FSC Part 2 Mathematics Notes - MatricNotes

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Chapter 1: Integration

Integration as Anti-Derivative

∫f(x)dx = F(x) + C
Where F'(x) = f(x)

Standard Integrals

∫xⁿdx = xⁿ⁺¹/(n+1) + C (n ≠ -1)
∫dx/x = ln|x| + C
∫e^x dx = e^x + C
∫a^x dx = a^x/ln(a) + C
∫sinx dx = -cosx + C
∫cosx dx = sinx + C
∫sec²x dx = tanx + C
∫cosec²x dx = -cotx + C

Integration by Parts

∫u dv = uv - ∫v du
ILATE: Inverse, Log, Algebraic, Trigonometric, Exponential

Integration by Substitution

Make substitution to simplify integral.

Definite Integrals

∫ₐᵇ f(x)dx = F(b) - F(a)

Chapter 2: Differential Equations

Order and Degree

Order: Highest order derivative
Degree: Power of highest order derivative (must be polynomial)

Formation

Given general solution, eliminate constants.

Solutions

Variable Separable

dy/dx = f(x)g(y)
dy/g(y) = f(x)dx

Homogeneous

dy/dx = f(y/x)
Substitute v = y/x

Linear

dy/dx + Py = Q
Integrating Factor: I.F. = e^{∫P dx}
Solution: y(I.F.) = ∫Q(I.F.)dx + C

Chapter 3: Linear Programming

Linear Inequations

Inequalities in two variables.

Feasible Region

Common region satisfying all constraints.

Corner Point Method

Graph all constraints
Find corner points
Evaluate objective function at each corner
Maximum/minimum at corner

Chapter 4: Vectors

Types

Equal Vectors: Same magnitude and direction
Null Vector: Zero magnitude
Unit Vector: Magnitude 1
Parallel Vectors: Same or opposite direction

Operations

a + b = (a₁+b₁)i + (a₂+b₂)j + (a₃+b₃)k
ka = ka₁i + ka₂j + ka₃k

Dot (Scalar) Product

a·b = |a||b|cosθ
a·b = a₁b₁ + a₂b₂ + a₃b₃

Cross (Vector) Product

a × b = |a||b|sinθ n̂
a × b = | i j k |
| a₁ a₂ a₃ |
| b₁ b₂ b₃ |

Scalar Triple Product

a·(b × c) = Volume of parallelepiped

Chapter 5: Conic Sections

Circle

(x-h)² + (y-k)² = r²
Center: (h,k), Radius: r

Parabola

y² = 4ax (horizontal)
x² = 4ay (vertical)
Focus: (a,0), Directrix: x = -a

Ellipse

x²/a² + y²/b² = 1
c² = a² - b²
Foci: (±c,0), Vertices: (±a,0)

Hyperbola

x²/a² - y²/b² = 1
c² = a² + b²
Foci: (±c,0), Asymptotes: y = ±(b/a)x

Chapter 6: Statistics

Measures of Dispersion

Variance

σ² = Σ(xᵢ - x̄)² / n (Population)
s² = Σ(xᵢ - x̄)² / (n-1) (Sample)

Standard Deviation

σ = √Variance

Combined Mean

x̄ = (n₁x̄₁ + n₂x̄₂) / (n₁ + n₂)

Correlation

r = Σ(x-x̄)(y-ȳ) / √[Σ(x-x̄)² Σ(y-ȳ)²]

Regression Lines

y on x: y - ȳ = r (σ_y/σ_x) (x - x̄)
x on y: x - x̄ = r (σ_x/σ_y) (y - ȳ)

Chapter 7: Probability

Basic Terms

Experiment: Action with uncertain outcome
Sample Space (S): All possible outcomes
Event: Subset of sample space

Probability

P(A) = n(A)/n(S)

Addition Theorem

P(A∪B) = P(A) + P(B) - P(A∩B)

Multiplication Theorem

P(A∩B) = P(A) × P(B|A)

Bayes' Theorem

P(A|B) = P(B|A) P(A) / P(B)

Binomial Distribution

P(r) = nCᵣ pʳ qⁿ⁻ʳ
Mean = np, Variance = npq