General Mathematics (MATH01) – Comprehensive Bullet-Point Study Notes

Course Overview

• Course Code: MATH01 – General Mathematics

• Credit: 1 Unit

• Prerequisite/Co-requisite: None

• Global framing

  • Anchored on Mapúa University’s Vision (socio-economic growth via innovation) and Mission (global competitiveness, research, industry solutions).

• End-goal: Equip learners to model and solve real–life problems with rational, exponential, logarithmic, financial and sequence/series tools.

Course Outcomes (CO)

• CO1: Illustrate & manipulate functions and inverse functions comprehensively.

• CO2: Solve real-life problems with rational functions/equations.

• CO3: Apply exponential & logarithmic functions/equations in context.

• CO4: Use sequences, series & binomial expansion for applications.

• CO5: Evaluate simple & general annuities plus simple/compound interest with business instruments.

• CO6: Judge appropriateness & proper use of stocks, bonds and consumer/business loans.

Modular Design & Weekly Flow (Outcome-Based)

• Each module = cluster of weeks + performance task + written work + quarterly assessment.

• Digital tools: ALEKS, Cengage WebAssign, GeoGebra, Zoom recordings.

• Assessment codes: WW (Written Work), PT (Performance Task), QA (Quarterly Assessment).

• All modules delivered via mixed Guided Learning Lectures, classroom/F2F, Zoom & asynchronous outputs.

Module 1 – Functions (Weeks 0-3, CO1)

• Review core skills

• Laws of Exponents

• Special products & factoring patterns.
  • Relations vs. Functions (definition, vertical-line test).
  • Notation: $f(x),\,g:x\mapsto …$
  • Function Types & graphs

• Constant, Linear, Quadratic, Absolute-value, Piecewise.
  • Domain & Range (algebraic and graphical extraction).
  • Operations on Functions: $\big(f\pm g\big)(x),\,(fg)(x),\,\left(\frac f g\right)(x)$
  • Function Evaluation: numeric & algebraic inputs.
  • Applications: cost, revenue, temperature conversion, projectile motion, etc.
  • One-to-One criterion (horizontal-line test) → ensures inverses.
  • Inverse Functions

• Definition: $f^{-1}(f(x))=x$ for all $x$ in domain of $f$.

• Algebraic derivation (swap x & y then solve).

• Graphical symmetry about $y=x$.

• Domain–range interchange.

• Real-world: undoing unit conversions, decoding encryption keys.

Module 2 – Rational Equations & Functions (Weeks 3-5, CO2)

• Rational function: form $R(x)=\dfrac{P(x)}{Q(x)}$ where $P, Q$ are polynomials & $Q\neq0$.
• Domain: exclude roots of $Q(x)=0$.
• Intercepts & zeroes

• $x$-intercepts: roots of $P(x)=0$ not canceled.

• $y$-intercept: $R(0)$ if defined.
  • Asymptotes

• Vertical: zeros of $Q(x)$.

• Horizontal: compare degrees; slant (oblique) via polynomial division.
  • Graphing strategy: combine intercepts, asymptotes, sign analysis.
  • Inverse of a rational function (when 1-1): may require restriction of domain.
  • Rational equations: clear denominators → solve polynomial → reject extraneous roots.
  • Applications:

• Combined work problems, average speed, mixture concentration, economic break-even scenarios.

Module 3 – Exponential & Logarithmic Functions / Equations (Weeks 5-9, CO3)

Exponential Functions

• Definition: f(x)=ab^{x},\;a\neq0,\;b>0,\;b\neq1.

• Domain $(-\infty,\infty)$; Range $(0,\infty)$ if a>0.

• Key features: $y$-intercept $(0,a)$, horizontal asymptote $y=0$.

• Transformations: $f(x)=ab^{k(x-h)}+c$ gives
  • Horizontal shift $h$, vertical shift $c$, stretch/compression factor $k$, reflection if a<0.

• Exponential equations solved via common bases or logarithms.

• Applications: population growth $P(t)=P_0e^{rt}$, radioactive decay, continuously compounded interest $A=Pe^{rt}$.

Logarithmic Functions

• Definition: $g(x)=\log_b(x)$ is inverse of $b^{x}$.

• Domain $(0,\infty)$; Range $(-\infty,\infty)$.

• Key points: $x$-intercept $(1,0)$, vertical asymptote $x=0$.

• Laws of Logs
  • Product: $\log<em>b(MN)=\log</em>bM+\log<em>bN$ • Quotient: $\log</em>b\dfrac MN=\log<em>bM-\log</em>bN$
  • Power: $\log<em>b(M^{k})=k\log</em>bM$
  • Change of base: $\log<em>bM=\dfrac{\log</em>kM}{\log_kb}$.

• Logarithmic equations solved via exponentiation or properties above.

• Real-world: pH scale ($\text{pH}=-\log<em>{10}[H^+]$), Richter magnitude, information entropy $H=-\sum p</em>i\log<em>2p</em>i$.

Module 4 – Sequences & Series (Weeks 10-11, CO4)

• Sequence: ordered list (a1,a2,\dots); Series: sum $S<em>n=\sum</em>{k=1}^na<em>k$. • Sigma notation $\sum</em>{k=m}^n a_k$.
• Arithmetic sequence

• Formula: $a<em>n=a</em>1+(n-1)d$.

• Partial sum: $S<em>n=\dfrac n2(2a</em>1+(n-1)d)$.
  • Geometric sequence

• Formula: $a<em>n=a</em>1r^{n-1}$.

• Partial sum (finite): $S<em>n=a</em>1\dfrac{1-r^{n}}{1-r},\;r\neq1$.

• Infinite sum if |r|<1: $S<em>{\infty}=\dfrac{a</em>1}{1-r}$.
  • Harmonic sequence: reciprocal of arithmetic; harmonic mean use.
  • Binomial Theorem

• Expansion: $(x+y)^n=\sum_{k=0}^{n}\binom{n}{k}x^{n-k}y^{k}$.

• $r^{\text{th}}$ term: $T_{r+1}=\binom n r x^{n-r}y^{r}$.
  • Applications: amortization scheduling, probability modeling (Bernoulli trials), network routing paths.

Module 5 – Interest (Weeks 12-13, CO5)

• Simple Interest

• Interest: $I=Prt$.

• Maturity value: $F=P(1+rt)$.

• Exact (365-day) vs. Ordinary (360-day) time.

• Simple Discount: $D=Fdt$, proceeds $P=F-D$.
  • Compound Interest

• Periodic compounding: $A=P\left(1+\dfrac rn\right)^{nt}$.

• Continuous: $A=Pe^{rt}$.

• Nominal vs. Effective annual rate: $i_{\text{eff}}=\left(1+\dfrac r n\right)^{n}-1$.
  • Time-Value-of-Money diagrams, cash-flow equations of value using focal dates.
  • Annual Percentage Yield (APY) calculation for consumer comparison.

Module 6 – Annuities, Sinking Fund & Amortization (Weeks 14-16, CO6)

• Annuity: stream of equal payments at regular intervals.

• Occurrence types: Ordinary (end-of-period), Due (beginning), Deferred.
  • Future value of ordinary annuity: $F=R\dfrac{(1+i)^n-1}{i}$.
  • Present value of ordinary annuity: $P=R\dfrac{1-(1+i)^{-n}}{i}$.
  • Sinking fund: periodic deposits to accumulate a target amount; same formulas as FV.
  • Amortization

• Loan repayment schedule; each payment covers interest $I=P_{\text{prev}}i$ plus principal reduction.

• Amortization table columns: Period, Payment, Interest, Principal, Balance.

  • Practical evaluation of consumer loans, mortgages, bonds using above models.

Assessment & Grading Structure

• Two grading periods each worth 50 % of final grade.

• Each period:
  • Performance Tasks (PT) ≈ 50 % total weight (PT1–PT6).
  • Written Works (WW) ≈ 25 %.
  • Quarterly Assessment (QA) ≈ 25 %.

• Minimum satisfactory average: 60 % raw.

• Detailed grade equivalents (100 → 60) provided for transparency.

• Pass criterion for individual tasks: ≥ 20 % of weight.

Learning Resources

• Prescribed e-books / courseware
  • Cengage WebAssign: “Algebra & Trigonometry” by Larson.
  • Wiley eText: “Algebra & Trigonometry” by Young.
  • McGraw-Hill ALEKS adaptive course.
  • Cengage: “Contemporary Mathematics for Business & Consumers”.

• Supplementary texts
  • Young, Algebra & Trigonometry 4e.
  • Alhabeeb, Mathematical Finance.
  • Connally, Functions Modeling Change 5e.

• Instructor-provided: syllabus, PowerPoints, lesson modules, micro-videos, Zoom recordings.

Course Policies & Ethics

• Attendance: > 16 h absences ⇒ automatic failing mark.
• English communication required; language quality affects grades.
• Course Portfolio: submit lowest/median/highest marked outputs in PDF to instructor.
• Dress & Grooming: follow university code.
• Academic Integrity

• Strict ban on plagiarism, unauthorized login, exam leaks, contract cheating, massive pre-meditated online collusion.

• Sanctions range: zero on task → expulsion.
  • AI-Tool Policy

• Instructors state allowed uses; similarity index ≤ 15 % via Turnitin.

• Students may use AI for quality check but not full content generation; attribution mandatory.

• Misuse = academic dishonesty.
  • Appeal system: one-week window per decision; escalates to Program Chair/Dean whose ruling is final.

Lifelong Learning & Professional Relevance

• Builds quantitative literacy essential for STEM, business, finance & data-driven fields.

• Enhances analytical thinking, decision-making, and adaptability to technological innovation (AI, digital finance).

• Prepares students for higher tertiary mathematics, actuarial studies, engineering problem-solving, entrepreneurial valuation.

Quick Formula Reference (cheat-sheet)

• Simple Interest: $I=Prt$
• Compound Interest: $A=P\left(1+\dfrac rn\right)^{nt}$; continuous $A=Pe^{rt}$
• Simple Discount: $D=Fdt$
• Arithmetic $a<em>n=a</em>1+(n-1)d$, $S<em>n=\dfrac n2(2a</em>1+(n-1)d)$
• Geometric $a<em>n=a</em>1r^{n-1}$, $S<em>n=a</em>1\dfrac{1-r^{n}}{1-r}$
• Future value annuity: $F=R\dfrac{(1+i)^n-1}{i}$
• Present value annuity: $P=R\dfrac{1-(1+i)^{-n}}{i}$
• Binomial: $(x+y)^n=\sum<em>{k=0}^{n}\binom n k x^{n-k}y^{k}$ • Change of base: $\log</em>bM=\dfrac{\log<em>kM}{\log</em>kb}$
• Effective rate: $i_{\text{eff}}=\left(1+\dfrac r n\right)^{n}-1$

End of organized study notes. Replace reading of the 14-page syllabus with this concise yet detailed reference.