newton's method, mean value theorems, integration techniques, and applications of integration to area, volume, work, centroids and RMS values.

• Limits — Limits describe what a function approaches as its input approaches a value. Learn the intuitive and formal ideas, one-sided and two-sided limits, infinite limits and limits at infinity, algebraic techniques (factorising, rationalising), standard limits such as sin x/x, indeterminate forms and L'Hôpital's rule.
• Continuity — A continuous function has no breaks, jumps or holes. Learn the three-part definition, left and right continuity, continuity on intervals, types of discontinuity (removable, jump, infinite, oscillating), the intermediate value theorem, and engineering interpretations such as switching events and material interfaces.
• Differentiation — The derivative measures instantaneous rate of change and slope. Learn the limit definition, geometric and physical meaning, standard derivatives, product, quotient and chain rules, higher derivatives, implicit, parametric, logarithmic and inverse-function differentiation, and an introduction to partial derivatives.
• Applications of Derivatives — Derivatives solve real design problems. Learn increasing/decreasing tests, stationary points, local and global extrema, concavity and inflection points, curve sketching, optimisation, related rates, Newton's method, linearisation, Taylor approximations and sensitivity analysis.
• Mean Value Theorems — Mean value theorems connect average and instantaneous rates. Learn Rolle's theorem, Lagrange's mean value theorem and Cauchy's generalised form, their assumptions, geometric interpretation, and applications to bounds, error estimates, uniqueness of roots and speed enforcement.
• Integration — Integration accumulates quantities and reverses differentiation. Learn antiderivatives and indefinite integrals, definite integrals as Riemann sums and areas, the fundamental theorem of calculus, standard integrals, substitution, integration by parts, partial fractions, trigonometric integrals and substitutions, and improper integrals.
• Applications of Integration — Integrals compute physical totals. Learn area between curves, volumes of revolution (disc and shell), arc length and surface area, work done by variable forces, mass, centroids and moments, average and RMS values, and charge and energy accumulation.