Mathematics with Maple: Exploring Functions Beyond Polynomials
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Chapter 1: Introduction to Functions
In earlier discussions (posts #8 and #9), we examined 'Linear' and 'Polynomial' functions alongside their graphical representations on the coordinate plane using real numbers. In this article, we will touch upon various commonly utilized functions and introduce a few advanced special functions. Due to the brevity of this post, we won't be able to cover all the details comprehensively, but we will provide an overview of these functions and their graphs for real numbers. Additionally, a few properties will be highlighted as space allows.
Starting from this post, we will be utilizing the 'Interactive Development Front End', a layer built on the Maple software, commonly referred to as 'Maple Learn'. This application serves as an open canvas that comprehends mathematical concepts. In prior posts, we utilized 'GeoGebra' for basic function graphing, which has an almost negligible learning curve. However, only a select few mathematical software options, including Maple, can handle special functions effectively. Maple Learn can graph functions on a coordinate plane similarly to GeoGebra. While there is a learning curve associated with Maple Learn, those familiar with Microsoft Word or similar software will find the interface quite intuitive. The menus and ribbon icons in Maple Learn are tailored to its mathematical functions.
As an added incentive for users, Maple Learn is available for a nominal monthly fee, with no long-term commitments required. Additionally, since it operates in the cloud, there is no need to purchase or download the Maple or Maple Learn software.
Commonly Used Functions
A variety of commonly utilized functions exist, most of which can be categorized based on the following types:
- Exponential function of a variable 'x'
- Trigonometric function of a variable 'x'
- Logarithmic function of a variable 'x'
- Irrational square root function of a variable 'x'
- Inverse function of a variable 'x'
Special Advanced Functions
There are several advanced functions used in engineering and science that have been developed through extensive mathematical exploration. These advanced functions can often be challenging to grasp simply by reviewing their formulas, yet many can be expressed in series forms derived from the aforementioned commonly used functions. Future posts will delve into some of these special functions and their series representations.
Exponential Function: e^x
This function is one of the most crucial in mathematics, frequently appearing in scientific and engineering contexts.
Trigonometric Functions: sin(x), cos(x), tan(x)
These fundamental trigonometric functions originate from the unit circle and form the foundation for other trigonometric identities. The exponential function exp(x) can be expressed using sin(x) and cos(x) through Euler's formula: exp(x) = cos(x) + i*sin(x).
Logarithmic Function: ln(x), log(x)
The logarithmic function is inversely related to the exponential function. This inverse is not a reciprocal but rather signifies a reflection about the line x=y, creating a mirror effect.
Irrational Square Root Function: sqrt(x)
All square root functions are categorized as irrational, with sqrt(x) being the simplest form among them.
Inverse Function: 1/x
Gamma Function: Gamma(x)
This advanced function plays a significant role in various mathematical applications.
Gauss Function: integrate(exp(-x²,{x,0,inf})
The Gauss function is defined in Maple in terms of an error function. Its application is widely recognized in statistics among engineers and scientists.
Final Thoughts
This post serves as an introduction to the foundational exponential and trigonometric functions, along with two notable advanced functions. There are numerous other functions based on these commonly used functions and the two special functions mentioned. Future posts will introduce more functions as they pertain to engineering and scientific applications.
Next Time: Our upcoming discussion will focus on three key operations in Calculus, starting with the basic use of simple functions and advancing towards differential equations and their solutions.
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