Calculus Notes: Increasing and Decreasing Functions, Concavity, and Rates of Change

Mathematical Precision in Increasing and Decreasing Functions

Basic Visual Definition:
- An increasing function moves upwards from left to right on a graph.
- A decreasing function moves downwards from left to right on a graph.
Precise Calculus Definition via Rate of Change:
- Rate of change corresponds to the slope of a tangent line drawn at a specific point on a graph.
- Tangent lines represent the instantaneous rate of change at a single point.
- If the slope of the tangent line is positive ( $> \text{zero}$ ), the function is increasing at that point.
- If the slope of the tangent line is negative ( $< \text{zero}$ ), the function is decreasing at that point.
Point-by-Point Evaluation on Curves:
- Point $E$ : The function is increasing because the tangent slope is positive ( $> \text{zero}$ ).
- Point $F$ : The function is decreasing because it lies on a downward slope where the tangent slope is negative ( $< \text{zero}$ ).
- Point $G$ : The function is decreasing because the tangent slope is negative ( $< \text{zero}$ ).
- Point $H$ : The function is increasing because the tangent slope is positive ( $> \text{zero}$ ).

Fundamentals of Concavity and Rate of Change

Visual Check for Concavity:
- A graph that opens upwards is concave up.
- A graph that opens downwards is concave down.
Calculus Definition of Concavity:
- Concavity measures whether the rate of change (slope of the tangent line) itself is increasing or decreasing.
- Concavity tracks the behavior of the slopes, which is distinct from tracking whether the function's output values are increasing or decreasing.
Tangent Slopes at Extrema:
- At local maximums and local minimums, the tangent line is horizontal, meaning the slope is exactly equal to $0$ .
Numerical Trends of Slopes Relative to Zero:
- Negative values moving toward $0$ (such as $-10 \rightarrow -5 \rightarrow 0$ ): The rate of change is increasing, because values closer to $0$ from the negative side are mathematically larger.
- Positive values moving toward $0$ : The rate of change is decreasing.
- Moving from $0$ to negative values (such as $0 \rightarrow -5 \rightarrow -10$ ): The rate of change is decreasing.

Concavity Types and the Four Function Behaviors

Concave Down:
- Occurs when the rate of change (slope) is decreasing.
- Symbolized by curves that open downwards.
Concave Up:
- Occurs when the rate of change (slope) is increasing.
- Symbolized by curves that open upwards.
Combination 1: Increasing at an Increasing Rate:
- Function behavior: Increasing (positive slope, line going up).
- Rate of change behavior: Increasing (concave up).
- Graphical representation: The right side of a concave-up cup curve.
Combination 2: Increasing at a Decreasing Rate:
- Function behavior: Increasing (positive slope, line going up).
- Rate of change behavior: Decreasing (concave down).
- Graphical representation: The left side of a concave-down hill curve approaching a maximum, where positive slopes get flatter and closer to $0$ .
Combination 3: Decreasing at a Decreasing Rate:
- Function behavior: Decreasing (negative slope, line going down).
- Rate of change behavior: Decreasing (concave down).
- Graphical representation: The right side of a concave-down hill curve leaving a maximum, moving from $0$ to increasingly negative slopes (e.g., $0 \rightarrow -2 \rightarrow -5$ ).
Combination 4: Decreasing at an Increasing Rate:
- Function behavior: Decreasing (negative slope, line going down).
- Rate of change behavior: Increasing (concave up).
- Graphical representation: The left side of a concave-up cup curve approaching a minimum, where negative slopes become less steep and closer to $0$ (e.g., $-5 \rightarrow -2 \rightarrow 0$ ).

Point of Inflection

Definition:
- A point of inflection is the precise location on a graph where the concavity changes from concave down to concave up, or from concave up to concave down.
- At this point, the rate of change switches from decreasing to increasing, or from increasing to decreasing.
Rate of Change Speed:
- Near local maximums and minimums where slope is near $0$ , the rate of change changes slowly.
- The graph changes most rapidly at the point of inflection, representing the point with the largest rate of change magnitude (steepest positive or steepest negative slope).

Constructing a Graph from Interval Specifications

Interval and Point Data:
- Interval $(-\infty, -2)$ : Function $f$ is increasing; Rate of change is decreasing (Concave down).
- Point at $x = -2$ : Output value is $f(-2) = 4$ .
- Interval $(-2, 3)$ : Function $f$ is decreasing; Rate of change is decreasing (Concave down).
- Point at $x = 3$ : Output value is $f(3) = -1$ .
- Interval $(3, \infty)$ : Function $f$ is decreasing; Rate of change is increasing (Concave up).
Analysis of Key Points on the Graph:
- At $(-2, 4)$ : Represents a local maximum where slope equals $0$ . The curve reaches a peak of $4$ and turns downward while maintaining concave-down behavior.
- At $(3, -1)$ : Represents a point of inflection. The function continues to decrease beyond $x = 3$ , but concavity flips from concave down (decreasing rate of change) to concave up (increasing rate of change).

Homework and Applied Problem Solutions

Section 1.2, Question 3c — Crunchy Cookie Company Profit Model:
- Objective: Determine when Crunchy Cookie Inc. makes zero profit.
- Procedure: Locate the zero (x-intercept) of the graph where the profit output equals $0$ .
- Graphic Note: The textbook logo covers the x-intercept area on the graph (near an approximate value of $204,000$ ), preventing an exact visual reading directly from the image.
Section 1.2, Question 6e — Museum Visitor Entry Rate:
- Context: The graph displays the rate at which visitors enter a museum over time (measured in hours).
- Graph Progression:
- Initial slow increase in entry rate.
- Rapid increase in entry rate up to a peak maximum.
- Constant entry rate for a period.
- Decreasing entry rate during hours $6$ to $8$
- Claim Analysis: Holly states that the total number of people inside the museum is decreasing between hours $6$ and 8$.\n * Verdict: Inaccurate / Not enough information.\n * Explanation: The graph displays the rate of visitors *entering* the museum. Between hours 6 $and$ 8, visitors are still entering, albeit at a slower rate (the entry rate remains positive). Because no data is provided regarding the rate at which visitors are *leaving* the museum, the total population inside the museum cannot be calculated.\n\n# Supplemental Information\n\n* **Upcoming Content:**\n * Further questions from Section 1.3$$ will be addressed in subsequent class sessions.