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11.1: Introduction

  • Page ID
    90301
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    There are two main topics in calculus: derivatives and integrals. You learned that derivatives are useful in providing rates of change in either time or space. Integrals provide areas under curves, but also are useful in providing other types of sums over continuous bodies, such as lengths, areas, volumes, moments of inertia, or flux integrals. In physics, one can look at graphs of position versus time and the slope (derivative) of such a function gives the velocity. (See Figure \(\PageIndex{1}\).) By plotting velocity versus time you can either look at the derivative to obtain acceleration, or you could look at the area under the curve and get the displacement: \[ x = \int^t_{t_0} v \, dt \label{eq:1} \] This is shown in Figure \(\PageIndex{2}\).

    clipboard_e6d9c85c5e7248fa88bf14f46992161f0.png
    Figure \(\PageIndex{1}\): Plot of position vs time.
    clipboard_e1fd00a973eefe97d1333f71e867702eb.png
    Figure \(\PageIndex{2}\): Plot of velocity vs time.

    Of course, you need to know how to differentiate and integrate given functions. Even before getting into differentiation and integration, you need to have a bag of functions useful in physics. Common functions are the polynomial and rational functions. You should be fairly familiar with these. Polynomial functions take the general form \[ f(x) = a_nx^n+a_{n-1}x^{n-1}+...+ a_1x+a_0 \label{eq:2}\] where \(a_n \neq 0\). This is the form of a polynomial of degree \(n\). Rational functions, \(f (x) = \frac{g(x)}{h(x)}\), consist of ratios of polynomials. Their graphs can exhibit vertical and horizontal asymptotes.

    Next are the exponential and logarithmic functions. The most common are the natural exponential and the natural logarithm. The natural exponential is given by \(f (x) = e^x\), where \(e ≈ 2.718281828 . . . \). The natural logarithm is the inverse to the exponential, denoted by \(\ln x\). (One needs to be careful, because some mathematics and physics books use \(\underline{\log}\) to mean natural exponential, whereas many of us were first trained to use this notation to mean the common logarithm, which is the ‘\(\text{log base 10}\)’. Here we will use \(\ln x\) for the natural logarithm.)

    The properties of the exponential function follow from the basic properties for exponents. Namely, we have: \[ \begin{align} e^0 \quad &= \quad 1\label{eq:3} \\ e^{-a} \quad &= \quad \dfrac{1}{e^a}\label{eq:4} \\ e^ae^b \quad &= \quad e^{a+b}\label{eq:5} \\ (e^a)^b \quad &= \quad e^{ab}\label{eq:6} \end{align} \]

    The relation between the natural logarithm and natural exponential is given by \[ y = e^x \Leftrightarrow x = \ln y \label{eq:7}\]

    Some common logarithmic properties are \[ \begin{align} \ln 1\quad &= \quad 0\label{eq:8} \\ \ln \dfrac{1}{a} \quad &= \quad -\ln a\label{eq:9} \\ \ln (ab) \quad &= \quad \ln a + \ln b\label{eq:10} \\ \ln \dfrac{a}{b} \quad &= \quad \ln a - \ln b\label{eq:11} \\ \ln \dfrac{1}{b} \quad &= \quad -\ln b\label{eq:12} \end{align} \]

    We will see applications of these relations as we progress through the course.


    This page titled 11.1: Introduction is shared under a CC BY-NC-SA 3.0 license and was authored, remixed, and/or curated by Russell Herman via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.