Crash introduction to infinitesimal calculus.
- Limits and Derivatives - this file
- Integrals
- Ordinary Differential Equations
This session provides a crash introduction to limits and derivatives. Without going into details of a rigorous definition of the limit of a real function, first section should provide an intuitive idea of the limit, provide some examples of pathological cases, and a bridge to the derivative. The second section should provide an illustration of the geometrical interpretation of the derivative, introducing the definition, and showing the way to evaluate the derivative of some fundamental functions.
A list of the missing steps for a complete understanding of the topics is presented at the end of this section. These steps will be covered in the following sessions.
The interactive html blocks have been designed with the fundamental help of Gemini.
Introduction to Limits
See: Matematica: Calcolo infinitesimale: Introduzione al Calcolo: Limiti
Before going into the details of a rigorous definition, this section aims at
providing an intuitive idea of the concept of limit, in 4 cases listed in the SELECT LIMIT CASE dropdown menu. In the finite/finite example for the continuous function \(f(x) = \tfrac{x^2-1}{2}\), taking the limit for \(x \rightarrow 2\) just means evaluating the function \(f(x)\) for \(x = 2\), and thus \(\lim_{x\rightarrow 2} f(x) = f(2) = \tfrac{3}{2}\)
showing some pathological cases: a function “with jumps” (a discontinuous function), an “increasingly fast” oscillating function (\(\sin(1/x)\) for \(x \rightarrow 0\))
presenting an example of indeterminate form \(\tfrac{0}{0}\). The function \(f(x) = \tfrac{x^2-4}{x-2}\) is the ratio of two polynomial functions whose values are zero for \(x=2\). The existence conditions for \(f(x)\) read \(x \ne 2\), otherwise the denominator becomes equal to zero (and we know that’s not possible to divide by zero). Anyway, the limit operation allows to study the behavior of the function as \(x\) approaches \(2\). Since \(x^2-4 = (x-2)(x+2)\), for all the \(x \ne 2\) the function \(f(x) = \tfrac{\cancel{(x-2)}(x+2)}{\cancel{x-2}}\) is equivalent to \(x+2\). Now, \(x+2\) is a continuous function - as in the first bullet of this item list - that gives no issues at all when evaluated for \(x = 2\).
\[\lim_{x \rightarrow 2} \frac{(x-2)(x+2)}{x-2} = \lim_{x \rightarrow 2} \left( x + 2 \right) = 4 \ .\]
Introduction to Derivatives
See: Matematica: Calcolo infinitesimale: Derivata
A geometrical interpretation is presented first, then the definition is provided along with the fundamental derivatives.
Geometrical Interpretation
In the previous sessions, we tried to get familiar with real functions and their graph. Now, we provide the geometrical interpretation of the derivative \(f'(x)\) of a function \(f(x)\). Given a function \(f(x)\), the function \(f'(x)\) provides the slope of the tangent line to the graph of the function in point \((x, f(x))\).
Definition
The derivative \(f'(x)\) of function \(f(x)\), evaluated for the generic value \(x\) of the independent variable can be interpreted as the slope of the tangent line in \((x, f(x))\). Using the interactive tab, this is equivalent to - for a given point \(A \equiv (x, f(x))\), i.e. for a given value \(x\) - to make the point \(B \equiv ( x + \Delta x, f(x+\Delta x))\) approach the point \(A\), i.e. to make \(\Delta x \rightarrow 0\). As the slope of the secant line through \(A\) and \(B\) is defined as the incremental ratio
\[m_{AB}^{sec} = \frac{y_B - y_A}{x_B-x_A} = \frac{f(x+\Delta x) - f(x)}{\Delta x} \ ,\]
the slope of the tangent line in \(A\) is just the limit for \(\Delta x \rightarrow 0\) of the incremental ratio,
\[m_{A}^{tan} = \lim_{\Delta x \rightarrow 0} \frac{f(x+\Delta x) - f(x)}{\Delta x} =: f'(x) \ .\]
As the point \(B\) approaches \(A\), the the secant approaches the tangent line (if there’s a tangent line in \(A\), and thus if the derivative in \(x\) exists).
Derivative of fundamental functions
See: Matematica: Calcolo infinitesimale: Derivata: Derivata delle funzioni fondamentali
Missing steps
Limits and continuous functions.
- Formal definition of limit for revisiting the intuitive introduction
- Operations and theorems
- Fundamental limits
- Comparison of infinites and infinitesimals (i.e. functions approaching infinite, or zero)
Derivatives.
- Derivation rules, allowing to use the derivatives of fundamental functions as the elementary blocks to evaluate the derivative of any arbitrary function
- Theorems
- Applications: approximations, studies of a function (local minima, local maxima, inflection points,…),…
Next steps
Integrals. Given the fundamental theorem of calculus, the integral is the opposite operation of the derivative. We need to understand that, and take all the steps taken with derivatives backwards: integrals of fundamental functions, rules of integration,…
Ordinary differential equations. Lots of systems of interest are governed by differential equations, especially in physics, classical mechanics and engineering. A differential equation is nothing but an equation containing a unknown function and its derivatives. We’ll introduce and solve the simplest ones, that nonetheless can be used as the mathematical model of lots of systems. Just as an example,
under some circumstances, the temperature of a body \(T(t)\) with mass \(m\), heat capacity \(C\), in an environment with temperature \(T_e(t)\) and that can exchange heat with the body, is governed by the differential equation
\[m C T'(t) + h T(t) = h T_e(t) \ .\]
Here, with all the other parameters and functions given, \(T(t)\) is the unknown of the problem.
under some circumstances, the position of a mass connected to the ground with a spring and a damper, and subject to an external force \(F_e(t)\) is governed by the differential equation
\[m x''(t) + c x'(t) + k x(t) = F_e(t) \ .\]
Here, with all the other parameters and functions given, \(x(t)\) is the unknown of the problem.
As we all learned how to solve algebraic equations, we’ll learn how to solve the simplest ordinary differential equations.