College Physics 1: Lecture 1 - Mathematics Review

Jul 17, 2024

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College Physics 1: Lecture 1 - Mathematics Review

Overview

Lecture covers basic math skills required for College Physics, focusing on exponents, fractions, and solving equations. Emphasis is placed on algebra-based physics as opposed to calculus-based physics.

Key Points

  • Audience: Intended for students in the class, but made publicly available.
  • Math Skills: Fundamental for succeeding in physics; ensures focus is on physics concepts rather than struggling with math.

Exponents

Basics

  • Superscript notation: x raised to the nth power.
  • 4^3 = 4 * 4 * 4.
  • Fractional exponents: Example, x^(1/2) is the square root of x.
  • *Rules for Exponents:
  1. Product Rule: x^m * x^n = x^(m+n).
    • Example: 3^3 * 3^2 = 3^(3+2) = 3^5 = 243.
  2. Quotient Rule: x^m / x^n = x^(m-n).
    • Example: 3^3 / 3^2 = 3^(3-2) = 3^1 = 3.
  3. Power Rule: (x^m)^n = x^(mn).
    • Example: 2^2^3 = 2^6 = 64.
  4. Distributive Rule: (xy)^m = x^m * y^m.
    • Example: (3*2)^4 = 3^4 * 2^4 = 81 * 16 = 1296.
  5. Negative Exponent Rule: x^-n = 1/x^n.
    • Example: 2^-1 = 1/2 = 0.5.
  6. Roots Rule: x^(1/n) = n-root of x.
    • Example: 4^(1/2) = √4 = 2.

Fractions

  1. Multiplication: Multiply straight across.
    • Example: a/b * c/d = ac/bd.
  2. Division: Multiply by the reciprocal.
    • Example: a/b / c/d = a/b * d/c = ad/bc.
  3. Addition/Subtraction: Find common denominators.
    • Example: a/b ± c/d = (ad ± bc)/bd.

Solving Equations

  1. Basic Principle: Perform the same operation on both sides of the equation.
  2. *Examples:
    • y = x/5 → x = 5y (multiply by 5)
    • ax - y = 2y → x = 3y/a (move terms, divide by a)
    • a(x + b) = c → x = c/a - b (divide by a, move terms)
    • y/x + a = b → x = y/(b - a) (move terms, multiply by x)
    • √x + 3 - y = 0 → x = y^2 - 3 (move terms, square both sides)
    • a(x - y) = b(x + y) → x = y(a + b)/(a - b) (distribute, factor, solve)

Quadratic Equations

  • Standard Form: ax^2 + bx + c = 0.
  • Quadratic Formula: x = (-b ± √(b^2 - 4ac)) / 2a
  • *Steps:
    1. Rearrange into standard form.
    2. Identify coefficients a, b, c.
    3. Substitute into quadratic formula.
    4. Solve for x.
    • Example: 2x^2 = 3x + 5 → 2x^2 - 3x - 5 = 0 → x = (-(-3) ± √((-3)^2 - 4(2)(-5))) / 2(2) = x = 2.5 or x = -1.

Conclusion

  • Importance of learning to rearrange equations before plugging in numbers.
  • Use of quadratic formula for solving specific types of equations.
  • Next lecture will continue with further math review, introducing relationships, geometric concepts, and trigonometric functions.