College Physics 1 - Lecture 1: Mathematics Review

Introduction

• Intended Audience: Geared towards students of the lecturer's class, but available publicly.
• Clarification:
  • College Physics: Algebra-based.
  • University Physics: Calculus-based.
• Note: Public viewers may not find all required material.

Importance of Math in Physics

• Basic math skills are vital for understanding physics at this level.
• Initial lectures cover essential math topics to prevent distraction from unfamiliar math during physics concepts.
• Topics in this lecture: Exponents, Fractions, Solving Equations.

Exponents

• Definition: A value with a superscript, where the superscript is the exponent indicating the power to which the base is raised.
  • Example: 4³ is 4 raised to the 3rd power, equivalent to 4 * 4 * 4.
  • Fractional Exponents: Represent roots.
    • Example: x^(1/2) = √x.

Rules of Exponents

1. Product Rule: x^m * x^n = x^(m+n).
   • Example: 3³ * 3² = 3^(3+2) = 3^5 = 243.
2. Quotient Rule: x^m / x^n = x^(m-n).
   • Example: 3³ / 3² = 3^(3-2) = 3¹ = 3.
3. First Power Rule: (x^m)^n = x^(mn).
   • Example: (2²)³ = 2^(2*3) = 2^6 = 64.
4. Distributive Rule:
   • Multiplication: (xy)^n = x^n * y^n.
   • Division: (x/y)^n = x^n / y^n.
   • Example: (3*2)^4 = 3^4 * 2^4 = 81 * 16 = 1296.
5. Negative Exponent Rule: x^(-m) = 1 / x^m.
   • Example: 2^(-1) = 1 / 2^1 = 1/2.
6. Roots Rule: x^(1/m) = m-th root of x.
   • Example: 4^(1/2) = √4 = 2.

Fractions

• Multiplication: Multiply the numerators together and the denominators together.
  • Example: (a/b) * (c/d) = (ac) / (bd).
• Division: Multiply by the reciprocal of the denominator.
  • Example: (a/b) ÷ (c/d) = (a/b) * (d/c) = (ad) / (bc).
• Addition/Subtraction: Find a common denominator.
  • Example: (a/b) ± (c/d) = (ad ± bc) / bd.

Solving Equations

• Perform the same operation on both sides of the equation to maintain equality.

Examples

1. Simple Equation: y = x/5
   • Multiply both sides by 5: x = 5y.
2. Addition: ax - y = 2y
   • Add y to both sides, then divide by a: x = 3y / a.
3. Factoring: a(x + b) = c
   • Divide by a, then subtract b: x = (c / a) - b.
4. Fractions: y/x + a = b
   • Subtract a, then multiply by x, then isolate x: x = y / (b - a).
5. Square Roots: √(x + 3) - y = 0
   • Add y, then square both sides, then subtract 3: x = y² - 3.
6. Multiple x Terms: a(x - y) = b(x + y)
   • Distribute, rearrange to isolate x terms, factor out x, then isolate x: x = y(a + b) / (a - b).

Direct and Indirect Methods

• Expert problem solvers solve using variables first and substitute numbers last.
• This method reduces errors and provides a better general understanding.

Quadratic Equations

• Form: ax² + bx + c = 0
• Solution: Use the quadratic formula.
  • Formula: x = [-b ± √(b² - 4ac)] / 2a
  • Example: 2x² = 3x + 5 rearranges to 2x² - 3x - 5 = 0.
  • Identify a, b, c and substitute in formula.

Conclusion

• Next lecture will cover: Direct and indirect relationships, equation of a line, plane geometry, areas and volumes, and trigonometric functions.
• Emphasis on understanding math concepts to ease understanding physics concepts.