Suppose that the speedometer on your car reads 60 miles per hour at 1 PM and 70 miles per hour at 2 PM. Using math, argue why your speedometer must have read 65 miles per hour at some point between 1 PM and 2 PM.
+129850 Let's assume that we can represent your speed at 1 PM by the point (1,60) on a graph and that your speed at 2 PM can be represented by the point (2, 70).....
Since the time [ represented by all the x values between x =1 and x = 2 ] is continuous between these two points, an associated speed at each time [represented by a y value] can be associated wth that particular time. And a continuous curve [ of some type] consisting of all these (x,y) values can be drawn connecting the two points mentioned above.......And since the curve is continous, it is obvious that no matter how fast you travel or how slowly you travel between x = 1 and x = 2 [ 0 being the lower bound, of course] that at some x value, your speed will = 65 mph
In strict math terms this says that, given that f(a) and f(b) exist on some continuous curve.......then that curve will, at the very least, take on all y values between f(a) and f(b)
BTW - This is known in Calculus as the " Intermediate Value Theorem"
