Average Velocity and Secant Line Slope in Calculus

Motion Along a Line: Average Velocity

Definition and Formula

In calculus, the average velocity of an object moving along a straight line over a time interval is a fundamental concept used to describe the rate of change of position. It is calculated as the change in position divided by the change in time.

• Position function: $s(t)$ gives the position of the object at time $t$.

• Average velocity over $[a, b]$:

$\text{Average velocity} = \frac{s(b) - s(a)}{b - a}$

• $s(a)$: position at initial time $a$

• $s(b)$: position at final time $b$

• $b - a$: length of the time interval

Example Calculation

Suppose $s(2) = 150$ and $s(5) = 240$. Find the average velocity over $[2, 5]$:

• $\frac{s(5) - s(2)}{5 - 2} = \frac{240 - 150}{3} = \frac{90}{3} = 30$

Interpretation: The object moves at an average velocity of 30 units per time interval from $t=2$ to $t=5$.

Secant Line Slope

Definition and Formula

The slope of the secant line between two points on the graph of a function represents the average rate of change of the function over that interval. For a function $f(x)$, the secant line through points $(a, f(a))$ and $(b, f(b))$ has slope:

• $\text{Slope of secant line} = \frac{f(b) - f(a)}{b - a}$

This is the same formula as average velocity when $f(x)$ is a position function.

Connection to Tangent Line

• As $b$ approaches $a$, the secant line approaches the tangent line at $a$.

• The slope of the tangent line is the instantaneous rate of change (the derivative).

$\text{Slope of tangent line at } t = a: \quad \lim_{h \to 0} \frac{s(a+h) - s(a)}{h}$

Worked Examples: Average Velocity for Quadratic Position Functions

Example 1: Interval [0, 8]

Given $s(t) = -4.9t^2 + 34t + 25$, find the average velocity over $[0, 8]$:

• $s(8) = -4.9(8)^2 + 34(8) + 25 = -313.6 + 272 + 25 = -313.6 + 297 = -16.6$

• $s(0) = -4.9(0)^2 + 34(0) + 25 = 25$

• Average velocity: $\frac{-16.6 - 25}{8 - 0} = \frac{-41.6}{8} = -5.2$

Interpretation: The negative average velocity indicates the object is moving in the opposite direction over the interval.

Example 2: Interval [0, 6]

• $s(6) = -4.9(6)^2 + 34(6) + 25 = -176.4 + 204 + 25 = 52.6$

• $s(0) = 25$

• Average velocity: $\frac{52.6 - 25}{6 - 0} = \frac{27.6}{6} = 4.6$

Interpretation: The positive average velocity shows the object is moving forward over this interval.

Example 3: Interval [0, 4]

• $s(4) = -4.9(4)^2 + 34(4) + 25 = -78.4 + 136 + 25 = 82.6$

• $s(0) = 25$

• Average velocity: $\frac{82.6 - 25}{4 - 0} = \frac{57.6}{4} = 14.4$

Interpretation: The average velocity increases as the interval shortens and the object moves forward.

Summary Table: Average Velocity Calculations

Interval [a, b]	$s(a)$	$s(b)$	Average Velocity
[0, 8]	25	-16.6	-5.2
[0, 6]	25	52.6	4.6
[0, 4]	25	82.6	14.4

Key Concepts and Applications

• Average velocity is a measure of the overall change in position per unit time over an interval.

• Secant line slope generalizes average rate of change for any function, not just position.

• These concepts are foundational for understanding instantaneous rate of change and the derivative in calculus.

• Applications include physics (motion), economics (rate of change of cost or revenue), and biology (population growth rates).