Calculus Preliminaries Quiz

Preliminaries

• Calculus involves real numbers that can be expressed as decimals.
• Real numbers include natural numbers, integers, and rational numbers.
• Natural numbers: 1, 2, 3, 4, ...
• Integers: 0, ±1, ±2, ±3, ...
• Rational numbers: can be expressed as m/n, where m and n are integers and n ≠ 0.
• Examples of rational numbers: 200/67, 13/3, (5 - 5)/67, 1/3, 0/0, 3/0
• Division by zero is undefined.
• Division by zero results in undefined expressions.
• Real numbers can be represented on a number line; this is called the real line.

Intervals

• Interval: a set of real numbers.
• Open interval: (a, b) represents all numbers between a and b, excluding a and b.
• Closed interval: [a, b] represents all numbers between a and b, including a and b.
• Other variations include half-open intervals: (a, b] and [a, b).

Inequalities

• Solving inequalities involves finding the range of numbers that satisfy a specific inequality in the variable 'x.'
• "=" means "implies."
• Inequalities can be solved by manipulating both sides using the following rules:
  • Adding or subtracting the same number to both sides doesn't change the solution.
  • Multiplying or dividing both sides by the same positive number doesn't change the solution.
  • Multiplying or dividing both sides by the same negative number reverses inequality signs.
• Example:
  • 2x + 4 < 12 is solved by subtracting 4 from both sides: 2x < 8.
  • Then divide both sides by 2: x < 4. Therefore, the solution is all x < 4.

Solving Inequalities

• Example 1:
  • 3x + 12 < 0 can be solved by subtracting 12 from both sides: 3x < -12.
  • Dividing both sides by 3 gives x < -4.
  • The solution set is the open interval (-∞, -4), which means all numbers less than -4.
• Example 2:
  • 7x + 3 > -18 can be solved by subtracting 3 from both sides: 7x > -21.
  • Then dividing both sides by 7 gives x > -3.
  • The solution set is the open interval (-3, ∞), which means all numbers greater than -3.
• Example 3:
  • 6/(x - 1) ≥ 5 can hold only if x > 1.
  • Multiplying both sides by (x - 1) results in 6 ≥ 5(x - 1).
  • This simplifies to 6 ≥ 5x - 5.
  • Adding 5 to both sides provides 11 ≥ 5x.
  • Dividing both sides by 5 gives x ≤ 11/5.
  • The solution set is the half-open interval (1, 11/5], which includes all numbers greater than 1 but less than or equal to 11/5.