Calculus Readiness

Prerequisites

Content

• The meaning and proper use of the equals sign \(=\)
• Implication arrows (\(\implies\), \(\iff\)) vs. equals signs
• Working vertically to show logical algebraic progression
• Set notation (\(x \in \mathbb{R}\)) and interval notation
• Leaving answers as exact values vs. decimal approximations

Prerequisites

Content

• Distance and midpoint formulas
• Slope: \(m = \frac{\Delta y}{\Delta x}\)
• Equations of lines: Point-slope form \(y - y_1 = m(x - x_1)\) vs. slope-intercept form
• Parallel and perpendicular lines relationship
• Finding equations of secant lines

Prerequisites

Content

• Review of BEDMAS/PEMDAS: Brackets, Exponents, Division/Multiplication, Addition/Subtraction
• The fraction bar as a grouping symbol: \(\frac{x+1}{x-2}\) means \((x+1)\div(x-2)\), not \(x + \frac{1}{x} - 2\)
• Nested grouping: parentheses inside brackets, working from the inside out
• How exponents bind: \(-x^2 \neq (-x)^2\) and \(2x^3 \neq (2x)^3\)
• Distributing vs. not distributing: when the minus sign applies to one term vs. an entire expression
• Reading complex rational expressions without ambiguity: \(\frac{1}{x+h} - \frac{1}{x}\) vs. \(\frac{1}{x+h-\frac{1}{x}}\)

Prerequisites

Content

• Integer and rational exponents
• Product, quotient, and power rules
• Converting between radical and exponential forms
• Rationalizing numerators and denominators (conjugates)

Prerequisites

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• Greatest Common Factor (GCF)
• Factoring by grouping
• Difference of squares: \(a^2 - b^2 = (a-b)(a+b)\)
• Factoring quadratic trinomials (Product-Sum method)
• Factor Theorem, Long Division, and Synthetic Division
• Factoring fractional/negative exponents (e.g., \(x^{-1/2} + x^{1/2}\))

Prerequisites

Content

• Identifying domain restrictions (denominator \(\neq 0\))
• Simplifying by factoring numerators and denominators
• Multiplication and division of rational expressions
• Addition and subtraction of rational expressions: building a common denominator for algebraic (not just numeric) expressions — finding the LCD, rewriting each fraction, then combining numerators carefully
• Simplifying complex fractions (fractions within fractions)

Prerequisites

Content

• Definition of a function and the Vertical Line Test
• Evaluating functions: \(f(a)\), \(f(x+h)\)
• Function operations: \((f+g)(x)\), \((fg)(x)\)
• Composition of functions: \((f \circ g)(x) = f(g(x))\)
• Decomposing composite functions: identifying what is "inside" and what is "outside" in expressions like \(\sqrt{x^2+1}\) or \(\sin(3x^2)\)

Prerequisites

Content

• Algebraic domain restrictions: denominators (\(\neq 0\)), even roots (\(\geq 0\)), logarithmic arguments (\(> 0\))
• Combining multiple restrictions (e.g., a function with both a root and a denominator)
• Expressing domains using interval notation and inequality notation
• Finding the inverse: \(f^{-1}(x)\) by swapping \(x\) and \(y\) and solving; the fundamental property \(f(f^{-1}(x)) = x\)
• Domain of \(f\) vs. Range of \(f^{-1}\)

Prerequisites

Content

• Sketching from memory: shape, domain, range, key points, and behaviour of all 7 base functions: \(x^2, x^3, \sqrt{x}, \frac{1}{x}, |x|, e^x, \ln x\)
• Key features for each base graph: intercepts, asymptotes (if any), end behaviour, symmetry (even/odd)

Prerequisites

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• Piecewise function definition and notation
• Evaluating piecewise functions at specific values and at boundary points
• Graphing piecewise functions (open vs. closed circles at boundaries)
• Absolute value as a piecewise definition: \(|x| = x\) if \(x \geq 0\), and \(-x\) if \(x < 0\)
• Graphing absolute value functions using the piecewise definition
• Continuity at boundary points (intuitive, preparing for formal limits)

Prerequisites

Content

• Definition of logarithms: \(y = \log_b(x) \iff b^y = x\)
• The natural base \(e\) and natural logarithm \(\ln(x)\)
• Laws of logarithms: Product, Quotient, and Power rules
• Change of base formula
• Solving exponential and logarithmic equations algebraically

Prerequisites

Content

• Radians vs. degrees and conversions
• The Unit Circle: special angles (\(\pi/6, \pi/4, \pi/3, \pi/2\)) and exact values
• Sine, Cosine, Tangent and their reciprocals (Csc, Sec, Cot)
• Evaluating trig functions using reference angles (ASTC rule)
• Pythagorean identity: \(\sin^2\theta + \cos^2\theta = 1\)

Prerequisites

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• Base graphs of \(\sin x\), \(\cos x\), \(\tan x\)
• Amplitude, period \(\frac{2\pi}{|b|}\), phase shift, and vertical shift
• Applying transformations: \(a\sin(b(x-c))+d\)
• Graphing reciprocal trig functions (csc, sec, cot) from their parents
• Identifying vertical asymptotes for tangent, cotangent, secant, cosecant

Prerequisites

Content

• Pythagorean identities: \(\sin^2\theta + \cos^2\theta = 1\) and derived forms
• Quotient and reciprocal identities
• Sum and difference formulas (overview)
• Double angle formulas (\(\sin(2\theta)\) and \(\cos(2\theta)\))
• Solving linear and quadratic trigonometric equations over \([0, 2\pi)\)
• Using factoring techniques from ALG-3 on trig expressions (e.g., \(2\sin^2\theta - \sin\theta - 1 = 0\))

Prerequisites

Content

• Zero Product Property
• Solving quadratics using factoring and the quadratic formula
• The discriminant \(b^2 - 4ac\) and nature of roots
• Solving rational equations by multiplying by the LCD
• Identifying and discarding extraneous solutions

Prerequisites

Content

• Linear vs. non-linear inequalities
• Creating sign charts for polynomial and rational inequalities
• Finding critical numbers: zeros of numerator and undefined points of denominator
• Absolute value inequalities: \(|x| < a\) vs. \(|x| > a\)

Prerequisites

Content

• Sigma notation: \(\sum_{i=1}^{n} a_i\) — index, bounds, general term
• Expanding sigma notation and writing sums in sigma form
• Properties of finite sums: linearity (\(\sum(a_i + b_i)\), \(\sum c \cdot a_i\)), splitting ranges
• Standard formulas: \(\sum_{i=1}^{n} i = \frac{n(n+1)}{2}\), \(\sum_{i=1}^{n} i^2 = \frac{n(n+1)(2n+1)}{6}\)
• Telescoping sums (preview)
• Connection to calculus: sigma notation as the language of Riemann sums and series

Prerequisites

Content

• The difference quotient \(\frac{f(x+h)-f(x)}{h}\) — what it represents geometrically (slope of secant line, connecting back to COM-2)
• Evaluating \(f(x+h)\) carefully for various function types
• Simplifying the difference quotient for polynomial functions (expand, subtract, cancel \(h\))
• Simplifying for rational functions (common denominator technique from ALG-4)
• Simplifying for radical functions (rationalizing the numerator using conjugates from ALG-2)
• Simplifying for exponential functions: \(\frac{e^{x+h}-e^x}{h} = e^x \cdot \frac{e^h-1}{h}\)
• The limit idea: "what happens as \(h \to 0\)?" (intuitive preview)