📚 Complete reference for Class Test 1 and beyond
Test Scope: Ch 1.1 → 3.4 (Functions, Limits, Continuity, Differentiation, Extreme Values)
Format: Closed book — memorize this, don’t bring it!
Chapter 1: Functions
1.1 Domain & Range
Domain Rules
| Function Type | Domain Restriction | Example |
|---|---|---|
| Polynomial | All real numbers | $f(x) = x^3 - 2x + 1$ → $\mathbb{R}$ |
| Rational | Denominator ≠ 0 | $f(x) = \frac{1}{x-2}$ → $x \neq 2$ |
| Square root | Inside ≥ 0 | $f(x) = \sqrt{x-3}$ → $x \geq 3$ |
| √ in denominator | Inside > 0 (strict!) | $f(x) = \frac{1}{\sqrt{x-3}}$ → $x > 3$ |
| Logarithm | Inside > 0 | $f(x) = \ln(x-1)$ → $x > 1$ |
Multiple Restrictions
Example: $f(x) = \frac{\sqrt{x-3}}{x^2-4}$
- Square root: $x - 3 \geq 0$ → $x \geq 3$
- Denominator: $x^2 - 4 \neq 0$ → $x \neq \pm 2$
- Combine: $x \geq 3$ (already excludes ±2)
Domain: $[3, \infty)$
Tricky Case: Multiple Regions
Example: $f(x) = \sqrt{x^2 - 9}$
Need: $x^2 - 9 \geq 0$ → $x^2 \geq 9$ → $|x| \geq 3$
Domain: $(-\infty, -3] \cup [3, \infty)$ — TWO regions!
1.2 Transformations
The Golden Rule
$$\text{Inside } f() = \text{Horizontal (opposite)} \quad | \quad \text{Outside } f() = \text{Vertical (same)}$$
Transformation Table
| Transformation | Formula | Effect |
|---|---|---|
| Vertical shift up | $f(x) + k$ | Move up by $k$ |
| Vertical shift down | $f(x) - k$ | Move down by $k$ |
| Horizontal shift right | $f(x - h)$ | Move right by $h$ (opposite!) |
| Horizontal shift left | $f(x + h)$ | Move left by $h$ (opposite!) |
| Vertical stretch | $a \cdot f(x)$, $a > 1$ | Stretch vertically |
| Vertical compress | $a \cdot f(x)$, $0 < a < 1$ | Compress vertically |
| Horizontal compress | $f(bx)$, $b > 1$ | Compress horizontally (opposite!) |
| Horizontal stretch | $f(bx)$, $0 < b < 1$ | Stretch horizontally (opposite!) |
| Reflect over x-axis | $-f(x)$ | Flip vertically |
| Reflect over y-axis | $f(-x)$ | Flip horizontally |
Example
$g(x) = 2f(x-3) + 1$
Reading order (inside out):
- $x - 3$: shift RIGHT 3 (horizontal, opposite)
- $2f(…)$: stretch vertically by 2
- $… + 1$: shift UP 1
1.3 Function Types
Classification
| Type | Definition | Examples |
|---|---|---|
| Polynomial | $a_nx^n + a_{n-1}x^{n-1} + … + a_0$ | $x^3 - 2x + 1$ |
| Rational | $\frac{P(x)}{Q(x)}$ where P, Q are polynomials | $\frac{x^2+1}{x-1}$ |
| Algebraic | Built from polynomials + roots | $\sqrt{x^2 + 1}$ |
| Transcendental | NOT algebraic (trig, exp, log) | $\sin x$, $e^x$, $\ln x$ |
Rule: Any transcendental part → whole function is transcendental
Example: $f(x) = x^2 + \sin x$ → Transcendental (has $\sin x$)
1.4 Special Functions
Inverse Functions
$$f^{-1}(f(x)) = x \quad \text{and} \quad f(f^{-1}(x)) = x$$
| Function | Inverse | Domain of Inverse |
|---|---|---|
| $e^x$ | $\ln x$ | $x > 0$ |
| $\ln x$ | $e^x$ | $\mathbb{R}$ |
| $\sin x$ | $\arcsin x$ | $[-1, 1]$ |
| $\cos x$ | $\arccos x$ | $[-1, 1]$ |
| $\tan x$ | $\arctan x$ | $\mathbb{R}$ |
⚠️ Ch1 Common Mistakes
| Mistake | Wrong | Correct |
|---|---|---|
| √ in denominator | $x \geq 3$ | $x > 3$ (strict inequality!) |
| Horizontal shift direction | $f(x-3)$ shifts left | $f(x-3)$ shifts RIGHT (opposite!) |
| $\sqrt{x^2-9}$ domain | $x \geq 3$ only | $x \leq -3$ OR $x \geq 3$ (two regions!) |
| Numerator = 0 | Not allowed | $\frac{0}{5} = 0$ is fine! |
Chapter 2: Limits & Continuity
2.1 Limit Laws
Basic Laws
If $\lim_{x \to a} f(x) = L$ and $\lim_{x \to a} g(x) = M$, then:
| Law | Formula | Condition |
|---|---|---|
| Sum | $\lim[f + g] = L + M$ | — |
| Difference | $\lim[f - g] = L - M$ | — |
| Product | $\lim[f \cdot g] = L \cdot M$ | Both limits must be finite |
| Quotient | $\lim\frac{f}{g} = \frac{L}{M}$ | $M \neq 0$ |
| Power | $\lim[f^n] = L^n$ | — |
| Constant | $\lim[c \cdot f] = c \cdot L$ | — |
Division Patterns
| Form | Result | Intuition |
|---|---|---|
| $\frac{5}{0}$ | $\pm\infty$ (blows up 🚀) | Dividing by tiny number → huge |
| $\frac{0}{5}$ | $0$ | Zero divided by anything = 0 |
| $\frac{0}{0}$ | Indeterminate (不定式 🤔) | Could be anything — must simplify! |
| $\frac{\infty}{\infty}$ | Indeterminate | Must simplify |
| $0 \cdot \infty$ | Indeterminate | Must rewrite |
2.2 Indeterminate Forms
The 7 Indeterminate Forms
$$\frac{0}{0}, \quad \frac{\infty}{\infty}, \quad 0 \cdot \infty, \quad \infty - \infty, \quad 0^0, \quad 1^\infty, \quad \infty^0$$
Solving $\frac{0}{0}$
Method 1: Factor and Cancel
$$\lim_{x \to 2} \frac{x^2 - 4}{x - 2} = \lim_{x \to 2} \frac{(x-2)(x+2)}{x-2} = \lim_{x \to 2} (x+2) = 4$$
Method 2: Conjugate (for radicals)
$$\lim_{x \to 0} \frac{\sqrt{x+4} - 2}{x} = \lim_{x \to 0} \frac{(\sqrt{x+4} - 2)(\sqrt{x+4} + 2)}{x(\sqrt{x+4} + 2)} = \lim_{x \to 0} \frac{x+4-4}{x(\sqrt{x+4} + 2)} = \frac{1}{4}$$
2.3 Limits at Infinity
Degree Rules for Rational Functions
$$\lim_{x \to \infty} \frac{a_nx^n + …}{b_mx^m + …}$$
| Condition | Result | Memory Trick |
|---|---|---|
| $n < m$ (top smaller) | $0$ | Top “loses” |
| $n = m$ (same degree) | $\frac{a_n}{b_m}$ | Leading coefficients |
| $n > m$ (top bigger) | $\pm\infty$ | Top “wins” |
Example
$$\lim_{x \to \infty} \frac{3x^2 + 1}{x^2 - 5} = \frac{3}{1} = 3 \quad \text{(same degree)}$$
$$\lim_{x \to \infty} \frac{2x + 1}{x^3 - 1} = 0 \quad \text{(top smaller)}$$
Special Trig Limits
$$\lim_{x \to 0} \frac{\sin x}{x} = 1 \qquad \lim_{x \to 0} \frac{1 - \cos x}{x} = 0$$
2.4 Continuity
3-Part Test for Continuity at $x = a$
A function is continuous at $x = a$ if:
- $f(a)$ exists (defined at the point)
- $\lim_{x \to a} f(x)$ exists (limit exists)
- $\lim_{x \to a} f(x) = f(a)$ (they match)
4 Types of Discontinuity
| Type | Condition | Visual |
|---|---|---|
| Removable (Hole) | $\lim_{x \to a^-} f = \lim_{x \to a^+} f \neq f(a)$ | Single point missing |
| Jump | $\lim_{x \to a^-} f \neq \lim_{x \to a^+} f$ | Left ≠ Right |
| Infinite | $\lim_{x \to a} f = \pm\infty$ | Vertical asymptote |
| Oscillatory | Limit doesn’t exist (oscillates) | e.g., $\sin(1/x)$ at 0 |
Hole Detection
- WHERE: Check ORIGINAL function for where denominator = 0
- WHAT VALUE: Simplify, then substitute to find the y-coordinate of hole
2.5 Intermediate Value Theorem (IVT)
Statement
If $f$ is continuous on $[a, b]$ and $N$ is between $f(a)$ and $f(b)$, then there exists $c \in (a, b)$ such that $f(c) = N$.
Root Finding Application
If $f$ is continuous on $[a, b]$ with $f(a) < 0$ and $f(b) > 0$ (or vice versa), then there exists a root in $(a, b)$.
Intuition: “Can’t teleport through floors” — continuous function must cross zero.
Example
Show $f(x) = x^3 - x - 1$ has a root in $[1, 2]$.
- $f(1) = 1 - 1 - 1 = -1 < 0$
- $f(2) = 8 - 2 - 1 = 5 > 0$
- $f$ is polynomial (continuous everywhere)
- By IVT, there exists $c \in (1, 2)$ where $f(c) = 0$ ✓
⚠️ Ch2 Common Mistakes
| Mistake | Wrong | Correct |
|---|---|---|
| $5/0$ | “undefined” or “0” | $\pm\infty$ (blows up!) |
| $0/0$ | “0” | Indeterminate — must simplify! |
| Product law with $\infty$ | $0 \cdot \infty = 0$ | Indeterminate — can’t use product law! |
| Hole vs Jump | Both “don’t match” | Hole: L=R≠f(a), Jump: L≠R |
| Divide by highest power | Always do it | Only for $x \to \infty$, not $x \to a$ |
Chapter 3: Differentiation
3.1 Derivative Definition
$$f’(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}$$
Interpretation
- Geometric: Slope of tangent line at $(x, f(x))$
- Physical: Instantaneous rate of change
Tangent Line Equation
At point $(a, f(a))$:
$$y - f(a) = f’(a)(x - a)$$
3.2 Basic Differentiation Rules
Power Rule
$$\frac{d}{dx}[x^n] = nx^{n-1} \quad \text{for ALL } n \in \mathbb{R}$$
Examples:
$$\frac{d}{dx}[x^5] = 5x^4 \qquad \frac{d}{dx}[x^{-2}] = -2x^{-3} \qquad \frac{d}{dx}[\sqrt{x}] = \frac{d}{dx}[x^{1/2}] = \frac{1}{2}x^{-1/2}$$
Constant Rule
$$\frac{d}{dx}[c] = 0 \qquad \frac{d}{dx}[cf(x)] = c \cdot f’(x)$$
Sum/Difference Rule
$$\frac{d}{dx}[f \pm g] = f’ \pm g’$$
Product Rule
$$\frac{d}{dx}[f \cdot g] = f’g + fg’$$
Example:
$$\frac{d}{dx}[x^2 \sin x] = 2x \cdot \sin x + x^2 \cdot \cos x$$
Quotient Rule
$$\frac{d}{dx}\left[\frac{f}{g}\right] = \frac{f’g - fg’}{g^2}$$
Memory: “Lo d-Hi MINUS Hi d-Lo, over Lo-Lo” (low × d(high) − high × d(low))
Example:
$$\frac{d}{dx}\left[\frac{x^2}{x+1}\right] = \frac{2x(x+1) - x^2(1)}{(x+1)^2} = \frac{2x^2 + 2x - x^2}{(x+1)^2} = \frac{x^2 + 2x}{(x+1)^2}$$
Chain Rule
$$\frac{d}{dx}[f(g(x))] = f’(g(x)) \cdot g’(x)$$
Memory: “Derivative of outside (keep inside) × Derivative of inside”
Example:
$$\frac{d}{dx}[\sin(x^2)] = \cos(x^2) \cdot 2x = 2x\cos(x^2)$$
Triple Product Rule
$$\frac{d}{dx}[fgh] = f’gh + fg’h + fgh’$$
Example:
$$\frac{d}{dx}[x^2 \cos x \cdot e^x] = 2x \cos x \cdot e^x + x^2(-\sin x) e^x + x^2 \cos x \cdot e^x$$
Trig Derivatives
The 6 Trig Derivatives
| Function | Derivative | Memory |
|---|---|---|
| $\sin x$ | $\cos x$ | Sin → Cos (positive) |
| $\cos x$ | $-\sin x$ | Cos → negative Sin |
| $\tan x$ | $\sec^2 x$ | Squared family |
| $\cot x$ | $-\csc^2 x$ | Squared family (CO = negative) |
| $\sec x$ | $\sec x \tan x$ | Multiply family |
| $\csc x$ | $-\csc x \cot x$ | Multiply family (CO = negative) |
Pattern Families
Squared Family: tan and cot give squared derivatives $$\frac{d}{dx}[\tan x] = \sec^2 x \qquad \frac{d}{dx}[\cot x] = -\csc^2 x$$
Multiply Family: sec and csc multiply with their buddy $$\frac{d}{dx}[\sec x] = \sec x \tan x \qquad \frac{d}{dx}[\csc x] = -\csc x \cot x$$
CO = NEGATIVE: cot and csc always have negative signs!
With Chain Rule
$$\frac{d}{dx}[\sin(u)] = \cos(u) \cdot u’ \qquad \frac{d}{dx}[\tan(u)] = \sec^2(u) \cdot u’$$
Example:
$$\frac{d}{dx}[\sin(3x)] = \cos(3x) \cdot 3 = 3\cos(3x)$$
$$\frac{d}{dx}[\cos^2(x)] = 2\cos(x) \cdot (-\sin x) = -2\cos(x)\sin(x)$$
3.3 Inverse Functions & Special Derivatives
Exponential
$$\frac{d}{dx}[e^x] = e^x \qquad \frac{d}{dx}[e^u] = e^u \cdot u’$$
Memory: $e^x$ is “immortal” — derivative can’t kill it!
Examples:
$$\frac{d}{dx}[e^{3x}] = e^{3x} \cdot 3 = 3e^{3x}$$
$$\frac{d}{dx}[e^{x^2}] = e^{x^2} \cdot 2x = 2xe^{x^2}$$
Logarithm
$$\frac{d}{dx}[\ln x] = \frac{1}{x} \qquad \frac{d}{dx}[\ln u] = \frac{u’}{u}$$
Memory: ln “flips” — becomes 1/x
Examples:
$$\frac{d}{dx}[\ln(x^2)] = \frac{2x}{x^2} = \frac{2}{x}$$
Shortcut: $\ln(x^2) = 2\ln(x)$ → $\frac{d}{dx} = 2 \cdot \frac{1}{x} = \frac{2}{x}$
$$\frac{d}{dx}[\ln(\sin x)] = \frac{\cos x}{\sin x} = \cot x$$
Implicit Differentiation
When to Use
When you can’t (or don’t want to) solve for $y$ explicitly.
Method
- Differentiate both sides with respect to $x$
- For any $y$ term, use chain rule: $\frac{d}{dx}[f(y)] = f’(y) \cdot \frac{dy}{dx}$
- Collect all $\frac{dy}{dx}$ terms on one side
- Solve for $\frac{dy}{dx}$
Example 1: Circle
$$x^2 + y^2 = 25$$
Differentiate: $$2x + 2y\frac{dy}{dx} = 0$$
Solve: $$\frac{dy}{dx} = -\frac{x}{y}$$
Example 2: Product with y
$$x^2 + xy + y^2 = 7$$
Differentiate: $$2x + \left(1 \cdot y + x \cdot \frac{dy}{dx}\right) + 2y\frac{dy}{dx} = 0$$
Note: For $xy$, use product rule! $\frac{d}{dx}[xy] = 1 \cdot y + x \cdot \frac{dy}{dx}$
Solve: $$2x + y + x\frac{dy}{dx} + 2y\frac{dy}{dx} = 0$$ $$\frac{dy}{dx}(x + 2y) = -2x - y$$ $$\frac{dy}{dx} = \frac{-2x - y}{x + 2y}$$
Example 3: Trig
$$\sin(x) + \cos(y) = 1$$
Differentiate: $$\cos(x) + (-\sin(y)) \cdot \frac{dy}{dx} = 0$$
Solve: $$\frac{dy}{dx} = \frac{\cos(x)}{\sin(y)}$$
3.4 Extreme Values
Critical Points
A critical point of $f$ is where:
- $f’(c) = 0$, OR
- $f’(c)$ does not exist (but $f(c)$ exists)
Extreme Value Theorem (EVT)
If $f$ is continuous on closed bounded interval $[a, b]$, then $f$ has both an absolute maximum and minimum on $[a, b]$.
Closed Interval Method
To find absolute max/min of $f$ on $[a, b]$:
- Find all critical points in $(a, b)$
- Evaluate $f$ at critical points AND endpoints $a, b$
- Largest value = absolute max, smallest = absolute min
Example
Find absolute max/min of $f(x) = x^3 - 6x^2 + 9x + 2$ on $[-1, 4]$.
Step 1: Find critical points $$f’(x) = 3x^2 - 12x + 9 = 3(x^2 - 4x + 3) = 3(x-1)(x-3)$$ $$f’(x) = 0 \Rightarrow x = 1, 3$$
Step 2: Evaluate at critical points and endpoints
| $x$ | $f(x)$ |
|---|---|
| $-1$ | $(-1)^3 - 6(-1)^2 + 9(-1) + 2 = -1 - 6 - 9 + 2 = -14$ |
| $1$ | $1 - 6 + 9 + 2 = 6$ |
| $3$ | $27 - 54 + 27 + 2 = 2$ |
| $4$ | $64 - 96 + 36 + 2 = 6$ |
Step 3: Compare
- Absolute max: $6$ at $x = 1$ and $x = 4$
- Absolute min: $-14$ at $x = -1$
Increasing/Decreasing Intervals
| $f’(x)$ | $f(x)$ is… |
|---|---|
| $f’(x) > 0$ | Increasing ↗ |
| $f’(x) < 0$ | Decreasing ↘ |
| $f’(x) = 0$ | Potential local max/min |
First Derivative Test
At critical point $c$:
- $f’$ changes from + to − → Local maximum
- $f’$ changes from − to + → Local minimum
- $f’$ doesn’t change sign → Neither (inflection point)
Concavity (Second Derivative)
| $f’’(x)$ | $f(x)$ is… |
|---|---|
| $f’’(x) > 0$ | Concave up ∪ |
| $f’’(x) < 0$ | Concave down ∩ |
| $f’’(x) = 0$ | Potential inflection point |
Inflection Points
Where concavity changes (f’’ changes sign).
⚠️ Ch3 Common Mistakes
My 11 Errors from Mixed Derivatives Hell!
| # | Error | Fix |
|---|---|---|
| 3 | Read $\tan(x^3)$ as $\tan^3(x)$ | Read notation TWICE! $\tan(x^3) \neq [\tan(x)]^3$ |
| 9 | Wrote $-\csc(2x)$ instead of $-\csc^2(2x)$ | $\frac{d}{dx}[\cot u] = -\csc^2(u) \cdot u’$ — SQUARED! |
| 10 | $u = x^3 + x \to u’ = 3x^2$ (forgot +1) | Differentiate EVERY term: $u’ = 3x^2 + 1$ |
| 12 | Used $\tan(x)$ instead of $\sec^2(x)$ | Product rule: check you’re using $v’$, not $v$ |
| 13 | Quotient rule with + instead of − | “Lo d-Hi MINUS Hi d-Lo” — always MINUS! |
| 14 | $\cos^3(2x)$: got $6\cos^2(2x)$ | Missing $-\sin(2x)$! Complete ALL chain layers |
| 17 | Triple product got messy | Simplify first: $\sin x \cos x \tan x = \sin^2 x$ |
| 20 | Got $(x - 2y)$ instead of $(x + 2y)$ | Collect terms carefully, check signs! |
| 24 | Inner deriv: $(3x^2 - x)$ not $(3x^2 - 2)$ | $\frac{d}{dx}[-2x] = -2$, not $-x$ |
| 26 | Read $\sin(x^2) \cdot \cos(x^2)$ as division | Read · vs / carefully! |
| 28 | $\sqrt{\sin x}$: wrote $\frac{1}{2}\sin(x)\cos(x)$ | Power drops: $[\sin x]^{1/2} \to \frac{1}{2}[\sin x]^{-1/2}$ |
General Error Prevention Protocol
Before each problem (10 seconds):
- ✓ Read the problem TWICE
- ✓ Identify which rules needed
- ✓ Count chain rule layers
After each problem (5 seconds):
- ✓ Did I complete ALL chain rule layers?
- ✓ Did I use $v’$ not $v$?
- ✓ Are my signs correct?
Quick Reference Card
All Derivatives in One Table
| Function | Derivative |
|---|---|
| $c$ (constant) | $0$ |
| $x^n$ | $nx^{n-1}$ |
| $e^x$ | $e^x$ |
| $\ln x$ | $\frac{1}{x}$ |
| $\sin x$ | $\cos x$ |
| $\cos x$ | $-\sin x$ |
| $\tan x$ | $\sec^2 x$ |
| $\cot x$ | $-\csc^2 x$ |
| $\sec x$ | $\sec x \tan x$ |
| $\csc x$ | $-\csc x \cot x$ |
Rules
| Rule | Formula |
|---|---|
| Product | $(fg)’ = f’g + fg'$ |
| Quotient | $\left(\frac{f}{g}\right)’ = \frac{f’g - fg’}{g^2}$ |
| Chain | $[f(g(x))]’ = f’(g(x)) \cdot g’(x)$ |
| Triple | $(fgh)’ = f’gh + fg’h + fgh'$ |
Trig Identities (Simplify First!)
| Expression | Simplifies To |
|---|---|
| $\tan x$ | $\frac{\sin x}{\cos x}$ |
| $\cot x$ | $\frac{\cos x}{\sin x}$ |
| $\sec x$ | $\frac{1}{\cos x}$ |
| $\csc x$ | $\frac{1}{\sin x}$ |
| $\sin x \cos x$ | $\frac{1}{2}\sin(2x)$ |
| $\sin^2 x + \cos^2 x$ | $1$ |
Limit Patterns
| Form | Result |
|---|---|
| $\frac{n}{0}$ | $\pm\infty$ |
| $\frac{0}{n}$ | $0$ |
| $\frac{0}{0}$ | Indeterminate — simplify! |
- Created: 2026-02-05
- Coverage: TCX2101 Ch 1.1 - 3.4
- Purpose: 宝典 for Class Test 1 and beyond
Source: NUS TCX2101 Calculus and Linear Algebra, prepared during CT1 prep week
GitHub: Session notes