Notes on Average Rate of Change, Intervals, and Limits

Average Rate of Change over an Interval

Context: You have a function of time that gives displacement or position, and you want the average velocity between two times.
If the displacement is a function f(t), the average rate of change (average velocity) on the time interval from a to b is:
\bar{v} = \frac{f(b) - f(a)}{b - a}
Intuition: You take the change in output (displacement) divided by the change in input (time). It gives the slope of the secant line connecting the two endpoint points ((a,f(a))) and ((b,f(b))) on the graph of f.
Concrete example (from the transcript): on the interval from 1 to 3,
\text{Average velocity} = \frac{f(3) - f(1)}{3 - 1} = \frac{f(3) - f(1)}{2}.
Points to remember:
- The average rate of change depends only on the endpoint values, not on the details of the function in the interior of the interval.
- The outputs are y-values of the function (displacement); the inputs are x-values (time).
- In calculus, letters like f are used for generic functions; in physics, displacement is often denoted by s(t) or h(t). These letters are placeholders and can be changed as needed.
Notation clarifications:
- The phrase “average velocity on an interval” is tied to an interval [a,b].
- The order is typically left-to-right (from a to b) when writing an interval.

Brackets vs parentheses indicate whether endpoints are included:
- Closed interval: [a,b] includes endpoints a and b, i.e., all numbers x with a \le x \le b.
- Open interval: (a,b) excludes endpoints, i.e., all numbers x with a < x < b.
Important distinction:
- The same notation that denotes an interval can look like the notation for a point (a, b) in the plane. Context matters:
- An interval is a set of numbers on the number line between a and b.
- A coordinate pair (a,b) denotes a point in the plane.
Examples from the transcript:
- The interval from 1 to 3, including endpoints: [1,3].
- The interval from 1 to 3, excluding endpoints: (1,3).
Practical note:
- When you see a bracket notation with two numbers, interpret it as an interval unless context clearly indicates a point or coordinate.
- In calculus, endpoints are often included when considering a closed interval for certain theorems or definitions; otherwise, open intervals are used to indicate exclusion of endpoints.

Replacing specific numbers with placeholders:
- Let the interval be from time a to time b, where a and b are real numbers (a may be less than b, usually we take a < b).
- The displacement function can be denoted f (or s, h, etc.). The placeholders emphasize that the same formulas work for any endpoints.
General average rate of change over [a,b]:
\bar{v}_{[a,b]} = \frac{f(b) - f(a)}{b - a}
When you introduce a small time increment h (a common practice in calculus):
- If you start at time t and move forward by h, the average rate of change over the interval [t, t+h] is:
  \frac{f(t+h) - f(t)}{h}
Relationship to instantaneous rate of change:
- The instantaneous rate of change (the derivative) is obtained in the limit as h → 0 of the average rate of change:
  \frac{df}{dt}(t) = \lim_{h \to 0} \frac{f(t+h) - f(t)}{h}
Summary: The first expression gives the average rate over a finite interval; the limit of this expression as the interval shrinks to zero gives the instantaneous rate.

What a limit means: The limit of a function f(x) as x approaches a value a is the value that f(x) gets arbitrarily close to as x gets arbitrarily close to a (from allowed directions).
One-sided limits:
- Left-hand limit: \lim_{x \to a^-} f(x)
- Right-hand limit: \lim_{x \to a^+} f(x)
Two-sided limit exists iff the left-hand and right-hand limits exist and are equal to the same value L:
\lim{x \to a} f(x) = L \quad\text{iff}\quad \lim{x \to a^-} f(x) = \lim_{x \to a^+} f(x) = L.
Examples discussed in the transcript:
- When x approaches -8, the output approaches -6 from both sides; hence the limit exists and equals -6.
- When x approaches 6, the left-hand limit and the right-hand limit may exist but be different, so the two-sided limit does not exist.
- When x approaches 10, the limit exists if the left- and right-hand limits are equal (the transcript indicates this as an example of a limit that does exist, even though details are cut off).
Practical approach to limits (without a graph):
- Evaluate f(x) at values close to a from the left and from the right, and compare:
- For instance, to estimate a limit near 0 for a function like e^x, you can plug in x values near 0 from both sides to see the approaching value.

Consider the exponential function f(x) = e^x, where e ≈ 2.71828…
Key limit:
\lim_{x \to 0} e^x = 1
Demonstrating from both sides:
- If x = -0.1, then e^x ≈ e^{-0.1} ≈ 0.9048.
- If x = 0.1, then e^x ≈ e^{0.1} ≈ 1.1052.
The base being positive (≈ 2.71) is not the point here; the limit concerns the value as x tends to 0, which is 1 for e^x.
This example illustrates the limit process described in the transcript: approaching a value from either direction and observing the resulting y-values.

The average rate of change is the slope of the secant line between two points on the graph of f.
The instantaneous rate of change is the slope of the tangent line at a single point, obtained as a limit of average rates of change as the interval shrinks to zero:
\frac{df}{dt}(t) = \lim_{h \to 0} \frac{f(t+h) - f(t)}{h}
This connects to the derivative concept taught later: instantaneous velocity is the derivative of the displacement function with respect to time.
The interval notation and endpoint conventions (closed vs open) are essential when defining functions, integrals, and limits; the same symbols can denote endpoints of an interval or a coordinate pair, so always use context to disambiguate.

To compute average velocity on [a,b], use \bar{v} = \frac{f(b) - f(a)}{b - a}.
When labeling intervals, remember:
- Closed interval [a,b] includes endpoints; open interval (a,b) excludes endpoints.
- Bracket notation is for intervals on the real line, not necessarily the same as a coordinate pair in the plane.
Endpoints a and b are placeholders; you can replace them with any real numbers, e.g., a = 1, b = 3.
The quantity h often represents a small time increment; the average rate over [t, t+h] is \frac{f(t+h) - f(t)}{h}.
The instantaneous rate of change is the derivative: \frac{df}{dt}(t) = \lim_{h \to 0} \frac{f(t+h) - f(t)}{h}.
Limits quantify behavior of a function as x approaches a; a two-sided limit exists only if both one-sided limits exist and are equal.
Demonstrative example: For e^x, \lim_{x \to 0} e^x = 1; values from both sides approach 1.