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.
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$$t$$ seconds is given by s(t)=96t−16t2$$s(t) = 96t - 16t^2$$.
96t$$96t$$ 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=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.
Average velocity = change in position / elapsed time.
Formula: $$v{avg} = (s(t2) - s(t1)) / (t2 - t_1)$$
Example: Find the average velocity between t=1$$t = 1$$ and t=3$$t = 3$$.
s(3)=−16(32)+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=80$$s(1) = -16(1^2) + 96(1) = -16 + 96 = 80$$
vavg=(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=1$$t = 1$$ and t=2$$t = 2$$.
s(2)=−16(22)+96(2)=−16(4)+192=−64+192=128$$s(2) = -16(2^2) + 96(2) = -16(4) + 192 = -64 + 192 = 128$$
vavg=(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.
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.
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$$t = 1$$ 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$$t = 1$$ is 64 feet per second.
Instantaneous velocity notation: limt→1t−1s(t)−s(1)$$\lim_{t \to 1} \frac{s(t) - s(1)}{t - 1}$$
The limit as t$$t$$ approaches 1 of the average velocity.
t$$t$$ represents the changing time interval.
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.
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.
Limits: Introduction to Average and Instantaneous Velocity