The displacement (in meters) of a particle moving in a...

The displacement (in meters) of a particle moving in a straight line is given by $s=t^2-8 t+18$, where $t$ is measured in seconds. (a) Find the average velocity over each time interval: (i) $[3,4]$ (ii) $[3.5 .4]$ (iii) $[4,5]$ (iv) $[4,4.5]$ (b) Find the instantaneous velocity when $t=4$. (c) Draw the graph of $s$ as a function of $t$ and draw the secant lines whose slopes are the average velocities in part (a) and the tangent line whose slope is the instantaneous velocity in part (b).

The displacement (in meters) of a particle moving in a straight line is given by $s=t^2-8 t+18$, where $t$ is measured in seconds.
(a) Find the average velocity over each time interval:
(i) $[3,4]$
(ii) $[3.5 .4]$
(iii) $[4,5]$
(iv) $[4,4.5]$
(b) Find the instantaneous velocity when $t=4$.
(c) Draw the graph of $s$ as a function of $t$ and draw the secant lines whose slopes are the average velocities in part (a) and the tangent line whose slope is the instantaneous velocity in part (b).

Key Concepts

Displacement Function

A displacement function represents the position of a particle along a line as a function of time. It is fundamental in kinematics for describing how an object's position changes over time and sets the stage for analyzing motion by providing a mathematical model of the particle's trajectory.

Average Velocity

Average velocity over a time interval is calculated as the change in displacement divided by the change in time. This concept gives a measure of the overall rate of motion between two points in time, serving as an important introductory idea for the study of rates of change in calculus.

Instantaneous Velocity

Instantaneous velocity is the velocity of a particle at a specific moment in time and is obtained by taking the derivative of the displacement function with respect to time. This concept bridges the study of average rates of change to the precise, moment-to-moment analysis of motion using calculus.

Derivative

The derivative of a function measures how that function changes as its input changes. In the context of motion, the derivative of the displacement function with respect to time gives the instantaneous velocity, highlighting the fundamental connection between differentiation and rates of change.

Secant Line

A secant line is a straight line that intersects a curve at two or more points, and its slope represents the average rate of change of the function between those points. In kinematics, the slope of the secant line between two points on a displacement-time graph provides the average velocity over that interval.

Tangent Line

A tangent line touches a curve at a single point and has a slope equal to the instantaneous rate of change of the function at that point. In the context of motion, the slope of the tangent line on a displacement-time graph at a given time represents the instantaneous velocity, connecting geometric interpretations with calculus.

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