Course Title: EN112 - Engineering Mathematics I
Department: Mathematics and Computer Science
Institution: The Papua New Guinea University of Technology
Semester: Semester 1, 2025
Definition: A function f is a rule that associates each input x to a unique output f(x).
Functions can be expressed:
Numerically (tables)
Geometrically (graphs)
Algebraically (formulas)
Illustrates independent (input) and dependent (output) variables with an example y = 1/x.
A curve represents a function if a vertical line intersects it at most once.
Example: The circle defined by x² + y² = 25 fails the test; thus, it's not a function of x.
It can be viewed as two semi-circles: y = √(25 - x²) and y = -√(25 - x²).
Domain: All input numbers x that a function can process.
Range: The corresponding collection of output numbers y.
Example: For y = √(1 - x²), the domain is -1 ≤ x ≤ 1, and the range is 0 ≤ y ≤ 1.
Functions can be chained; the output from one function becomes the input of another.
Notation: f ◦ g ◦ h = f(g(h(x))).
Example: For f(x) = √x, g(x) = 1/x, h(x) = x³, the composition yields (f ◦ g ◦ h)(x) = 1/x^(3/2).
Inverse functions reverse the operations of the original function, denoted f⁻¹(x).
Examples:
f(x) = kx (linear)
f(x) = k/x (inverse proportional)
Examples include Boyle's Law P = kV and Newtonian viscosity τx = µ ∂u/∂y.
Common polynomial types include linear (y = mx + c), quadratic (y = ax² + bx + c), and cubic (y = x³).
2nd-order polynomials' roots can be found using the quadratic formula:
x = [-b ± √(b² - 4ac)] / 2a.
Completed square form of a quadratic function is derived as follows:
(x + d)² = x² + 2dx + d².
Example: For f(x) = x² + 6x - 3:
Completed to (x + 3)² - 12.
Examples of cubic functions include f(x) = x³, f(x) = (x − 3)³, and others.
Functions might also include absolute, square root, and trigonometric functions (SOH CAH TOA).
Recall index laws for exponents.
Differences between polynomial (f(x) = xᵐ) and exponential functions (f(x) = m^x) are crucial for calculation speed.
The Euler number e ≈ 2.718 is the basis for natural exponential functions.
Derived from compound interest: e = lim(n→∞) (1 + 1/n)ⁿ.
The inverse of exponential functions, denoted log (base 10) or ln (base e).
For example, log₁₀(100) = 2, ln(1) = 0.
Change of base formula: logₐ x = ln x / ln a.
To solve (log₂81)(log₃32):
Use the identities from previous pages to simplify without a calculator, yielding a final value of 20.
Perform calculations for log values yielding 1.908485 / 0.30103 and 0.47712 resulting in a total of approximately 20.
Defined via circumference-to-diameter ratio.
1 radian = 180/π degrees.
Conversion formulas include x radians = (180/π) × x°.
Recap of trigonometric ratios: sine, cosine, and tangent (SOH CAH TOA).
Inverse functions (arcsin, arccos, arctan) provide angle outputs.
All functions are periodic, plotted with specific properties.
General form: y = f(x) = Asin(Bx) or A*cos(Bx).
Factors include amplitude (A), period (T = 2π/|B|), and frequency (f = |B|/2π).
Extended form: y = Asin(Bx − C) or y = Acos(Bx − C), allowing phase shifts based on C.
Relevant in wave studies, including frequency and bandwidth discussions.
Fundamental equations like sin²θ + cos²θ = 1 are essential for simplifying expressions.
These identities are provable and necessary for operations in math.
Defined as expressions involving e and e⁻.
Similar properties to trigonometric functions including certain identities.
Common hyperbolic functions: sinh, cosh, tanh.
Defined in terms of natural logarithms:
For example, sinh⁻¹ x = ln(x + √(x² + 1)).
Functions expressed in terms of parameters (often time) for modeling.
Example: Circle defined by x = cosθ, y = sinθ shows the relation x² + y² = 1.
Expressed using parameters, resulting in curves described by x² − y² = 1.
Example: x = cosh(t), y = sinh(t) represents a unit hyperbola.