MATH1044 Chapter 1 Review Summary

Definition and Requirements of Continuity

• A function $f(x)$ is continuous at $x = a$ if and only if all three conditions are met:
  • $f(a)$ is defined.
  • $\lim_{x \rightarrow a} f(x)$ exists.
  • $\lim_{x \rightarrow a} f(x) = f(a)$.
• Discontinuities occur when any of the three conditions fail, resulting in removable, jump, or infinite discontinuities.

Graphical Analysis of Limits and Continuity

• The limit $\lim_{x \rightarrow a} f(x)$ exists if both one-sided limits match: $\lim_{x \rightarrow a^-} f(x) = \lim_{x \rightarrow a^+} f(x)$.
• Continuity requires a smooth, uninterrupted graph at $x = a$ without holes, jumps, or asymptotes.

Business and Economic Models

• Cost Function: $C(q) = \text{Fixed Costs} + (\text{Variable Cost per unit}) \times q$
• Revenue Function: $R(q) = p \times q$
• Profit Function: $P(q) = R(q) - C(q)$
• Average Cost Function: $\bar{C}(q) = \frac{C(q)}{q}$
• Break-Even Point: The quantity $q$ where total revenue equals total cost ($R(q) = C(q)$).
• Interpretation of Slope: Represents the rate of change of the output with respect to input (e.g., cost per pound of waste or annual profit increase per year).

Exponential Growth, Decay, and Financial Mathematics

• Growth and Decay Models:
  • Percentage growth model (e.g., Alphatown with population $17000$ increasing at $2.4\%$ per year): $A(t) = 17000(1 + 0.024)^t$
  • Linear growth model (e.g., Betatown with population $17000$ increasing by $240$ people per year): $B(t) = 17000 + 240t$
  • Radioactive decay half-life model: $Q(t) = Q_0 \left(\frac{1}{2}\right)^{\frac{t}{t_{1/2}}}$
  • Percentage decay model: $Q(t) = Q_0 (1 - r)^t$, such as a $20\,mg$ sample decaying at $1.5\%$ per hour: $Q(t) = 20(1 - 0.015)^t$
• Percent Change: $\text{Percent Change} = \frac{\text{New Value} - \text{Old Value}}{\text{Old Value}} \times 100\%$
• Interest Models:
  • Monthly compounding: $A(t) = P\left(1 + \frac{r}{12}\right)^{12t}$
  • Continuous compounding: $A(t) = P e^{r t}$
  • Annual Percentage Yield: $\text{APY} = \left(1 + \frac{r}{n}\right)^n - 1$

Composite Functions and Discrete Evaluation

• Function Operations:
  • Algebraic combinations: $(f \pm g)(x) = f(x) \pm g(x)$
  • Composite functions: $(f \circ g)(x) = f(g(x))$
• Evaluation of composite operations is performed using numerical coordinate tables for $f(x)$ and $g(x)$.