Worksheet for Section 2.1

Section 2.1 considers the problem of measuring speed, in particular instantaneous velocity.

Example: Consider an object falling from a height of 10 feet. Acceleration due to gravity is constant, $32$ft$/$sec$^2$, downward, so to model the motion of the object, use a function $s(t)$ representing the height of the object above the ground at time $t$: $s(t)=10-16t^2$. How fast is the object travelling when it hits the ground?

As a first approximation for this calculation, consider that the object is at height when you drop it (at time $t=0$, $s(0)=10$). Now find the value of $t$ when the object hits the ground (at height 0, so find $t$ when $s(t)=0$). Now since speed=distance/time, compute the speed of the object as $$\frac{(10)-(0)}{(0)-(?)}$$ (where ``$?$'' is filled in with the time the object hits the ground). This is not a particularly good approximation, though -- it amounts to finding the average speed of the object as it falls.

This computation of average speed corresponds to finding the slope of a secant line to the graph of the function $s(x)$. In general, for a function $f(x)$, the slope of the line through the points $(a,f(a))$ and $(b,f(b))$ on the graph of $f(x)$ is given by:

$$m_{sec}=\frac{f(b)-f(a)}{b-a}=\frac{f(a+h)-f(a)}{h}$$   (where $b=a+h$)

For another example of a secant line, sketch the graph of $f(x)=x^2$, and draw the line through the points on the graph corresponding to $x=-1$ and $x=2$. What is the slope of that line? Now apply this formula to the falling object problem above, with $s(t)=10-16t^2$, $a=0$, and $h$ the time when the object hits the ground.

To get a more accurate computation of the speed of the object when it hits the ground, you need to find the slope of a tangent line to the graph of the function, at the point on the graph corresponding to when the object hits. In the above computations, then, you will need to take a limit of the slope expression as the length of time $h$ approaches 0. Start with the function $f(x)=x^2$ above -- let $a=2$ and take a limit in the secant line slope expression as $h$ approaches 0. To do this, first generate a table of values of the slope expression for values of $h$ near 0 (the Limits applet will be useful here), then evaluate the limit algebraically using techniques from Section 1.8. Next, use the function $s(t)=10-16t^2$, and find the limit of the slope expression as $h$ approaches 0, at the time when the object hits the ground. This is the instantaneous velocity of the object.