How to determine the instantaneous velocity at a time t=2s?

Question

im trying to teach myself physics and trying to figure out this problem.

i already calculated the average velocity for the time t=1.50 s to t=4.00 s and got -2.4m/s.

then i have to find the instantaneous velocity at t=2s. the answer in my book gives -3.8m/s. im having trouble understanding the book's directions as well as finding any good explanations online. can someone pleeeease show me step-by-step how to arrive at -3.8m/s?! Note: there is actually a line going from 13m down to 3.5s and hitting the parabola at the point 6m/2s, but you can't see it here.

here is the link to the graph: http://www.ux1.eiu.edu/~cfadd/1350/Hmwk/Ch02/Images/FigP2.09a.gif

Answer 1

The instantaneous velocity would be the derivative of x with respect to time , or on a graph this would be the slope of the line.  To get the exact answer you would need the equation for the parabola graphed which is

x = (t-4)^2 + 2

now the instantaneous velocity at any time is

x' = v = 2(t-4) so at t = 2 seconds I get -4 not -3.8  they are very close and the difference is probably my equation for x is not exactly correct.

Answer 2

Okay. Instantaneous velocity at t = 2 is simply a measure of the slope of the tangent line to the curve at t = 2s. Allow me to explain further:

A tangent line is a line that touches the function only once - at the point where you wish to measure. It is perpendicular to the function at that point. There are other properties of tangent lines, but most of them are irrelevant for this purpose.

Now, you draw this line at t = 2 and extend it through two known points (usually some fixed numbers on the graph's grid). Once you have these two points, since the tangent line is straight, you find the slope of it by calculating the change in rise over the change in run:

slope_tangent = (y2 - y1)/(x2 - x1)

That will give you instantaneous velocity. Hope I've helped you out.

EDIT: rscanner gave you the helpful way to do this problem using calculus. If you are not familiar with calculus do not fret: both his and my methods are correct (though calculus is more accurate because everybody draws lines differently, thus resulting in a measured error).