Intervention Calculus Comprehensive Study Notes - Dr. Lopez

Real Number System

The real number system is a hierarchical classification of numbers used in calculus and mathematics. It is categorized as follows:

• Real Numbers: The overarching set containing all rational and irrational numbers.
• Rational Numbers: Defined as the ratio of two integers. These numbers are either terminating or repeating decimals.
  • Examples provided: $5$, $33$, $0.5$, $1.23$, $3.3$
• Integers: Counting numbers including zero and their negative counterparts.
  • Set representation: $\{ …, -3, -2, -1, 0, 1, 2, 3, … \}$
• Whole Numbers: Positive counting numbers including zero.
  • Set representation: $\{ 0, 1, 2, 3, … \}$
• Natural Numbers: Positive counting numbers starting from 1 (excludes 0).
  • Set representation: $\{ 1, 2, 3, … \}$
• Irrational Numbers: Numbers that cannot be written as a fraction. They are non-terminating and non-repeating.
  • Examples: $\pi$, $\sqrt{2}$, $\sqrt{3}$, $\sqrt{5}$

Basic Laws of Natural Numbers and Equality

Laws for Addition

• Closure Property: If $a$ and $b$ are natural numbers, then $a + b = c$, where $c$ is also a natural number.
• Commutative Property: The order of addition does not change the sum: $a + b = b + a$.
• Associative Property: The grouping of numbers does not change the sum: $a + (b + c) = (a + b) + c$.

Laws for Multiplication

• Closure Property: If $a$ and $b$ are natural numbers, then $a \times b = c$, where $c$ is also a natural number.
• Commutative Property: The order of multiplication does not change the product: $a \times b = b \times a$.
• Associative Property: The grouping of factors does not change the product: $a \times (b \times c) = (a \times b) \times c$.
• Distributive Property: Multiplication distributes over addition: $a \times (b + c) = (a \times b) + (a \times c)$.

Basic Laws of Equality

• Reflexive Property: A value is always equal to itself: $a = a$.
• Symmetric Property: If the first value equals the second, the second equals the first: If $a = b$, then $b = a$.
• Transitive Property: If two values are both equal to a third value, they are equal to each other: If $a = b$ and $b = c$, then $a = c$.
• Addition Property of Equality: If $a = b$ and $c = d$, then $a + c = b + d$.
• Multiplication Property of Equality: If $a = b$ and $c = d$, then $a \times c = b \times d$.

Laws of Exponents

Exponents follow specific rules for simplification and manipulation:

• Product Rule: $a^m \times a^n = a^{m+n}$
• Quotient Rule: $\frac{a^m}{a^n} = a^{m-n}$
• Power of a Power Rule: $(a^m)^n = a^{mn}$
• Power of a Product Rule: $(ab)^m = a^m b^m$
• Power of a Quotient Rule: $(\frac{a}{b})^m = \frac{a^m}{b^m}$
• Zero Exponent Rule: $a^0 = 1$
• Negative Exponent Rule: $a^{-m} = \frac{1}{a^m}$
• Fractional Exponent Rule: $a^{\frac{m}{n}} = \sqrt[n]{a^m}$

Properties of Logarithms

Logarithms are the inverses of exponential functions and follow these operational rules:

• Product Rule: $\log_b(xy) = \log_b x + \log_b y$
• Quotient Rule: $\log_b(\frac{x}{y}) = \log_b x - \log_b y$
• Power Rule: $\log_b(x^n) = n \log_b x$
• Change of Base Rule: $\log_b x = \frac{\log_a x}{\log_a b}$ or $\log_b x \times \log_a b = \log_a x$
• Zero Rule: $\log_b(1) = 0$
• Identity Rule: $\log_b(b) = 1$
• Equality Rule: If $\log_b x = \log_b y$, then $x = y$
• Inverse Rules: $b^{\log_b x} = x$ and $\log_b(b^x) = x$
• Reciprocal Rule: $\frac{1}{\log x} = -\log(x^{-1})$ (Note: Slide lists this as $\frac{1}{\log x} = -\log(x)$, representing the internal reciprocal property).

Properties of Radicals

• Product of Radicals: $\sqrt[n]{ab} = \sqrt[n]{a} \sqrt[n]{b}$
• Quotient of Radicals: $\sqrt[n]{\frac{a}{b}} = \frac{\sqrt[n]{a}}{\sqrt[n]{b}}$
• Root of a Root: $\sqrt[n]{\sqrt[m]{a}} = \sqrt[nm]{a}$
• Power of a Radical: $\sqrt[n]{a^m} = (\sqrt[n]{a})^m$
• Index Power Cancellation: $\sqrt[n]{a^n} = a$
• Odd Index Negative Root: $\sqrt[n]{-a} = -\sqrt[n]{a}$ for odd numbers $n$

Algebraic Terms and Polynomials

Key Definitions

• Variable: A letter that denotes a real number.
• Algebraic Expression: Formed by combining variables and real numbers using addition, subtraction, multiplication, division, powers, or roots.
• Polynomial: An expression of the form $a_n x^n + a_{n-1} x^{n-1} + … + a_1 x + a_0$, where coefficients are real numbers.
• Degree: The highest power of the variable present in the polynomial.

Classification Examples

• $3x^2$: Degree 2
• $7x + 5$: Degree 1
• $x^3 + 3x^2 - x + 1$: Degree 3
• $6x^2 - 4x + 7$: Degree 2
• $x$: Degree 1

Classification by Terms

• Monomial: One term (e.g., $5x^7$)
• Binomial: Two terms (e.g., $8x^6 - 1$)
• Trinomial: Three terms (e.g., $-3x^2 + x - 1$)
• Polynomial: Many terms (e.g., $5x^3 - 2x^2 + 3x - 6$)

Classification by Degree Names

• Constant: Degree 0 (e.g., $5$)
• Linear: Degree 1 (e.g., $2x + 1$)
• Quadratic: Degree 2 (e.g., $3x^2 - 5x - 3$)
• Cubic: Degree 3 (e.g., $x^3 - x^2 + x + 1$)
• Fourth-degree: Degree 4 (e.g., $7x^4 + 3x^3 - 7x + 8$)

Algebraic Operations

Rules for Signage (Assuming $a > b$)

• Addition:
  • $+a + (+b) = +(a + b)$
  • $-a + (-b) = -(a + b)$
  • Numbers with opposing signs are subtracted, and the result carries the sign of the larger number.
• Subtraction:
  • $+a - (+b) = +(a - b)$
  • $-a - (-b) = -(a - b)$
  • $+a - (-b) = +(a + b)$
  • $-a - (+b) = -(a + b)$
• Multiplication:
  • (+a)(+b) = +(a & b)
  • (-a)(-b) = +(a & b)
  • (+a)(-b) = -(a & b)
  • (-a)(+b) = -(a & b)
• Division:
  • $(+a) / (+b) = +(a/b)$
  • $(-a) / (-b) = +(a/b)$
  • $(+a) / (-b) = -(a/b)$
  • $(-a) / (+b) = -(a/b)$

Polynomial Operations

Addition and Subtraction

• Addition: Combine like terms.
  • Example: $(x^2 - x + 5) + (6x^2 + 2x - 10) = 7x^2 + x - 5$
• Subtraction: Rewrite the subtraction as addition by changing signs of terms in the subtrahend, then combine like terms.
  • Example: $(3x^2 - 8x + 7) - (2x^2 - 6x + 12) = (3x^2 - 8x + 7) + (-2x^2 + 6x - 12) = x^2 - 2x - 5$

Multiplication

• Involves distributive property across all terms.
  • Example: $(4x - 3)(5x^2 - 2x - 1)$
  • Step 1: $4x(5x^2 - 2x - 1) = 20x^3 - 8x^2 - 4x$
  • Step 2: $-3(5x^2 - 2x - 1) = -15x^2 + 6x + 3$
  • Step 3: Combine: $20x^3 - 23x^2 + 2x + 3$

Division

• Can be performed via long division.
  • Example: $(x^3 + 7x^2 + 10x - 6) \div (x + 3)$
  • $(x^3 + 3x^2)$ subtracted from top results in $4x^2 + 10x$
  • $(4x^2 + 12x)$ subtracted results in $-2x - 6$
  • $-(-2x - 6)$ results in $0$
  • Quotient: $x^2 + 4x - 2$

Factorization and Special Products

Special Products

• Product of sum and difference: $(x + y)(x - y) = x^2 - y^2$
• Square of a binomial (sum): $(x + y)^2 = x^2 + 2xy + y^2$
• Square of a binomial (difference): $(x - y)^2 = x^2 - 2xy + y^2$
• Cube of a binomial (sum): $(x + y)^3 = x^3 + 3x^2y + 3xy^2 + y^3$
• Cube of a binomial (difference): $(x - y)^3 = x^3 - 3x^2y + 3xy^2 - y^3$

Factoring Techniques

• Difference of Squares: $x^2 - y^2 = (x + y)(x - y)$
• Perfect Square Trinomials:
  • $x^2 + 2xy + y^2 = (x + y)^2$
  • $x^2 - 2xy + y^2 = (x - y)^2$
• Sum of Cubes: $x^3 + y^3 = (x + y)(x^2 - xy + y^2)$
• Difference of Cubes: $x^3 - y^3 = (x - y)(x^2 + xy + y^2)$

Practice Exercises

Exponent Practice

1. $3 \times 4^3$
2. $4x^3 \times 2x^3$
3. $x^5 \times x^3$
4. $2x^3 \times 2x^2$
5. $6^5 / 6^3$
6. $x^5 y^6 / x y^2$
7. $x^7 y^5 / x y^4$
8. $8^0$
9. $-(9x)^0$
10. $(x^2 y)^4$
11. $16 x y^4 / 8 x^4 y^2$
12. $20 x y^5 / 5 x^3 y$
13. $(2cd^4)^2 (cd)^5$
14. $(2fg^4)^4 (fg)^6$

Logarithm Practice

1. $4 \log 3 - 4 \log 8$
2. $\log 2 + \log 11 + \log 7$
3. $\ln x - 4 \ln y$
4. $6 \log 3 + 6 \log 3^v$

Factorization Practice

1. $3x^2 - 2x - 5$
2. $10m^2 + 17m + 6$
3. $121a^2 - 66ab + 9b^2$
4. $10u^3 - 130u^2 + 400u$

Complex Simplification

1. $3x - \frac{1}{3}x + 3(x - 1)$
2. $5x - 2(x^2 - 7x^2)$
3. $\frac{x^2 - 10x + 25}{x - 5}$
4. $\frac{x^2 - 4x - 5}{x + 1}$
5. $\frac{4x^3 - 6x^2}{2x}$
6. $\frac{x^2 - 2x - 3}{x^2 + 6x + 5}$