Limits: Introduction to Average and Instantaneous Velocity

Limits: An Introduction

Introduction

• Chapter two covers limits, which form the basis of calculus.
• The concept of limits will be applied in various ways throughout the course.
• The lesson plan involves finding average velocity and the slope of tangent lines.
• These two seemingly unrelated concepts will be shown to be related.

Average Velocity

• A rock is launched vertically upward from the ground with a speed of 96 feet per second.
• Air resistance is neglected.
• The position of the rock after $t$ seconds is given by $s(t) = 96t - 16t^2$.
  • $96t$ represents the initial upward velocity.
  • $-16t^2$ represents the effect of gravity.
• The graph of $s(t)$ is a downward-opening parabola.
• $s = 0$ corresponds to the ground.
• The parabola illustrates the velocity over time, not the trajectory of the rock.
• The rock moves straight up and down.
• Average velocity is the change in position divided by the change in time.
  • This is equivalent to the slope of a line.

Calculating Average Velocity

• Average velocity = change in position / elapsed time.
• Formula: $v<em>{avg} = (s(t</em>2) - s(t<em>1)) / (t</em>2 - t_1)$
• Example: Find the average velocity between $t = 1$ and $t = 3$.
  • $s(3) = -16(3^2) + 96(3) = -16(9) + 288 = -144 + 288 = 144$
  • $s(1) = -16(1^2) + 96(1) = -16 + 96 = 80$
  • $v_{avg} = (144 - 80) / (3 - 1) = 64 / 2 = 32$ feet per second.
• The average velocity varies depending on the time interval.
• Example: Find the average velocity between $t = 1$ and $t = 2$.
  • $s(2) = -16(2^2) + 96(2) = -16(4) + 192 = -64 + 192 = 128$
  • $v_{avg} = (128 - 80) / (2 - 1) = 48 / 1 = 48$ feet per second.
• On the interval from one to two seconds, the average velocity is 48 feet per second.

Generic Form and Secant Lines

• The generic form to find average velocity is represented using a slope triangle with secant lines.
• Change in position: $s(t<em>1) - s(t</em>0)$
• Change in time: $t<em>1 - t</em>0$
• Secant Line: A line that intersects a curve at two or more points.

Instantaneous Velocity

• Average velocity is found between two distinct points.
• Instantaneous velocity is the velocity at a single point in time.
• To find instantaneous velocity, consider smaller and smaller intervals around that point.
• This concept is related to limits.
• Example: Estimating instantaneous velocity at $t = 1$ second.
  • Interval from 1 to 2 seconds: 48 feet per second.
  • Interval from 1 to 1.5 seconds: 56 feet per second.
  • Interval from 1 to 1.1 seconds: 62.4 feet per second.
  • Interval from 1 to 1.01 seconds: 63.84 feet per second.
  • Interval from 1 to 1.001 seconds: 63.984 feet per second.
• As the interval gets smaller, the average velocity approaches 64 feet per second.
• Instantaneous velocity at $t = 1$ is 64 feet per second.

Notation

• Instantaneous velocity notation: $\lim_{t \to 1} \frac{s(t) - s(1)}{t - 1}$
• The limit as $t$ approaches 1 of the average velocity.
• $t$ represents the changing time interval.

Graphical Illustration

• As the interval gets smaller, the average velocity approaches the instantaneous velocity.
• When the interval is long, we could be far away from the instantaneous velocity.
• As the interval shrinks, we get closer to the instantaneous velocity.

Secant Lines and Tangent Lines

• A secant line intersects the curve at two points.
• As the points on the curve get closer, the secant line gets closer to the curve.
• When the secant line touches the curve at only one point, it becomes a tangent line.
• The slope of the secant lines approaches the slope of the tangent line.
• This illustrates instantaneous velocity graphically.