In this video, speed, defined as the distance traveled in one unit of time, is represented by rectangles on the distance-time (distance as a function of time) graph. At first, the examples stick to a constant speed per unit of time, and illustrate that within each time unit, the slope of the line segment and the area of the rectangle (which has unit width, and the difference in corresponding outputs between the maximum and minimum bounds of that unit input as height) are equal, up to sign. For example, if a unit width rectangle has a height of 3, then both its area, and the unsigned slope of the line segment connecting a pair of its diagonal corners must equal 3. Of course, distance-time graphs don’t always adhere to a constant speed per time unit (and most other calculus applications also involve continuous change). This is shown in the video at 7:42 - 8:39, although their continuous function still seems artificially well-suited to the use of unit length rectangles. In general, rectangle length can’t remain unit; we must shrink it indefinitely in order to produce indefinitely good curve/area approximations.
While I’m optimistic about using this illustration as an intuition for the fundamental theorem of calculus, associating slope to area on a single graph prior to moving the area to a speed-time graph, I am puzzled by some apparent slight of hand. A slope of 3 is not actually identical to an area of 3; it only appears this way by ignoring units. For instance, the area of a rectangle might be 3 inches squared, but the slope of a line could never be 3 inches squared. A slope in single variable calculus is almost always unitless, representing a ratio with cancelling units, such as y inches / x inches = 3. The fact that we’re taking the derivative entails that the ratio is all we care about; the actual values of [FONT=MathJax_Math-italic]y[/FONT] at [FONT=MathJax_Math-italic]x[/FONT] are lost in the process of shrinking them indefinitely. This jump between slope and area is near the very heart of calculus, but its justification seems shaky.
A second issue is that the association seems to fail when the rectangles are made non-unit length, but in general we need to shrink them to length dx. Suppose a distance of 10 is traversed between x = 2 and x = 4. The average slope would be 5, yet the area of a rectangle encompassing the region would be 20. Or suppose a distance of 10 is traversed between t =1 and t = 1.5. Here the average slope would be 20 while the corresponding area would be 5. In general, a slope equals the change in y divided by the change in x, while the area of our rectangle equals the change in y times the change in x. These quantities only coincide in the video because the change in x is intentionally set to the multiplicative identity, which is its own reciprocal. Otherwise, the association between slope and area appears to break down.
How are these problems overcome to understand the association between derivatives and antiderivatives/integrals?
While I’m optimistic about using this illustration as an intuition for the fundamental theorem of calculus, associating slope to area on a single graph prior to moving the area to a speed-time graph, I am puzzled by some apparent slight of hand. A slope of 3 is not actually identical to an area of 3; it only appears this way by ignoring units. For instance, the area of a rectangle might be 3 inches squared, but the slope of a line could never be 3 inches squared. A slope in single variable calculus is almost always unitless, representing a ratio with cancelling units, such as y inches / x inches = 3. The fact that we’re taking the derivative entails that the ratio is all we care about; the actual values of [FONT=MathJax_Math-italic]y[/FONT] at [FONT=MathJax_Math-italic]x[/FONT] are lost in the process of shrinking them indefinitely. This jump between slope and area is near the very heart of calculus, but its justification seems shaky.
A second issue is that the association seems to fail when the rectangles are made non-unit length, but in general we need to shrink them to length dx. Suppose a distance of 10 is traversed between x = 2 and x = 4. The average slope would be 5, yet the area of a rectangle encompassing the region would be 20. Or suppose a distance of 10 is traversed between t =1 and t = 1.5. Here the average slope would be 20 while the corresponding area would be 5. In general, a slope equals the change in y divided by the change in x, while the area of our rectangle equals the change in y times the change in x. These quantities only coincide in the video because the change in x is intentionally set to the multiplicative identity, which is its own reciprocal. Otherwise, the association between slope and area appears to break down.
How are these problems overcome to understand the association between derivatives and antiderivatives/integrals?