Notes for Section 1.1: Constant and Average Rates of Change

1.1 Constant and Average Rates of Change

Core idea: Understanding how quantities change in relation to each other. Distinguish between constant (uniform) rate of change and average rate of change over an interval.

1.1.1 Constant Rates of Change

Definition: Two quantities vary at a constant rate with respect to each other when the changes in one are a constant multiple of the corresponding changes in the other.
- This implies a predictable, linear relationship between the quantities.
Everyday relevance: Constant rates are common and provide a natural entry point to studying rates in calculus.
Example 1.1.1: Driving a car at a constant speed
- If speed is constant at 40 miles per hour, the distance D traveled after time t (in hours) is: $D = 40t$
- At t = 1 h: D = 40 miles; t = 2 h: D = 80 miles; t = 3 h: D = 120 miles.
- At 11 minutes: distance = 40 × (11/60) = 7.33 miles (approximately).
- Interpretation: The distance increases at a constant rate of 40 miles per hour.
- Graphical interpretation: This behavior is represented by a linear function; the slope of the graph (distance vs. time) equals the constant rate of change, computed as the ratio of any change in distance to the corresponding change in time: $m = \frac{\Delta y}{\Delta x}$ .
Example 1.1.2: Water filling a tank at a constant rate
- Flow rate: 10 liters per minute.
- Volume v (in liters) after t minutes: $v = 10t$
- At t = 1 min: v = 10 L; t = 2 min: v = 20 L; t = 3 min: v = 30 L; t = 4.7 min: v = 47 L.
- Conclusion: Volume increases at a constant rate of 10 liters per minute; the relationship is linear: v = 10t.
General form of a linear relationship
- If y varies at a constant rate with respect to x, then: $y = mx + b$
- y: dependent variable (e.g., distance, volume)
- m: constant rate of change (slope) (e.g., 60 mph, 10 L/min)
- x: independent variable (e.g., time)
- b: starting value (y-intercept), the value of y when x = 0
- When subtracting b from both sides of y = mx + b, the relationship can be written as: $y - b = m(x - 0)$
- Here, y − b represents the change in y from the starting value b, and x − 0 represents the change in x from 0.
- Point-slope form:
- $y - y1 = m(x - x1)$
 - y: dependent variable
 - m: constant rate of change (slope)
 - x: independent variable
 - (x1, y1): a fixed point on the graph
- This form expresses a proportional relationship between the changing y and changing x away from the fixed point (x1, y1).
- Graphical note: The slope m determines the rate of change for all linear functions; the graph is a straight line with slope m. See discussion of slope with reference to Figure 1.1.3 (rate of change represented graphically).
Examples of linear functions and slope intuition
- f(x) = -2x - 3: negative slope; as x increases, y decreases at a constant rate.
- g(x) = \frac{1}{2}x + 1: positive slope; y increases at half the rate of x.
- h(x) = 2: slope zero; y remains constant (horizontal line).
- k(x) = x^2: nonlinear; rate of change is not constant; the slope varies with x.
- Key takeaway: Slope m is the constant rate of change for linear functions; nonlinear functions have changing rates of change.
Notation recap
- Slope (constant rate of change): $m$
- Intercept (starting value): $b$
- Linear form: $y = mx + b$
- Point-slope form: $y - y1 = m(x - x1)$
Conceptual link to graphs
- For linear functions, the rate of change is constant and equals the slope.
- For nonlinear functions (e.g., k(x) = x^2), there is no single constant rate of change over the entire domain; the rate of change grows with x.
- This motivates the move to defining the average rate of change over an interval and, later, instantaneous rates (derivatives).

1.1.2 Average Rates of Change

Purpose: The average rate of change over an interval provides a single constant rate of change that yields the same overall change in y as the function f(x) experiences from x = x1 to x = x2, even if the instantaneous rate varies within the interval.
Formal definition (conceptual): The average rate of change of a function f from x1 to x2 is the constant rate of change of a linear function that has the same change in output as f over the interval [x1, x2]. In more concrete terms, if you construct a line whose endpoints match (x1, f(x1)) and (x2, f(x2)), that line has slope equal to the average rate of change of f over [x1, x2].
Key formula (average rate of change)
- If f is defined on [x1, x2], the average rate of change is:
- $m{ ext{avg}} = \frac{f(x2) - f(x1)}{x2 - x_1}$
Example 1.1.6: Plant growth
- Height grows from 2 meters to 6 meters over the interval from 2020 to 2028.
- Average rate of change: $\frac{6 - 2}{2028 - 2020} = \frac{4}{8} = 0.5 \text{ meters per year}$
- Interpretation: On average, the plant grows 0.5 meters each year over that interval.
Secant line as a geometric interpretation
- To approximate the rate of change of f at a point a, draw a secant line through the points ((a, f(a))) and ((x, f(x))) for nearby x.
- The slope of this secant line gives an estimate of the rate of change at x = a.
- The closer x is to a, the better the secant slope approximates the instantaneous rate of change at a.
Transition to instantaneous rate of change (preview)
- As the interval [x1, x2] shrinks (x2 → x1), the average rate of change (slope of the secant) approaches the instantaneous rate of change at x = a.
- The instantaneous rate of change is represented by the slope of the tangent line to the graph of f at x = a.
- A full treatment of instantaneous rates is deferred to Section 1.3.

1.1.3 Secant Lines and the Difference Quotient

Secant line through the points (a, f(a)) and (x, f(x))
- The secant line is the line joining these two points on the graph of f.
- Slope of the secant line (the average rate of change over [a, x]):
- $m_{ ext{sec}} = \frac{f(x) - f(a)}{x - a}$
- This expression is called the difference quotient because it is a quotient of differences (changes in values of f and x).
- In calculus, the difference quotient represents the average rate of change of y = f(x) over the interval from a to x.
Behavior as x → a
- As x gets arbitrarily close to a, the slope of the secant line approaches the slope of the tangent line at x = a (the instantaneous rate of change).
- The tangent line's slope at a is the limit of the secant slopes as x → a (discussion and formal treatment in Section 1.3).
Example: Slopes of secant lines for f(x) = x^2 at a = 1
- Consider the secant through (1, 1) and (2, 4):
- Slope: $m_{ ext{sec}} = \frac{4 - 1}{2 - 1} = 3$
- Consider the secant through (1, 1) and (\tfrac{3}{2}, \tfrac{9}{4})
- Slope: $m_{ ext{sec}} = \frac{\tfrac{9}{4} - 1}{\tfrac{3}{2} - 1} = \frac{\tfrac{5}{4}}{\tfrac{1}{2}} = \tfrac{5}{2} = 2.5$
- Observation: The secant slope with the point closer to (1,1) (i.e., x = 1.5) is closer to the tangent slope at (1,1), which is 2 for f(x) = x^2.
- Therefore, an estimate for the tangent slope at x = 1 is in the range around 2 to 2.5 (depending on the chosen nearby x).
Checkpoint style example (conceptual)
- Estimate the slope of the tangent line to f(x) = x^2 at x = 1 by computing the slopes of secant lines through (1,1) and another nearby point on the graph, such as (5/4, 25/16).
- This demonstrates how secant slopes converge to the tangent slope as the interval shrinks.

Quick reference of key formulas

Linear relationship (constant rate of change):
- $y = mx + b$
- Slope interpretation: $m = \frac{\Delta y}{\Delta x}$
Point-slope form (through (x1, y1)):
- $y - y1 = m(x - x1)$
Average rate of change over [x1, x2]:
- $m{ ext{avg}} = \frac{f(x2) - f(x1)}{x2 - x_1}$
Secant line slope through (a, f(a)) and (x, f(x)):
- $m_{ ext{sec}} = \frac{f(x) - f(a)}{x - a}$
Example of average rate (plant growth):
- $\frac{6 - 2}{2028 - 2020} = \frac{4}{8} = 0.5 \text{ m/year}$
Relationship to instantaneous rate (conceptual): as the interval shrinks, the secant slope approaches the tangent slope at a, which is the instantaneous rate of change.

Connections to broader ideas

Foundational principle: Linear functions have constant rates of change equal to their slopes; this underpins modeling with simple, predictable relationships in physics, chemistry, biology, and everyday life.
Nonlinear functions (e.g., k(x) = x^2) demonstrate changing rates of change; this motivates the need for instantaneous rates (derivatives) and limits in later sections.
The secant line and the difference quotient provide a bridge from average to instantaneous rates, leading to the derivative concept used throughout calculus.