Average and Instantaneous Velocity, and Limits in Calculus

Average and Instantaneous Velocity

Definitions and Concepts

Understanding velocity is fundamental in calculus, as it connects the concept of rates of change to the motion of objects. Two key types of velocity are average velocity and instantaneous velocity.

• Average velocity is the change in position (displacement) divided by the time interval over which the change occurs.

• Instantaneous velocity is the velocity of an object at a specific instant, defined mathematically as the limit of the average velocity as the time interval approaches zero.

Formulas:

• Average velocity over interval $[t_0, t_1]$: $v_{avg} = \frac{\Delta s}{\Delta t} = \frac{s(t_1) - s(t_0)}{t_1 - t_0}$

• Instantaneous velocity at time $t_0$: $v_{inst} = \lim_{t \to t_0} \frac{s(t) - s(t_0)}{t - t_0}$

Example Application:

• If a city bus passes milepost 100 miles at noon and 125 miles at 12:30 PM, the average velocity is: $v_{avg} = \frac{125 - 100}{12.5 - 12} = \frac{25}{0.5} = 50$ mi/hr

Key Point: The instantaneous velocity is measured by the slope of the tangent line to the position-time graph at a given point.

Worked Example: Vertical Motion

Consider a rock launched vertically upward with a speed of 96 ft/s. Neglecting air resistance, its position after $t$ seconds is given by $s(t) = 96t - 16t^2$.

• Average velocity between $t = 1$ and $t = 3$: $v_{avg} = \frac{s(3) - s(1)}{3 - 1} = \frac{(96 \times 3 - 16 \times 9) - (96 \times 1 - 16 \times 1)}{2} = \frac{288 - 144 - 96 + 16}{2} = \frac{128}{2} = 64$ ft/s

• Average velocity between $t = 1$ and $t = 2$: $v_{avg} = \frac{s(2) - s(1)}{2 - 1} = \frac{128 - 80}{1} = 48$ ft/s

Table: Average Velocity for Successively Smaller Intervals

Time Interval	Average Velocity (ft/s)
[1, 2]	48
[1, 1.5]	62.4
[1, 1.1]	69.6
[1, 1.01]	71.84
[1, 1.001]	71.984
[1, 1.0001]	71.9984

Observation: As the interval gets smaller, the average velocity approaches the instantaneous velocity at $t = 1$.

Limits and the Definition of a Limit

Preliminary Definition of a Limit

The concept of a limit is central to calculus. It describes the behavior of a function as the input approaches a particular value.

• If $f(x)$ is defined for all $x$ near $a$ (except possibly at $a$), and $f(x)$ approaches $L$ as $x$ approaches $a$, we write: $\lim_{x \to a} f(x) = L$

Graphical Interpretation: The value that $f(x)$ approaches as $x$ gets close to $a$ (from either side).

One-Sided Limits

• Left-sided limit: $\lim_{x \to a^-} f(x) = L$

• Right-sided limit: $\lim_{x \to a^+} f(x) = L$

Both one-sided limits must be equal for the two-sided limit to exist.

Example: Evaluating Limits from a Table

Given $f(x) = \frac{\sqrt{x} - 1}{x - 1}$, as $x$ approaches 1:

x	f(x)
0.9	0.537
0.99	0.503
0.999	0.5003
1.001	0.4998
1.01	0.497
1.1	0.453

Conjecture: As $x$ approaches 1, $f(x)$ approaches 0.5, so $\lim_{x \to 1} f(x) = 0.5$.

Relationship Between One-Sided and Two-Sided Limits

• The two-sided limit $\lim_{x \to a} f(x)$ exists if and only if both one-sided limits exist and are equal.

• If $\lim_{x \to a^-} f(x) = \lim_{x \to a^+} f(x) = L$, then $\lim_{x \to a} f(x) = L$.

Practice: Finding Limits from Graphs

• Use the graph of $h$ to find $h(2)$, $\lim_{x \to 2} h(x)$, $\lim_{x \to 4} h(x)$, $\lim_{x \to 1} h(x)$, $\lim_{x \to 5} h(x)$.

• Use the graph of $f$ to find $f(1)$, $\lim_{x \to 1} f(x)$, $\lim_{x \to 0} f(x)$.

Key Point: The value of the limit depends on the behavior of the function near the point, not necessarily the value at the point.

Summary Table: Average vs. Instantaneous Velocity

Type	Definition	Formula	Graphical Interpretation
Average Velocity	Change in position over time interval	$v_{avg} = \frac{s(t_1) - s(t_0)}{t_1 - t_0}$	Slope of secant line
Instantaneous Velocity	Velocity at a specific instant	$v_{inst} = \lim_{t \to t_0} \frac{s(t) - s(t_0)}{t - t_0}$	Slope of tangent line

Additional info:

• Instantaneous velocity is foundational for the derivative concept in calculus.

• Limits are used to rigorously define derivatives and continuity.