Key Formulas & Equations
Core Concepts
Function
A relation between a set of inputs (domain) and a set of possible outputs (codomain) where each input maps to exactly one output. Notation: f(x) = y.
Derivative
Measures the instantaneous rate of change of a function. Geometrically, it is the slope of the tangent line at a point. Defined as: f'(x) = lim(h→0) [f(x+h) - f(x)] / h
Integral
Represents the accumulation of quantities — the area under a curve. The indefinite integral ∫f(x)dx is the antiderivative. Definite integrals compute net area between bounds.
Limit
The value a function approaches as its input approaches some value. Fundamental to calculus: lim(x→a) f(x) = L means f(x) gets arbitrarily close to L as x approaches a.
Matrix
A rectangular array of numbers arranged in rows and columns. Used for solving systems of linear equations, transformations, and representing data. An m×n matrix has m rows and n columns.
Polynomial
An expression of the form aₙxⁿ + aₙ₋₁xⁿ⁻¹ + ... + a₁x + a₀, where coefficients are real numbers and n is a non-negative integer. The highest power n is the degree.
Complex Number
A number of the form a + bi where a and b are real numbers and i = √(-1). The real part is a and the imaginary part is b. |a+bi| = √(a² + b²).
Worked Examples
Example 1: Solve 2x² - 8x + 6 = 0
Step 1: Divide by 2: x² - 4x + 3 = 0
Step 2: Factor: (x - 1)(x - 3) = 0
Step 3: x - 1 = 0 or x - 3 = 0
Step 4: x = 1 or x = 3
Verification: 2(1)² - 8(1) + 6 = 2 - 8 + 6 = 0 ✓ | 2(3)² - 8(3) + 6 = 18 - 24 + 6 = 0 ✓
Example 2: Find the derivative of f(x) = 3x⁴ - 2x² + 5x - 7
Step 1: Apply power rule term by term.
Step 2: d/dx[3x⁴] = 3·4x³ = 12x³
Step 3: d/dx[-2x²] = -2·2x = -4x
Step 4: d/dx[5x] = 5
Step 5: d/dx[-7] = 0 (constant rule)
Answer: f'(x) = 12x³ - 4x + 5
Example 3: Find the line through (2, 3) and (5, 9)
Step 1: Find slope: m = (9-3)/(5-2) = 6/3 = 2
Step 2: Point-slope form: y - 3 = 2(x - 2)
Step 3: Simplify: y - 3 = 2x - 4
Step 4: Slope-intercept: y = 2x - 1
Check: When x=5: y = 2(5)-1 = 9 ✓
Example 4: Compute ∫(2x + 3) dx
Step 1: Integrate term by term.
Step 2: ∫2x dx = 2·(x²/2) = x²
Step 3: ∫3 dx = 3x
Step 4: Add constant of integration: x² + 3x + C
Verification: d/dx[x² + 3x + C] = 2x + 3 ✓
Practice Problems
Problem 1
Solve: x² - 5x + 6 = 0
Answer: (x-2)(x-3) = 0 → x = 2 or x = 3
Problem 2
Find the derivative of f(x) = x³ + 5x² - 3x + 2
Answer: f'(x) = 3x² + 10x - 3
Problem 3
What is the volume of a sphere with radius 3?
Answer: V = (4/3)π(3)³ = (4/3)π·27 = 36π ≈ 113.1 cubic units
Problem 4
Simplify: log₂(8) + log₂(4)
Answer: log₂(8) = 3 (since 2³=8), log₂(4) = 2 (since 2²=4). Sum = 5
Problem 5
Find the distance between (1, 2) and (4, 6).
Answer: d = √((4-1)²+(6-2)²) = √(9+16) = √25 = 5
Problem 6
Evaluate: ∫[0→2] (3x²) dx
Answer: Antiderivative = x³. F(2) - F(0) = 8 - 0 = 8
Problem 7
Solve the system: 2x + y = 7, x - y = -1
Answer: Add equations: 3x = 6 → x = 2. Then y = x+1 = 3. Solution: (2, 3)