Section 2.1 A preview of differential and integral calculus from kinematics
We start with a preview of what differential and integral calculus is all about. We start from the point of view of kinematics. We study how differential calculus, and the notion of derivative, is intimately connected to the ideas of rate of change and velocity. We then look at integral calculus, and the notion of integral, and see how it relates to the ideas of area and distance. In both cases, we realize that the notion of “limit” is key. Those are the foundations of calculus, which we will study throughout this course.
We are deliberately imprecise in this section: the aim is simply to give an overview of the foundational concepts of calculus, which we will develop more rigorously during the semester.
Objectives
You should be able to:
- Explain the difference between average and instantaneous velocity.
- Describe the limit process that arises in the computation of an instantaneous velocity.
- Describe and illustrate how to approximate the area under a curve using approximating rectangles.
- Describe the limit process that arises in the computation of the area under a curve.
- Relate the calculation of the area under a curve to the calculation of the distance covered in kinematics.
Subsection 2.1.1 Instructional videos
Subsection 2.1.2 Key concepts
Concept 2.1.1. Derivative, rate of change and velocity.
Given a function \(f\text{:}\)
- The average rate of change of \(f\) over the interval \([x_1, x_2]\) is given by\begin{equation*} \frac{f(x_2) - f(x_1)}{x_2- x_1}. \end{equation*}
- The instantaneous rate of change of \(f\) at \(x=c\) is called the derivative of \(f\) at \(x=c\) and is denoted by \(f'(c)\) or \(\frac{df}{dx}\Big|_{x=c}\text{.}\) It is given by the limit\begin{equation*} f'(c) = \lim_{x \to c} \frac{f(x) - f(c)}{x-c}. \end{equation*}The limit here means that we take the average rate of change over an interval \([x,c]\) with \(c\) as close as possible to \(x\text{.}\)
- If the function \(f\) is the position function of an object, then its average rate of change is the average velocity of the object over a time interval, while its instantaneous rate of change is its instantaneous velocity.
Concept 2.1.2. Integral, distance and area.
Let \(v(t)\) be the velocity function of an object, and assume that it is always positive (for simplicity):
- The distance covered by the object between \(t_1\) and \(t_2\) can be obtained by calculating the area under the graph of \(v(t)\) between \(t=t_1\) and \(t=t_2\text{.}\)
- An approximation can be obtained by slicing the area into rectangles of width \(\Delta t\) and summing over the areas of the rectangles.
- By taking the limit \(\Delta t \to 0\) (equivalently, by sending the number of rectangles to infinity), we obtain an exact expression for the area under the graph and the distance covered. The resulting expression is called the integral of the function \(v(t)\) between \(t=t_1\) and \(t=t_2\), and is written as\begin{equation*} \int_{t_1}^{t_2} v(t)\ dt. \end{equation*}It is the “inverse operation of differentiation”, as we will see later on when we study the Fundamental Theorem of Calculus.