The Main Idea

1. Definition:
   • Rate of change describes how an output quantity changes relative to the input quantity
   • Units: output units per input units
2. Average Rate of Change:
   • Calculated over an interval
   • Formula: [latex]\frac{\Delta y}{\Delta x} = \frac{f(x_2) - f(x_1)}{x_2 - x_1}[/latex]
3. Interpretation:
   • Positive rate: output increases as input increases
   • Negative rate: output decreases as input increases
4. Applications:
   • Population growth
   • Speed (distance per time)
   • Fuel efficiency (distance per volume)
   • Economic indicators (price changes over time)

Key Techniques

1. Calculating Average Rate of Change:
   • Identify the interval [latex][x_1, x_2][/latex]
   • Calculate change in output: [latex]\Delta y = f(x_2) - f(x_1)[/latex]
   • Calculate change in input: [latex]\Delta x = x_2 - x_1[/latex]
   • Divide: [latex]\frac{\Delta y}{\Delta x}[/latex]
2. Interpreting from Graphs:
   • Slope of secant line between two points
   • Vertical change divided by horizontal change
3. Working with Functions:
   • Evaluate function at endpoints of interval
   • Apply average rate of change formula

The Main Idea

• Increasing and Decreasing Intervals:
  • Increasing: function values increase as input increases
  • Decreasing: function values decrease as input increases
  • Constant: function values remain the same as input increases
• Local Extrema:
  • Local Maximum: where function changes from increasing to decreasing
  • Local Minimum: where function changes from decreasing to increasing
  • Collectively called local extrema or relative extrema
• Graphical Interpretation:
  • Increasing: graph slopes upward from left to right
  • Decreasing: graph slopes downward from left to right
  • Local maximum: highest point in a neighborhood
  • Local minimum: lowest point in a neighborhood
• Mathematical Definitions:
  • Increasing: [latex]f(b) > f(a)[/latex] for [latex]b > a[/latex] in an interval
  • Decreasing: [latex]f(b) < f(a)[/latex] for [latex]b > a[/latex] in an interval
  • Local maximum at [latex]x = b[/latex]: [latex]f(x) \leq f(b)[/latex] for all [latex]x[/latex] in some interval containing [latex]b[/latex]
  • Local minimum at [latex]x = b[/latex]: [latex]f(x) \geq f(b)[/latex] for all [latex]x[/latex] in some interval containing [latex]b[/latex]

Key Techniques

1. Identifying Increasing/Decreasing Intervals:
   • Observe graph from left to right
   • Note where slope changes from positive to negative or vice versa
2. Locating Local Extrema:
   • Look for “peaks” (local maxima) and “valleys” (local minima)
   • Confirm by checking neighboring points
3. Using Technology:
   • Utilize graphing calculators or software to visualize functions
   • Use built-in features to estimate extrema locations
4. Analyzing Complex Functions:
   • Break down the graph into smaller intervals
   • Analyze behavior within each interval

The Main Idea

• Constant Function: [latex]f(x) = c[/latex]
  • Behavior: Neither increasing nor decreasing
  • Graph: Horizontal line
• Identity Function: [latex]f(x) = x[/latex]
  • Behavior: Increasing on [latex](-\infty, \infty)[/latex]
  • Graph: Straight line through origin
• Quadratic Function: [latex]f(x) = x^2[/latex]
  • Increasing: [latex](0, \infty)[/latex]
  • Decreasing: [latex](-\infty, 0)[/latex]
  • Minimum: At [latex]x = 0[/latex]
  • Graph: Parabola opening upward
• Cubic Function: [latex]f(x) = x^3[/latex]
  • Behavior: Increasing on [latex](-\infty, \infty)[/latex]
  • Graph: S-shaped curve
• Reciprocal Function: [latex]f(x) = \frac{1}{x}[/latex]
  • Decreasing: [latex](-\infty, 0) \cup (0, \infty)[/latex]
  • Graph: Hyperbola with vertical asymptote at x = 0
• Reciprocal Squared: [latex]f(x) = \frac{1}{x^2}[/latex]
  • Increasing: [latex](-\infty, 0)[/latex]
  • Decreasing: [latex](0, \infty)[/latex]
  • Graph: U-shaped curve with vertical asymptote at x = 0
• Cube Root: [latex]f(x) = \sqrt[3]{x}[/latex]
  • Behavior: Increasing on [latex](-\infty, \infty)[/latex]
  • Graph: S-shaped curve, less steep than cubic
• Square Root: [latex]f(x) = \sqrt{x}[/latex]
  • Increasing: [latex](0, \infty)[/latex]
  • Domain: [latex][0, \infty)[/latex]
  • Graph: Curve starting at origin, opening upward
• Absolute Value: [latex]f(x) = |x|[/latex]
  • Increasing: [latex](0, \infty)[/latex]
  • Decreasing: [latex](-\infty, 0)[/latex]
  • Minimum: At [latex]x = 0[/latex]
  • Graph: V-shaped graph

The Main Idea

• Absolute Extrema vs. Local Extrema:
  • Absolute: Highest/lowest points over entire domain
  • Local: Highest/lowest points in a local region
• Absolute Maximum:
  • Highest point on the entire graph
  • [latex]f(c)[/latex] is absolute max if [latex]f(c) \geq f(x)[/latex] for all [latex]x[/latex] in domain
• Absolute Minimum:
  • Lowest point on the entire graph
  • [latex]f(d)[/latex] is absolute min if [latex]f(d) \leq f(x)[/latex] for all [latex]x[/latex] in domain
• Existence of Absolute Extrema:
  • Not all functions have absolute extrema
  • Example: [latex]f(x) = x^3[/latex] has neither absolute max nor min

Key Techniques

1. Graphical Identification:
   • Observe entire graph within function’s domain
   • Locate highest and lowest points
2. Comparing Extrema:
   • Compare all local maxima to find absolute maximum
   • Compare all local minima to find absolute minimum
3. Considering Domain:
   • Check domain boundaries for potential absolute extrema
   • Be aware of asymptotic behavior
4. Multiple Absolute Extrema:
   • Functions can have multiple absolute maxima or minima
   • These occur at same y-value but different x-values