CELE Engineering Mathematics — Differential CalculusStudy Notes

A **limit** describes the value that a function approaches as the independent variable moves toward a specified point. Formally, $\lim_{x\to a}f(x) = L$ means that for any small positive number $\epsilon$, there exists a small positive number $\delta$ such that whenever $0 < |x - a| < \delta$, we have $|f(x) - L| < \epsilon$. **Continuity at a point:** A function $f(x)$ is continuous at $x = a$ if and only if three conditions hold: 1. $f(a)$ is defined 2. $\lim_{x\to a}f(x)$ exists 3. $\lim_{x\to a}f(x) = f(a)$ **Indeterminate forms** arise when direct substitution yields $\frac{0}{0}$, $\frac{\infty}{\infty}$, $0 \cdot \infty$, $\infty - \infty$, $0^0$, $1^{\infty}$, or $\infty^0$. For these cases, **L'Hôpital's Rule** provides a systematic solution: if $\lim_{x\to a}\frac{f(x)}{g(x)}$ yields an indeterminate form and both $f'(x)$ and $g'(x)$ exist near $a$, then $$\lim_{x\to a}\frac{f(x)}{g(x)} = \lim_{x\to a}\frac{f'(x)}{g'(x)}$$ provisioning the latter limit exists. You may apply L'Hôpital's Rule repeatedly if necessary. **Limit laws (for limits that exist):** - Sum rule: $\lim[f(x) + g(x)] = \lim f(x) + \lim g(x)$ - Product rule: $\lim[f(x) \cdot g(x)] = [\lim f(x)] \cdot [\lim g(x)]$ - Quotient rule: $\lim\frac{f(x)}{g(x)} = \frac{\lim f(x)}{\lim g(x)}$ (provided $\lim g(x) \neq 0$) - Power rule: $\lim[f(x)]^n = [\lim f(x)]^n$ **Important standard limits:** $$\lim_{x\to 0}\frac{\sin x}{x} = 1, \quad \lim_{x\to 0}\frac{1 - \cos x}{x^2} = \frac{1}{2}, \quad \lim_{x\to \infty}\left(1 + \frac{1}{x}\right)^x = e$$ These limits appear frequently in engineering applications—for example, when analyzing small-angle approximations in structural dynamics or wave propagation.

The **derivative** of $f(x)$ at point $x$ measures the instantaneous rate of change of $f$ with respect to $x$. Geometrically, it is the slope of the tangent line to the curve $y = f(x)$ at that point. **Definition (limit form):** $$f'(x) = \lim_{h\to 0}\frac{f(x+h) - f(x)}{h}$$ Alternatively, at a specific point $x = a$: $$f'(a) = \lim_{x\to a}\frac{f(x) - f(a)}{x - a}$$ **Notation:** The derivative of $y = f(x)$ is denoted as $f'(x)$, $\frac{dy}{dx}$, $\frac{df}{dx}$, or $y'$. **Equation of the tangent line** at point $(x_0, f(x_0))$: $$y - f(x_0) = f'(x_0)(x - x_0)$$ **Differentiability:** A function is differentiable at $x = a$ if $f'(a)$ exists. Differentiability implies continuity, but continuity does not guarantee differentiability (e.g., $y = |x|$ is continuous at $x = 0$ but not differentiable there due to a sharp corner). **Physical interpretation in civil engineering:** If $y$ represents displacement and $x$ represents time, then $\frac{dy}{dx}$ is velocity. If stress $\sigma$ varies with strain $\epsilon$, then $\frac{d\sigma}{d\epsilon}$ is the elastic modulus. Derivatives quantify how quickly structural responses change under loading—critical for analyzing beam deflections, column buckling, and dynamic behavior.

Rather than use the limit definition repeatedly, we apply systematic rules to find derivatives efficiently. **Power Rule:** For any real constant $n$, $$\frac{d}{dx}x^n = nx^{n-1}$$ This applies even when $n$ is negative or fractional (e.g., $\frac{d}{dx}x^{-2} = -2x^{-3}$, $\frac{d}{dx}\sqrt{x} = \frac{d}{dx}x^{1/2} = \frac{1}{2}x^{-1/2}$). **Constant Multiple Rule:** $$\frac{d}{dx}[cf(x)] = c\frac{df}{dx}$$ **Sum and Difference Rule:** $$\frac{d}{dx}[f(x) \pm g(x)] = \frac{df}{dx} \pm \frac{dg}{dx}$$ **Product Rule:** For the product of two functions $u(x)$ and $v(x)$, $$(uv)' = u'v + uv'$$ Mnemonic: "first times derivative of second, plus second times derivative of first." **Quotient Rule:** For the quotient $\frac{u}{v}$ where $v \neq 0$, $$\left(\frac{u}{v}\right)' = \frac{u'v - uv'}{v^2}$$ Mnemonic: "low $d$-high minus high $d$-low, over low low" (denominator-derivative high minus high-derivative low, divided by low squared). **Chain Rule:** If $y = f(u)$ and $u = g(x)$, then $$\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx} = f'(g(x)) \cdot g'(x)$$ Alternatively, if $y = f(g(x))$, then $y' = f'(g(x)) \cdot g'(x)$. The chain rule is crucial for composite functions and appears constantly in engineering calculations. **Derivatives of trigonometric functions:** $$\frac{d}{dx}\sin x = \cos x, \quad \frac{d}{dx}\cos x = -\sin x, \quad \frac{d}{dx}\tan x = \sec^2 x$$ $$\frac{d}{dx}\cot x = -\csc^2 x, \quad \frac{d}{dx}\sec x = \sec x \tan x, \quad \frac{d}{dx}\csc x = -\csc x \cot x$$ **Derivatives of exponential and logarithmic functions:** $$\frac{d}{dx}e^x = e^x, \quad \frac{d}{dx}a^x = a^x \ln a \quad (a > 0, a \neq 1)$$ $$\frac{d}{dx}\ln x = \frac{1}{x}, \quad \frac{d}{dx}\log_a x = \frac{1}{x \ln a}$$ **Inverse trigonometric function derivatives:** $$\frac{d}{dx}\arcsin x = \frac{1}{\sqrt{1-x^2}}, \quad \frac{d}{dx}\arccos x = -\frac{1}{\sqrt{1-x^2}}, \quad \frac{d}{dx}\arctan x = \frac{1}{1+x^2}$$ These rules form the toolkit for all derivative calculations. Mastering them enables rapid solution of board-exam problems.

The **second derivative** $f''(x)$ (or $\frac{d^2y}{dx^2}$) is the derivative of the first derivative. It measures how quickly the rate of change is changing—i.e., the curvature or concavity of the graph. **Concavity interpretation:** - **Concave up** (convex): $f''(x) > 0$. The graph curves upward; tangent lines lie below the curve. Geometrically, the function is "holding water." - **Concave down**: $f''(x) < 0$. The graph curves downward; tangent lines lie above the curve. - **Inflection point**: A point where the concavity changes, where $f''(x) = 0$ (and the sign of $f''$ changes). **Second Derivative Test for Extrema:** At a critical point $x = c$ where $f'(c) = 0$: - If $f''(c) > 0$, then $f$ has a local minimum at $c$. - If $f''(c) < 0$, then $f$ has a local maximum at $c$. - If $f''(c) = 0$, the test is inconclusive; use the first derivative test instead. **Third and higher derivatives:** Successive differentiation yields $f'''(x)$, $f^{(4)}(x)$, etc. In engineering applications, the third derivative (often called "jerk" in motion analysis) represents the rate of change of acceleration and is important in vehicle dynamics and seismic analysis. Higher derivatives are rare in basic optimization but essential in Taylor series approximations. **Application in civil engineering:** When analyzing beam deflection under load, the deflection function $y(x)$ has a physical meaning: - $y'(x)$ = slope of the beam (angle of rotation) - $y''(x)$ = curvature (proportional to bending moment $M$; the beam equation is $M = EI\frac{d^2y}{dx^2}$ where $E$ is elastic modulus and $I$ is second moment of area) - $y'''(x)$ = shear force (related to internal shear $V$; $V = EI\frac{d^3y}{dx^3}$) - $y^{(4)}(x)$ = distributed load $q$ (the load equation is $q = EI\frac{d^4y}{dx^4}$) This direct relationship between derivatives and physical quantities makes differential calculus indispensable in structural mechanics.

**Finding extrema** (maximum and minimum values of a function) is among the most important applications of calculus in engineering. Optimal structural designs minimize cost or weight while meeting safety constraints—a fundamental principle in NSCP 2015 and modern engineering practice. **Critical points:** A critical point occurs at $x = c$ if: 1. $f'(c) = 0$ (stationary point), or 2. $f'(c)$ does not exist (e.g., sharp corner, cusp). Every local extremum in the interior of a domain must occur at a critical point. **First Derivative Test (always reliable):** 1. Find all critical points by solving $f'(x) = 0$ (or locating where $f'$ is undefined). 2. Examine the sign of $f'(x)$ on intervals around each critical point: - If $f'$ changes from positive to negative as $x$ increases through $c$, then $c$ is a local maximum. - If $f'$ changes from negative to positive as $x$ increases through $c$, then $c$ is a local minimum. - If $f'$ does not change sign, then $c$ is neither a maximum nor a minimum (saddle point or inflection). **Second Derivative Test (convenient, sometimes inconclusive):** At a critical point $c$ where $f'(c) = 0$: - If $f''(c) > 0$, then $c$ is a local minimum. - If $f''(c) < 0$, then $c$ is a local maximum. - If $f''(c) = 0$, the test is inconclusive; use the first derivative test. **Global extrema on a closed interval $[a,b]$:** Evaluate $f$ at: - All critical points in $(a, b)$ - The endpoints $f(a)$ and $f(b)$ The largest value is the global maximum, and the smallest is the global minimum. This is essential for constrained optimization in engineering design. **Optimization strategy:** 1. **Identify the quantity to optimize** and express it as a function of a single variable using constraint equations. 2. **Find the domain** (physical bounds on the variable). 3. **Differentiate** and set equal to zero to find critical points. 4. **Check endpoints** if the domain is closed. 5. **Verify using the second derivative test** or by examining sign changes in the first derivative. 6. **Interpret the result** physically—does it make sense given the context? **Example context in civil engineering:** Minimizing the cost of concrete in a rectangular foundation of fixed volume, or maximizing the load capacity of a beam of given cross-section—these are prototypical optimization problems solved via calculus.

**Related rates problems** involve quantities that change with respect to time and are connected by constraint equations. The goal is to find the rate of change of one quantity given the rates of others. **Strategy:** 1. **Draw and label a diagram** with all quantities and given rates. 2. **Identify the variables** and constants. Variables usually change with time; constants do not. 3. **Write the constraint equation** relating all variables (geometric formula, volume, etc.). 4. **Differentiate both sides with respect to time $t$**, treating all variables as functions of $t$ (use chain rule systematically). 5. **Substitute the known values** (rates, positions) to solve for the unknown rate. 6. **Verify units and sign**—positive rates mean increasing, negative means decreasing. **Key principle:** Differentiate *before* substituting numerical values. Substitution after differentiation prevents errors from missing rate terms. **Common constraint equations in related rates:** - Pythagorean theorem: $x^2 + y^2 = c^2$ (distances) - Circle area: $A = \pi r^2$; circumference: $C = 2\pi r$ - Sphere volume: $V = \frac{4}{3}\pi r^3$ - Cylinder volume: $V = \pi r^2 h$ - Cone volume: $V = \frac{1}{3}\pi r^2 h$ - Similar triangles: $\frac{x_1}{y_1} = \frac{x_2}{y_2}$ (proportional dimensions) **Application in civil engineering:** In traffic flow analysis, if the number of vehicles $N(t)$ passing a point increases at a known rate $\frac{dN}{dt}$, and the total distance traveled $D$ is related to $N$ by $D = L \cdot N$ (where $L$ is the route length), then the rate of distance increase is $\frac{dD}{dt} = L \frac{dN}{dt}$. Monitoring such rates helps engineers optimize traffic signal timing and estimate congestion dynamics.

The **tangent line** to a curve $y = f(x)$ at a point $(x_0, f(x_0))$ is a straight line that touches the curve at that point and has the same slope as the curve there. **Equation of the tangent line:** $$y - f(x_0) = f'(x_0)(x - x_0)$$ Alternatively, in slope-intercept form: $$y = f(x_0) + f'(x_0)(x - x_0)$$ The tangent line is the best linear approximation to $f$ near $x_0$. For values of $x$ close to $x_0$: $$f(x) \approx f(x_0) + f'(x_0)(x - x_0)$$ This is called **linear approximation** or **differential approximation**. The error $E = f(x) - [f(x_0) + f'(x_0)(x - x_0)]$ is small when $x$ is near $x_0$ and becomes negligible for very small intervals (e.g., small measurement uncertainties). **Differential notation:** The **differential** $dy$ represents the change in the tangent line over an interval $dx$: $$dy = f'(x)\,dx$$ This is useful for estimating small changes in $y$ due to small changes in $x$. For example, if a structural parameter changes slightly, engineers can estimate the resulting change in response without full recalculation. **Normal line:** The line perpendicular to the tangent at $(x_0, f(x_0))$ has slope $-\frac{1}{f'(x_0)}$ (negative reciprocal) and equation: $$y - f(x_0) = -\frac{1}{f'(x_0)}(x - x_0)$$ **Application in engineering:** In structural analysis, if a load parameter varies slightly from its design value, the engineer can use linear approximation to quickly estimate the change in deflection or stress without re-solving the full problem. This enables sensitivity analysis and uncertainty quantification.

**Complete curve sketching** synthesizes derivatives, critical points, concavity, and asymptotes to create an accurate graph. This skill is valuable for visualizing structural responses, force distributions, and optimization problems. **Curve-sketching procedure:** 1. **Domain:** Identify where the function is defined. 2. **Intercepts:** Find $x$-intercepts (set $y = 0$) and $y$-intercepts (set $x = 0$). 3. **Asymptotes:** - **Vertical:** Denominator = 0 (check left and right limits) - **Horizontal:** $\lim_{x \to \pm\infty} f(x)$ - **Oblique (slant):** Polynomial long division when degree of numerator exceeds denominator by 1 4. **First derivative:** Find critical points ($f'(x) = 0$), determine intervals of increase/decrease. 5. **Second derivative:** Find inflection points ($f''(x) = 0$), determine concavity. 6. **Sketch:** Plot intercepts, critical points, inflection points, and asymptotes; connect with smooth curves respecting concavity and monotonicity. **Radius of curvature:** The **curvature** $\kappa$ measures how sharply a curve bends. The **radius of curvature** $R$ is the reciprocal: $$\kappa = \frac{|f''(x)|}{[1 + (f'(x))^2]^{3/2}}, \quad R = \frac{1}{\kappa} = \frac{[1 + (f'(x))^2]^{3/2}}{|f''(x)|}$$ Geometrically, $R$ is the radius of the osculating circle (best-fitting circle) at a point on the curve. **Physical meaning:** - **Large $R$** → gentle curve (small curvature) - **Small $R$** → sharp curve (high curvature) - At inflection points where $f''(x) = 0$, the radius becomes infinite (curvature is zero). **Application in civil engineering:** Road and railway alignment design use curvature formulas. For a highway curve, the minimum radius of curvature is constrained by speed and friction to prevent skidding: $R_{\min} = \frac{v^2}{g(e + f)}$, where $v$ is speed, $e$ is banking angle, $f$ is friction coefficient, and $g$ is gravity. The radius of curvature of the designed centerline must exceed this minimum everywhere.

Many engineering problems involve functions of multiple independent variables. For example, the deflection of a beam depends on load, span length, material properties, and geometry—all varying independently. **Partial derivative notation:** The **partial derivative** of $f(x, y)$ with respect to $x$, denoted $\frac{\partial f}{\partial x}$ or $f_x$, is the derivative of $f$ with respect to $x$ alone, treating all other variables as constants. $$\frac{\partial f}{\partial x} = \lim_{h \to 0}\frac{f(x+h, y) - f(x, y)}{h}$$ Similarly, $\frac{\partial f}{\partial y} = f_y$ is the derivative with respect to $y$, treating $x$ as a constant. **Second-order partial derivatives:** - $\frac{\partial^2 f}{\partial x^2} = f_{xx}$: differentiate twice with respect to $x$ - $\frac{\partial^2 f}{\partial y^2} = f_{yy}$: differentiate twice with respect to $y$ - $\frac{\partial^2 f}{\partial x \partial y} = f_{xy}$: differentiate first with respect to $x$, then $y$ - $\frac{\partial^2 f}{\partial y \partial x} = f_{yx}$: differentiate first with respect to $y$, then $x$ **Schwarz's theorem:** If $f$ and its partial derivatives are continuous, then $f_{xy} = f_{yx}$ (mixed partials are equal). **Critical points for extrema:** A point $(x_0, y_0)$ is a critical point if $f_x(x_0, y_0) = 0$ and $f_y(x_0, y_0) = 0$ simultaneously. **Second Derivative Test (2D Hessian):** At a critical point $(x_0, y_0)$, define $$D = f_{xx}(x_0, y_0) \cdot f_{yy}(x_0, y_0) - [f_{xy}(x_0, y_0)]^2$$ Then: - If $D > 0$ and $f_{xx} > 0$: local minimum - If $D > 0$ and $f_{xx} < 0$: local maximum - If $D < 0$: saddle point (neither maximum nor minimum) - If $D = 0$: test is inconclusive **Application in civil engineering:** In foundation design optimization, the settlement $S$ depends on soil cohesion $c$ and angle of internal friction $\phi$. Minimizing cost $C(c, \phi)$ subject to settlement limits requires finding critical points via partial derivatives. This approach scales to problems with dozens of design variables.