CELE Engineering Mathematics — Differential CalculusSummary

The limit lim(x→a) f(x) = L means f(x) approaches L as x approaches a. A function is continuous at a if lim(x→a) f(x) = f(a). Indeterminate forms (0/0, ∞/∞) are resolved using L'Hôpital's rule: lim(f/g) = lim(f'/g'), provided the right-hand limit exists. This is foundational for understanding instantaneous rates of change.

The derivative f'(x) = df/dx represents the instantaneous rate of change of f with respect to x. Geometrically, it is the slope of the tangent line to the curve at x. Physically, for position s(t), the derivative ds/dt is velocity; the second derivative d²s/dt² is acceleration. The derivative encodes how a quantity responds to infinitesimal changes in its independent variable.

The power rule: d/dx(xⁿ) = nxⁿ⁻¹. For composite functions: (uv)' = u'v + uv' (product rule); (u/v)' = (u'v − uv')/v² (quotient rule); [f(g(x))]' = f'(g)·g'(x) (chain rule). These rules allow rapid differentiation without limits. Common derivatives: (sin x)' = cos x, (cos x)' = −sin x, (eˣ)' = eˣ, (ln x)' = 1/x.

At a local extremum (maximum or minimum), the first derivative is zero: f'(x) = 0, or f'(x) is undefined. These x-values are critical points. To classify: if f' changes from positive to negative at x = c, then f(c) is a local maximum; if f' changes from negative to positive, f(c) is a local minimum. The first derivative test relies on the sign of f' immediately left and right of c.

The second derivative f''(x) measures the curvature of the graph. If f''(x) > 0, the curve is concave up (shaped like ∪); if f''(x) < 0, concave down (shaped like ∩). At a critical point x = c: if f''(c) > 0, then f(c) is a local minimum; if f''(c) < 0, then f(c) is a local maximum; if f''(c) = 0, the test is inconclusive. Inflection points occur where f''(x) = 0 and the concavity changes.

To optimize (maximize or minimize) a quantity Q subject to a constraint: (1) Express Q as a function of a single independent variable using the constraint equation; (2) Find dQ/dx and set it equal to zero; (3) Solve for the critical point; (4) Verify it is a maximum or minimum using the second derivative or endpoint analysis; (5) Check boundary conditions. Example: maximize area A = xy subject to 2x + 2y = 40 (perimeter constraint) → y = 20 − x → A(x) = 20x − x² → dA/dx = 20 − 2x = 0 → x = 10 m, y = 10 m, Amax = 100 m².

When two or more quantities are related by an equation and both vary with time, differentiate the relating equation with respect to t to establish a relationship between their rates. For example, for a sphere V = (4/3)πr³, differentiate both sides with respect to t: dV/dt = 4πr² dr/dt. This links the rate of volume change to the rate of radius change. Substitute known rates and the current value of r to find the unknown rate.

The equation of the tangent line to y = f(x) at point (x₀, f(x₀)) is: y − f(x₀) = f'(x₀)(x − x₀). This tangent line provides a linear approximation: f(x) ≈ f(x₀) + f'(x₀)(x − x₀) for x near x₀. This is the basis for Newton's method for finding roots and for small-displacement approximations in mechanics.

Curvature κ measures how sharply a curve bends. For a curve y = f(x), the radius of curvature is: R = [1 + (dy/dx)²]^(3/2) / |d²y/dx²|. At a point where the tangent is nearly horizontal, R ≈ 1/|d²y/dx²|. A large R indicates gentle curvature; a small R indicates sharp bending. The curvature is related to the bending moment in beams: M = EI·κ (Euler-Bernoulli beam theory).

For a function z = f(x, y), the partial derivative ∂f/∂x is the derivative with respect to x, treating y as a constant. Similarly, ∂f/∂y treats x as constant. At a local extremum of f(x, y), both partial derivatives vanish: ∂f/∂x = 0 and ∂f/∂y = 0 simultaneously. The second partial derivative test (using the Hessian matrix) classifies the critical point. This extends optimization to problems with multiple design variables.

The Euler-Bernoulli beam equation relates curvature to bending moment: d²y/dx² = M(x)/(EI). The slope dy/dx is the angle of rotation; integrating d²y/dx² gives deflection y. Maximum moment occurs where dM/dx = 0 (the shear force); maximum deflection is found by setting dy/dx = 0. This is the standard approach for analyzing beam strength and serviceability under load.

Design of reinforced concrete columns, I-beams, and channels seeks to minimize cost or weight while meeting strength criteria. This typically involves maximizing the moment of inertia (I) or section modulus (Z) relative to material volume. Using calculus, engineers determine the optimal thickness of flanges, web height, and reinforcement ratio.

The Manning equation for flow rate Q = (1/n)A·R^(2/3)·S^(1/2) involves the cross-sectional area A and hydraulic radius R. For a trapezoidal channel of given roughness and slope, optimization of depth and width maximizes Q for a fixed wetted perimeter, or minimizes perimeter for a fixed Q. Differentiation determines the optimal hydraulic section.

As a concrete truck fills a cylindrical storage tank, the volume increases at a known rate (dV/dt, given by pump flow). The height h of concrete rises according to V = πr²h; thus dh/dt = (dV/dt)/(πr²). Engineers use this to estimate fill times and ensure overflow prevention. Similar analysis applies to excavation rate, sediment accumulation in reservoirs, and traffic flow dynamics.

Stress-strain curves (fundamental in mechanics of materials and ACI 318 for concrete) are analyzed using calculus. The modulus of elasticity E = dσ/dε (slope of the linear elastic region); the yield point is where d²σ/dε² changes sign (inflection point). Optimization of reinforcement ensures efficient use of steel and concrete within ductility constraints.

Heat conduction and pollutant diffusion are governed by the Fourier and Fick equations, which involve partial derivatives of temperature or concentration with respect to space and time: ∂T/∂t = α·∇²T. Solutions to these PDEs require mastery of partial derivatives and understanding of how derivatives encode physical processes.

Project cost functions C(x) often have a minimum at some optimal scale x (e.g., optimal batch size, crew size, or delivery frequency). Setting dC/dx = 0 and verifying d²C/dx² > 0 locates the most cost-effective operation. This is critical for construction management and resource planning.