Calculus for Engineers I — Functions & Pre-Calculus Foundations

Course Description

• Instruction Mode: Face-to-Face (F2F)

• Credit Units: 3

• Four (4) major chapters

• Focus: Core concepts of limits, continuity, differentiability for single- and multi-variable functions

  • Applications: curve tracing, optimization, rates of change, related rates, tangents & normals, approximations, partial differentiation

Grading & Assessment

• Components

  • Quizzes / Assignments: 40 %

  • Midterm Examination: 30 %

  • Final Examination: 30 %

• Transmutation Table (Final-grade scale)

  • $1.0 \;:\; 90\text{–}100$

  • $1.25 \;:\; 85\text{–}89$

  • $1.50 \;:\; 80\text{–}84$

  • $1.75 \;:\; 75\text{–}79$

  • $2.00 \;:\; 70\text{–}74$

  • $2.25 \;:\; 65\text{–}69$

  • $2.50 \;:\; 60\text{–}64$

  • $2.75 \;:\; 55\text{–}59$

  • $3.00 \;:\; 50\text{–}54$

  • $5.00 \;:\; 49 \text{ and below}$

Course Outline (Week-by-Week)

• Chapter 1 – Functions & Limits

  • Week 1: Review of Functions

  • Week 2: Linear, Quadratic, Exponential, Logarithmic, Trigonometric & Inverse-Trig Functions

  • Week 3: Intuitive Idea of Limits; Limit Theorems

  • Week 4: One-sided Limits, Infinite Limits, Limits at Infinity

  • Week 5: Continuity of a Function

• Chapter 2 – Differentiation

  • Week 6: Definition & Geometric Interpretation of the Derivative

  • Week 7: Differentiation Formulas (Algebraic & Transcendental)

  • Week 8: Implicit Differentiation, Higher-order Derivatives, Indeterminate Forms, $\text{L}\,'\text{Hôpital's Rule}$

  • Week 9: Midterm Examination

  • Week 10: Academic Break

  • Week 11: 1st-Derivative Test (Increasing/Decreasing), Concavity, 2nd-Derivative Test

  • Week 12: Curve Tracing (Polynomial Functions)

  • Week 13: Optimization & Related Rates

  • Week 14: Differential Approximation

• Chapter 3 – Parametric Equations

  • Week 15: Definition; Eliminating Parameters

  • Week 16: Derivatives, Tangent & Normal Lines for Parametric Curves

• Chapter 4 – Partial Differentiation

  • Week 17: Partial Derivatives

  • Week 18: Final Examination

1.1 Review on Functions

Definition

• A function $f$ assigns exactly one output $f(x)$ to each input $x$.

  • $x$ = independent variable; $y=f(x)$ = dependent variable.

Four Representations of Functions

1. Verbal (word description)

2. Numerical (table of values)

3. Visual (graph)

4. Algebraic (formula)

Vertical Line Test (Definition 1.1.2)

• A curve in the $xy$-plane is the graph of a function iff no vertical line intersects it more than once.

Example 1 – Function or Not?

1. "Course name → number of students" ✔︎ (Function)

2. (Item not fully shown) – context implies potentially not a function

3. "Your age on each birthday" ✔︎ (Function – one age per year) 4 & 5. Two tables interchange $x$ and $y$ values;

   • First table (distinct $x$ values) ✔︎ Function

   • Second table (repeated $x$ with multiple $y$) ✖︎ Not a function

1.1 Common Families of Functions

• Linear, Quadratic, Cubic (Polynomial)

• Exponential, Logarithmic

• Trigonometric, Inverse Trigonometric

Key Features to Analyze

• Domain & Range

• $x$- and $y$-intercepts

• Vertex (for quadratics)

• Absolute / Relative extrema

• Asymptotes

• End behavior

• Intervals of increase & decrease

• Points of inflection (cubic & higher)

1.2.1 Linear Function

General Form & Definition (Polynomial Degree 1)

• $P(x)=mx+b$ where $m$ = slope, $b$ = $y$-intercept.

Example 3 Analysis – $y = 3x - 2$

• Domain: $\mathbb R$ (all real numbers)

• Range: $\mathbb R$

• $x$-intercept: $\left(\frac{2}{3},0\right)$

• $y$-intercept: $(0,-2)$

• Monotonicity: Increasing on $(-\infty,\infty)$ (because m=3>0)

• End Behavior

  • As $x\to+\infty$, $y\to+\infty$

  • As $x\to-\infty$, $y\to-\infty$

Example 3 Analysis – $y = -2x + 4$ (summary)

• Domain/Range: $\mathbb R$

• Intercepts: $x$-intercept $(2,0)$, $y$-intercept $(0,4)$

• Decreasing (slope m=-2<0)

• End Behavior: $x\to+\infty$ ⇒ $y\to-\infty$; $x\to-\infty$ ⇒ $y\to+\infty$

1.2.2 Quadratic Function

General Form & Graph

• $P(x)=ax^2+bx+c$, $a\neq0$

• Parabola opens upward if a>0, downward if a<0.

Key Vocabulary

• Axis of Symmetry: vertical line $x=-\frac{b}{2a}$

• Vertex: $\left(-\frac{b}{2a},\;f!\left(-\frac{b}{2a}\right)\right)$; location of min/max value.

Canonical Example – $y=x^{2}$

• Domain: $\mathbb R$

• Range: $[0,\infty)$

• Intercepts: $x$- & $y$- intercept both at $(0,0)$

• Vertex: $(0,0)$; Axis: $x=0$

• End Behavior: $x\to\pm\infty$ ⇒ $y\to+\infty$

• Decreasing on $(-\infty,0]$; Increasing on $[0,\infty)$

Example 4 Tasks

1. $y = x^{2}-2x+1 = (x-1)^2$

   • Domain $\mathbb R$; Range $[0,\infty)$

   • Vertex $(1,0)$; Axis $x=1$

   • Intercepts: $x$- intercept $(1,0)$ (double root); $y$- intercept $(0,1)$

   • Increasing on $[1,\infty)$; Decreasing on $(-\infty,1]$

2. $y = -x^{2}-6x-5 = -(x+3)^2+4$

   • Domain $\mathbb R$; Range $(-\infty,4]$

   • Vertex $(-3,4)$; Axis $x=-3$

   • Intercepts: compute via factoring/quadratic formula; decreasing left of axis, increasing right (opens downward)

1.2.3 Cubic Function

General Form

• $P(x)=ax^{3}+bx^{2}+cx+d$ with $a\neq0$.

Prototype – $y = x^{3}$

• Domain & Range: $\mathbb R$

• Origin $(0,0)$ acts as both intercept and point of inflection (Definition 1.2.7).

• End Behavior: $x\to+\infty$ ⇒ $y\to+\infty$; $x\to-\infty$ ⇒ $y\to-\infty$

• Monotonic: Increasing on all $\mathbb R$ (for coefficient a>0).

General Features Example – $y=(x+1)(x-2)(x-3)$

• Zeros: $(-1,0), (2,0), (3,0)$

• $y$-intercept: $(0,6)$

• Local Max/Min occur between roots (approx. at $x\approx0$ & $x\approx2.5$)

• Intervals

  • Increasing: $(-\infty,0)\cup(2.5,\infty)$

  • Decreasing: $(0,2.5)$

1.2.4 Exponential Function

General Form

• $f(x)=a^{x}$, base a>0,\;a\neq1

Growth Case – $y = 2^{x}$

• Domain $\mathbb R$; Range $(0,\infty)$

• $y$-intercept: $(0,1)$

• Horizontal asymptote: $y=0$

• Increasing for all $x$ (because a>1)

• End Behavior: $x\to+\infty$ ⇒ $y\to+\infty$; $x\to-\infty$ ⇒ $y\to0^{+}$

Decay Case – $y = (0.5)^{x}$

• Same domain/range and asymptote $y=0$

• Decreasing for all $x$ (because 0<a<1)

1.2.5 Logarithmic Function

General Form

• $f(x)=\log_{a}x$ (inverse of $a^{x}$)

Example – $y=\log_{2}x$

• Domain: $(0,\infty)$

• Range: $\mathbb R$

• $x$-intercept: $(1,0)$ (since $\log_{a}1=0$)

• Vertical asymptote: $x=0$

• Increasing when a>1; decreasing when 0<a<1 (e.g., $a=0.5$)

1.2.6 Trigonometric Functions (Basics)

Sine – $y=\sin x$

• Domain: $(-\infty,\infty)$

• Range: $[-1,1]$

• Period: $2\pi$ (Definition 1.2.9)

• Symmetry: odd function (symmetric about origin)

• Zeros: $x=n\pi$, $n\in\mathbb Z$

• $y$-intercept: $(0,0)$

Other Basics

• Cosine similar but even symmetry, same period

• Tangent has period $\pi$, vertical asymptotes at $x=\frac{\pi}{2}+n\pi$

• Secant, Cosecant, Cotangent likewise defined

1.2.7 Inverse Trigonometric Functions

Notation Equivalents

• $\arcsin x = \sin^{-1}x$ etc.

Arccosine – $y=\arccos x$

• Domain $[-1,1]$; Range $[0,\pi]$

• $x$-intercept: $(1,0)$

• $y$-intercept: $\left(0,\frac{\pi}{2}\right)$

Arctangent – $y=\arctan x$

• Domain $\mathbb R$; Range $\left(-\frac{\pi}{2},\frac{\pi}{2}\right)$ (horizontal asymptotes at range endpoints)

• Intercepts: $(0,0)$

Parametric Equations (Preview – Chapter 3)

• A curve represented as $x=f(t),\;y=g(t)$.

• Eliminating parameter $t$ retrieves Cartesian form when possible.

• Derivative: $\frac{dy}{dx}=\frac{dy/dt}{dx/dt}$ (provided $dx/dt\neq0$).

• Tangent line slope at $t=t_{0}$ uses above derivative; normal line slope is negative reciprocal.

Partial Differentiation (Preview – Chapter 4)

• For $z=f(x,y)$, partial derivatives $\frac{\partial f}{\partial x},\;\frac{\partial f}{\partial y}$ treat the non-target variable as constant.

• Higher-order partials: $f<em>{xx},\,f</em>{yy},\,f<em>{xy}=f</em>{yx}$ (under certain continuity conditions – Clairaut’s theorem).

Connections & Significance

• Mastery of functions & their features provides the foundation for limits and derivatives in later chapters.

• Vertical line test, domain/range analysis, and asymptote identification are critical skills for curve sketching and optimization problems.

• Polynomial degree dictates graph behavior (even/odd symmetry, end behavior), influencing root-finding and engineering design tasks.

• Exponential & logarithmic models appear in growth/decay, signal attenuation, and finance; understanding inverses aids in solving for time variables.

• Trigonometric and inverse-trig functions model oscillatory motion, waves, and rotational kinematics.

• Parametric & partial differentiation extend single-variable calculus into motion along paths and multi-variable optimization—key for engineering applications.

Ethical / Practical Notes

• Proper application of calculus tools can optimize resources and safety in engineering projects; misuse or calculation errors may have significant real-world impacts.

• Awareness of assumptions (continuity, differentiability) is essential to avoid invalid conclusions.

These notes encapsulate every key definition, theorem, example, and conceptual linkage presented in the transcript, providing a stand-alone resource for exam preparation.