The Limitations of Distance = Rate × Time

Calculus: The Big Picture

In this course, Professor Paul Allen (Lewis & Clark College) explains the main ideas behind calculus, with the hope that we can keep this big picture in mind when learning calculus computation techniques. In the first mini-lecture, we discuss the limitations of the equation, distance = rate × time, and see how calculus provides a resolution to this limitation using a concept called linear approximation. In the second mini-lecture, we discuss linear approximation and cumulative effect using an example of going on a 2-hour walk. We see how calculating the cumulative effect becomes more and more accurate when using smaller and smaller intervals, and we assign mathematical notation to this concept (the definite integral). In the third mini-lecture, we continue with our example of going on a 2-hour walk to further discuss linear approximation and rates of change as the motivation behind the derivative. In the fourth mini-lecture, we tie together the concepts we’ve been learning in a discussion on the Fundamental Theorem of Calculus.

The Limitations of Distance = Rate × Time

In this mini-lecture, we introduce the equation, distance = rate × time, and consider its limitations. In particular, we discuss: (i) the equation, distance = rate × time, and consider some examples; (ii) the generalised form of this rate equation, amount = rate × input, and consider some examples; (iii) the limitations these rate equations when the rate is not constant; (iv) how calculus provides a resolution to these limitations with a concept called linear approximation, which is the idea that on small scales the rate can be approximated by a constant; and (v) ‘the limit’ as a tool used when applying the concept of linear approximation.