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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 tt$$t$$ seconds is given by s(t)=96t16t2s(t) = 96t - 16t^2$$s(t) = 96t - 16t^2$$.

    • 96t96t$$96t$$ represents the initial upward velocity.

    • 16t2-16t^2$$-16t^2$$ represents the effect of gravity.

  • The graph of s(t)s(t)$$s(t)$$ is a downward-opening parabola.

  • s=0s = 0$$s = 0$$ 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: $$v{avg} = (s(t2) - s(t1)) / (t2 - t_1)$$

  • Example: Find the average velocity between t=1t = 1$$t = 1$$ and t=3t = 3$$t = 3$$.

    • s(3)=16(32)+96(3)=16(9)+288=144+288=144s(3) = -16(3^2) + 96(3) = -16(9) + 288 = -144 + 288 = 144$$s(3) = -16(3^2) + 96(3) = -16(9) + 288 = -144 + 288 = 144$$

    • s(1)=16(12)+96(1)=16+96=80s(1) = -16(1^2) + 96(1) = -16 + 96 = 80$$s(1) = -16(1^2) + 96(1) = -16 + 96 = 80$$

    • vavg=(14480)/(31)=64/2=32v_{avg} = (144 - 80) / (3 - 1) = 64 / 2 = 32$$v_{avg} = (144 - 80) / (3 - 1) = 64 / 2 = 32$$ feet per second.

  • The average velocity varies depending on the time interval.

  • Example: Find the average velocity between t=1t = 1$$t = 1$$ and t=2t = 2$$t = 2$$.

    • s(2)=16(22)+96(2)=16(4)+192=64+192=128s(2) = -16(2^2) + 96(2) = -16(4) + 192 = -64 + 192 = 128$$s(2) = -16(2^2) + 96(2) = -16(4) + 192 = -64 + 192 = 128$$

    • vavg=(12880)/(21)=48/1=48v_{avg} = (128 - 80) / (2 - 1) = 48 / 1 = 48$$v_{avg} = (128 - 80) / (2 - 1) = 48 / 1 = 48$$ 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: $$s(t1) - s(t0)$$

  • Change in time: $$t1 - t0$$

  • 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=1t = 1$$t = 1$$ 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=1t = 1$$t = 1$$ is 64 feet per second.

Notation

  • Instantaneous velocity notation: limt1s(t)s(1)t1\lim_{t \to 1} \frac{s(t) - s(1)}{t - 1}$$\lim_{t \to 1} \frac{s(t) - s(1)}{t - 1}$$

  • The limit as tt$$t$$ approaches 1 of the average velocity.

  • tt$$t$$ 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.


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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 tt seconds is given by s(t)=96t16t2s(t) = 96t - 16t^2.
    • 96t96t represents the initial upward velocity.
    • 16t2-16t^2 represents the effect of gravity.
  • The graph of s(t)s(t) is a downward-opening parabola.
  • s=0s = 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: vavg=(s(t2)s(t1))/(t2t1)v{avg} = (s(t2) - s(t1)) / (t2 - t_1)
  • Example: Find the average velocity between t=1t = 1 and t=3t = 3.
    • s(3)=16(32)+96(3)=16(9)+288=144+288=144s(3) = -16(3^2) + 96(3) = -16(9) + 288 = -144 + 288 = 144
    • s(1)=16(12)+96(1)=16+96=80s(1) = -16(1^2) + 96(1) = -16 + 96 = 80
    • vavg=(14480)/(31)=64/2=32v_{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=1t = 1 and t=2t = 2.
    • s(2)=16(22)+96(2)=16(4)+192=64+192=128s(2) = -16(2^2) + 96(2) = -16(4) + 192 = -64 + 192 = 128
    • vavg=(12880)/(21)=48/1=48v_{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(t1)s(t0)s(t1) - s(t0)
  • Change in time: t1t0t1 - t0
  • 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=1t = 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=1t = 1 is 64 feet per second.

Notation

  • Instantaneous velocity notation: limt1s(t)s(1)t1\lim_{t \to 1} \frac{s(t) - s(1)}{t - 1}
  • The limit as tt approaches 1 of the average velocity.
  • tt 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.