Tags: #◇Computer Science #◇CSMATH1 #◇Differential Calculus #◇Reviewer #◇Finals

CSMATH1 Comprehensive Finals Reviewer

◇Overview

This reviewer consolidates the core concepts of Differential Calculus (CSMATH1): formal limits and continuity, differentiation rules, implicit differentiation, related rates of change, curve sketching (critical points, concavity, inflection), optimization problems, and Mean Value Theorem.

◇1. Fundamental Limit Theorems & Differentiation Rules

Rule Name	Formula	Notes
Limit of Indeterminate Forms	$\lim_{x \to c} \frac{f(x)}{g(x)} = \lim_{x \to c} \frac{f'(x)}{g'(x)}$	Applies ONLY when form is $\frac{0}{0}$ or $\frac{\pm \infty}{\pm \infty}$ (L'Hôpital's Rule)
Power Rule	$\frac{d}{dx}[x^n] = n x^{n-1}$	Holds for all real numbers $n$
Product Rule	$\frac{d}{dx}[u v] = u v' + v u'$	Both functions differentiated alternately
Quotient Rule	$\frac{d}{dx}\left[\frac{u}{v}\right] = \frac{v u' - u v'}{v^2}$	"Low d-High minus High d-Low over Low squared"
Chain Rule	$\frac{d}{dx}[f(g(x))] = f'(g(x)) \cdot g'(x)$	Composite function inner derivative factor
Implicit Differentiation	$\frac{d}{dx}[y^n] = n y^{n-1} \frac{dy}{dx}$	Treats $y$ as an implicit function of $x$

◇2. Curve Sketching & Derivative Tests

Critical Points: Points $c$ in the domain of $f$ where $f'(c) = 0$ or $f'(c)$ does not exist.
First Derivative Test:
- $f'$ changes from $+$ to $-$ at $c \implies$ Local Maximum.
- $f'$ changes from $-$ to $+$ at $c \implies$ Local Minimum.
Second Derivative Test:
- If $f'(c) = 0$ and $f''(c) > 0 \implies$ Concave Up $\implies$ Local Minimum.
- If $f'(c) = 0$ and $f''(c) < 0 \implies$ Concave Down $\implies$ Local Maximum.
- If $f''(c) = 0$ , the test is inconclusive.
Point of Inflection: A point where the concavity changes sign ( $f''$ changes sign) and the tangent line exists.

◇3. High-Yield Geometric Optimization Formulas

Open-Top Box: Sheet $L \times W$ , cut squares of side $x$ : $V(x) = x(L - 2x)(W - 2x)$ .
Cylindrical Can (Minimal Surface Area for Fixed Volume $V$ ):
$A = 2\pi r^2 + 2\pi r h = 2\pi r^2 + \frac{2V}{r} \implies \frac{dA}{dr} = 4\pi r - \frac{2V}{r^2} = 0 \implies h = 2r$
Optimal cylinder has height equal to diameter!

◇4. High-Yield Exam Checklist

Before applying L'Hôpital's Rule, did you check that it evaluates strictly to $\frac{0}{0}$ or $\frac{\infty}{\infty}$ ?
For related rates, did you differentiate with respect to time $t$ using the Chain Rule (e.g. $\frac{dr}{dt}$ )?
Did you check endpoints on closed intervals $[a, b]$ for Absolute Extrema (Extreme Value Theorem)?
For implicit differentiation, did you collect all $\frac{dy}{dx}$ terms to one side before factoring?
Is continuity guaranteed if a function is differentiable? YES (Differentiability $\implies$ Continuity).

Previous Topic: None
Next Topic: ◇CSMATH2
Prerequisites: ◇Differential Calculus, Limits, Derivatives
Cross-Curricular Connections: ◇STT101A

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