Essential Calculus and Probability Formulas

Probability Concepts

• Union of Events: P(A∪B) = P(A) + P(B) - P(A∩B)
• Complement of an Event: P(A^c) = 1 - P(A)
• Intersection of B and Complement of A: P(B∩A^c) = P(B) - P(A∩B)
• Conditional Probability: P(B|A) = P(A∩B) / P(A)
• Intersection of Complements: P(A^c∩B^c) = 1 - P(A∪B)
• Mutually Exclusive Events: P(A∩B) = 0
• Independent Events: P(A∩B) = P(A)·P(B)

Binomial Distribution Parameters

• Mean (μ): μ = n·p
• Standard Deviation (σ): σ = √(n·p·q)

Binomial Distribution

Notation: B(n, p)

Probability Mass Function: P(X=a) = (ⁿ_a) p^a q^n-a

Where:

• n = number of trials
• p = probability of success
• q = complement of p (q = 1-p)

Normal Distribution

Notation: X ∼ N(μ, σ)

Important Probabilities:

• P(Z < a) or P(Z > a) can be found using a standard normal table.
• P(Z > a) = 1 - P(Z < a)
• P(Z < -a) = P(Z > a)
• P(Z > -a) = P(Z < a)

Standardization (Z-score): Z = (X - μ) / σ

Example: To calculate P(X > 300) for a normal distribution, adjust for continuity (if approximating a discrete distribution) to P(X > 300.5), then standardize: P(Z > (300.5 - μ) / σ).

Function Domains

• Polynomial, Odd Root Irrational, Exponential, and Trigonometric Functions: Domain is all real numbers (ℜ).
• Logarithmic, Even Root Irrational, and Rational Functions:
  • For logarithmic functions, the argument must be strictly positive.
  • For even root irrational functions (e.g., square root), the radicand must be non-negative.
  • For rational functions, the denominator cannot be zero.

  To find the domain, set the expression inside the root/logarithm or the denominator to zero, and use a sign table to determine valid intervals.

Limits of Functions

• Indeterminate Form ∞/∞:
  • If the numerator's highest degree is greater, the limit is ±∞.
  • If the denominator's highest degree is greater, the limit is 0.
  • If the degrees are equal, the limit is the ratio of the leading coefficients.
• Indeterminate Form ∞ - ∞:
  • For expressions involving roots, multiply and divide by the conjugate.
• Indeterminate Form 1^∞ (or 0⁰, ∞⁰):
  • Use the formula: e^{lim [g(x) · ln(f(x))]}
• Indeterminate Forms 0/0 and 0 · ∞:
  • Apply L'Hôpital's Rule (if conditions are met).
  • Alternatively, rewrite the expression as a fraction to get 0/0 or ∞/∞ form.

Important Values and Properties

• e^∞ = ∞
• e¹ = e
• e⁰ = 1
• e^-∞ = 0
• k/∞ = 0 (for any finite constant k)
• ∞/k = ∞ (for any finite positive constant k)
• ln(∞) = ∞
• ln(e) = 1
• ln(1) = 0
• ln(0⁺) = -∞
• ln(negative number) = undefined (does not exist in real numbers)

Continuity of Functions

• Removable Discontinuity: Lateral limits are equal, but not equal to f(a), or f(a) is undefined.
• Jump Discontinuity (Finite): Lateral limits are different and result in finite numbers.
• Jump Discontinuity (Infinite): Lateral limits are different and result in ±∞.

Asymptotes

• Vertical Asymptote (VA): Occurs at x=a if lim_x→a f(x) = ±∞.
• Horizontal Asymptote (HA): Occurs at y=L if lim_x→±∞ f(x) = L (where L is a finite number).
• Oblique Asymptote (OA): Occurs at y=mx+n if:
  • m = lim_x→±∞ [f(x)/x] (m must be a finite, non-zero number)
  • n = lim_x→±∞ [f(x) - mx] (n must be a finite number)

Differentiability

A function f(x) is differentiable at a point 'a' if the limit of the difference quotient exists at that point: lim_h→0 [f(a+h) - f(a)] / h. Differentiability implies continuity at that point.

Monotonicity and Relative Extrema

To determine monotonicity and find relative extrema:

1. Calculate the first derivative, f'(x).
2. Set f'(x) = 0 and solve for x to find critical points. Also consider points where f'(x) is undefined or where the domain changes.
3. Create a sign table for f'(x), including critical points and domain boundaries.
4. Increasing: f(x) is increasing where f'(x) > 0.
5. Decreasing: f(x) is decreasing where f'(x) < 0.
6. Relative Extrema:
   • If f'(x) changes from positive to negative, there is a relative maximum.
   • If f'(x) changes from negative to positive, there is a relative minimum.
7. Substitute the x-values of relative extrema into the original function f(x) to find the corresponding y-values.

Curvature and Inflection Points

To determine curvature and find inflection points:

1. Calculate the second derivative, f''(x).
2. Set f''(x) = 0 and solve for x to find potential inflection points. Also consider points where f''(x) is undefined.
3. Create a sign table for f''(x).
4. Concave Up (Convex): f(x) is concave up where f''(x) > 0.
5. Concave Down (Concave): f(x) is concave down where f''(x) < 0.
6. Inflection Point: Occurs where f''(x) changes sign (from positive to negative or vice versa).

Tangent Line Equation

The equation of the tangent line to a function f(x) at a point (a, f(a)) is:

y - f(a) = f'(a)(x - a)

Key Calculus Theorems

Bolzano's Theorem (Intermediate Value Theorem for Roots)

If f(x) is a continuous function on the closed interval [a, b], and f(a) and f(b) have opposite signs (i.e., f(a)·f(b) < 0), then there exists at least one value c in the open interval (a, b) such that f(c) = 0.

Darboux's Theorem (Intermediate Value Theorem)

If f(x) is a continuous function on the closed interval [a, b], then for any value k between f(a) and f(b), there exists at least one value c in the open interval (a, b) such that f(c) = k.

Lagrange's Mean Value Theorem

If f(x) is a continuous function on the closed interval [a, b] and differentiable on the open interval (a, b), then there exists at least one value c in (a, b) such that:

f'(c) = (f(b) - f(a)) / (b - a)