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.