📌 Topic 1
Functions & Their Properties
A function maps each input to exactly one output. Key tests: Vertical Line Test (graph), one-to-one check for inverses.
Domain of f(x) = √(x−a): x ≥ a | Domain of 1/f(x): f(x) ≠ 0
💡Memorize: Composition (f∘g)(x) = f(g(x)) — always work inside-out. Inverse: swap x,y then solve for y.
▶ Quick Example
If f(x) = 2x+1 and g(x) = x², find (f∘g)(3).
✓ Answer: f(g(3)) = f(9) = 19
📌 Topic 2
Polynomials & Factoring
The Rational Root Theorem states that any rational root p/q has p | constant term and q | leading coefficient.
Remainder Theorem: f(a) = remainder when f(x) ÷ (x−a)
Factor Theorem: (x−a) is a factor ⟺ f(a) = 0
💡Memorize: Sum of roots = −b/a, Product of roots = c/a (for ax²+bx+c).
▶ Quick Example
Is (x−2) a factor of f(x) = x³ − 3x² + 4?
✓ f(2) = 8 − 12 + 4 = 0, so yes.
📌 Topic 3
Rational Functions & Asymptotes
Vertical asymptotes occur where denominator = 0 (and numerator ≠ 0). Horizontal asymptotes depend on degrees.
deg(num) < deg(den): HA at y=0
deg(num) = deg(den): HA at y = leading coefficients ratio
deg(num) > deg(den): no HA (oblique asymptote)
💡Memorize: A hole occurs when a factor cancels from both numerator and denominator.
▶ Quick Example
Find HA of f(x) = (3x²+1)/(x²−4).
✓ Same degree → HA: y = 3/1 = 3
📌 Topic 4
Exponential & Logarithmic Functions
Logs and exponentials are inverses. The natural log ln and base-e exponential are foundational in calculus.
log_b(xy) = log_b(x)+log_b(y) | log_b(x/y) = log_b(x)−log_b(y)
log_b(x^n) = n·log_b(x) | Change of base: log_b(x) = ln(x)/ln(b)
💡Memorize: b^(log_b x) = x and log_b(b^x) = x — they cancel each other.
▶ Quick Example
Solve: log₂(x) + log₂(4) = 5
✓ log₂(4x) = 5 → 4x = 32 → x = 8
📌 Topic 5
Trigonometry & Unit Circle
The unit circle defines sin and cos for all angles. Key identities must be memorized for exam success.
sin²θ + cos²θ = 1 | tan θ = sin θ/cos θ
sin(A±B) = sinA cosB ± cosA sinB
cos(A±B) = cosA cosB ∓ sinA sinB
sin(2θ) = 2 sinθ cosθ | cos(2θ) = cos²θ − sin²θ
💡Memorize: Special angles: sin30°=½, sin45°=√2/2, sin60°=√3/2. "All Students Take Calculus" — signs by quadrant (All+, Sin+, Tan+, Cos+).
▶ Quick Example
Find sin(2θ) if sinθ = 3/5 and θ is in Q1.
✓ cosθ = 4/5 → sin(2θ) = 2(3/5)(4/5) = 24/25
📌 Topic 6
Sequences & Series
Arithmetic sequences have constant differences; geometric sequences have constant ratios.
Arithmetic: aₙ = a₁ + (n−1)d | Sₙ = n(a₁+aₙ)/2
Geometric: aₙ = a₁·rⁿ⁻¹ | Sₙ = a₁(1−rⁿ)/(1−r)
Infinite Geometric Sum (|r|<1): S∞ = a₁/(1−r)
💡Memorize: For infinite series to converge, |r| must be strictly less than 1.
▶ Quick Example
Sum of infinite geometric series: a₁=6, r=1/3
✓ S∞ = 6/(1−1/3) = 6/(2/3) = 9
📌 Topic 7
Matrices & Systems
A 2×2 matrix inverse exists only when the determinant is non-zero.
det([[a,b],[c,d]]) = ad − bc
[[a,b],[c,d]]⁻¹ = (1/(ad−bc))·[[d,−b],[−c,a]]
💡Memorize: "AD minus BC" for 2×2 determinant. Singular matrix (det=0) has no inverse.
▶ Quick Example
Find det of [[3,2],[1,4]].
✓ det = 3·4 − 2·1 = 12 − 2 = 10
📌 Topic 8
Conic Sections
Four types: circle, ellipse, hyperbola, parabola — all derived from slicing a cone.
Circle: (x−h)² + (y−k)² = r²
Ellipse: x²/a² + y²/b² = 1 (a>b: horizontal major axis)
Hyperbola: x²/a² − y²/b² = 1 | Parabola: y = a(x−h)² + k
💡Memorize: Ellipse uses + (same sign), hyperbola uses − (opposite signs). The larger denominator tells you the major axis direction.
▶ Quick Example
Identify: x²/9 + y²/25 = 1
✓ Ellipse with vertical major axis (b=5 > a=3)
📌 Topic 9
Binomial Theorem & Pascal's Triangle
Expands (a+b)ⁿ using binomial coefficients C(n,k).
(a+b)ⁿ = Σ C(n,k)·aⁿ⁻ᵏ·bᵏ, k=0 to n
C(n,k) = n!/(k!(n−k)!) | (r+1)th term: C(n,r)·aⁿ⁻ʳ·bʳ
💡Memorize: The rth term (starting from r=1) uses k = r−1 in the formula. Powers always sum to n.
▶ Quick Example
3rd term of (x+2)⁵
✓ C(5,2)·x³·2² = 10·x³·4 = 40x³
📌 Topic 10
Introduction to Limits
A limit describes the value a function approaches as x approaches a given point — it may differ from the actual function value.
lim[x→a] f(x) = L means f(x) gets arbitrarily close to L
Limit at infinity: lim[x→∞] (aₙxⁿ + …)/(bₘxᵐ + …) — compare leading terms
💡Memorize: lim[x→∞] 1/xⁿ = 0 for n > 0. For 0/0 forms, factor and cancel, or use L'Hôpital's Rule (calculus).
▶ Quick Example
lim[x→2] (x²−4)/(x−2)
✓ = lim (x+2)(x−2)/(x−2) = lim(x+2) = 4
🎯 Practice Exam
20 multiple-choice questions · AP/SAT style · Select your answer to get instant feedback
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📋 Complete Answer Key & Solutions