- Definition and properties
- Exponential function
- Properties of the exponential function
- Logarithmic function
- Properties of the logarithm function
- Sine, Cosine, and Tangent Functions
- Derivatives and integrals
- Derivative of the exponential function
- Integral of the exponential function
- Table of derivatives and integrals of transcendent functions
- Examples
- Example 1
- Example 2
- References
The elementary transcendental functions are the exponential, logarithmic, trigonometric, inverse trigonometric functions, hyperbolic and inverse hyperbolic functions. That is, they are those that cannot be expressed by means of a polynomial, a quotient of polynomials or roots of polynomials.
The non-elementary transcendent functions are also known as special functions and among them the error function can be named. The algebraic functions (polynomials, quotients of polynomials and roots of polynomials) together with the elementary transcendental functions constitute what in mathematics are known as elementary functions.
Transcendent functions are also considered those that result from operations between transcendent functions or between transcendent and algebraic functions. These operations are: the sum and difference of functions, product and quotient of functions, as well as the composition of two or more functions.
Definition and properties
Exponential function
It is a real function of real independent variable of the form:
f (x) = a ^ x = a x
where a is a fixed positive real number (a> 0) called the base. The circumflex or superscript are used to denote the potentiating operation.
Let's say a = 2 then the function looks like this:
f (x) = 2 ^ x = 2 x
Which will be evaluated for several values of the independent variable x:
Below is a graph where the exponential function is represented for several values of the base, including the base e (Neper number e ≃ 2.72). The base e is so important that generally speaking of an exponential function we think of e ^ x, which is also denoted exp (x).
Figure 1. Exponential function a ^ x, for various values of the base a. (Own elaboration)
Properties of the exponential function
From figure 1 it can be seen that the domain of the exponential functions are the real numbers (Dom f = R) and the range or path is the positive reals (Ran f = R +).
On the other hand, regardless of the value of the base a, all exponential functions pass through the point (0, 1) and through the point (1, a).
When the base a> 1, then the function is increasing and when 0 <a <1 the function is decreasing.
The curves of y = a ^ x and y = (1 / a) ^ x are symmetric about the Y axis.
With the exception of the case a = 1, the exponential function is injective, that is, to each value of the image corresponds one and only one starting value.
Logarithmic function
It is a real function of real independent variable based on the definition of the logarithm of a number. The logarithm based on a number x is the number y to which the base must be raised to obtain the argument x:
log a (x) = y ⇔ a ^ y = x
That is, the logarithm function based on is the inverse function of the exponential function based on.
For example:
log 2 1 = 0, since 2 ^ 0 = 1
Another case, log 2 4 = 2, because 2 ^ 2 = 4
The root logarithm of 2 is log 2 √2 = ½, because 2 ^ ½ = √2
log 2 ¼ = -2, since 2 ^ (- 2) = ¼
Below is a graph of the logarithm function in various bases.
Figure 2. Exponential function for different values of the base. (Own elaboration)
Properties of the logarithm function
The domain of the logarithm function y (x) = log a (x) are the positive real numbers R +. The travel range or are real numbers R.
Regardless of the base, the logarithm function always passes through the point (1,0) and the point (a, 1) belongs to the graph of that function.
In the case that the base a is greater than unity (a> 1) the logarithm function is increasing. But if (0 <a <1) then it is a decreasing function.
Sine, Cosine, and Tangent Functions
The sine function assigns a real number and to each x value, where x represents the measure of an angle in radians. To obtain the value of the Sen (x) of an angle, the angle is represented in the unit circle and the projection of said angle on the vertical axis is the sine corresponding to that angle.
The trigonometric circle and sine for various angular values X1, X2, X3, and X4 are shown below (in Figure 3).
Figure 3. Trigonometric circle and the sine of various angles. (Own elaboration)
Defined in this way, the maximum value that the function Sen (x) can have is 1, which occurs when x = π / 2 + 2π n, where n is an integer (0, ± 1, ± 2,). The minimum value that the function Sen (x) can take occurs when x = 3π / 2 + 2π n.
The cosine function y = Cos (x) is defined in a similar way, but the projection of the angular positions P1, P2, etc. is carried out on the horizontal axis of the trigonometric circle.
On the other hand, the function y = Tan (x) is the quotient between the sine function and the cosine function.
Below is a graph of the transcendent functions Sen (x), Cos (x) and Tan (x)
Figure 4. Graph of the transcendent functions, Sine, Cosine and Tangent. (Own elaboration)
Derivatives and integrals
Derivative of the exponential function
The derivative y 'of the exponential function y = a ^ x is the function a ^ x multiplied by the natural logarithm of the base a:
y '= (a ^ x)' = a ^ x ln a
In the particular case of the base e, the derivative of the exponential function is the exponential function itself.
Integral of the exponential function
The indefinite integral of a ^ x is the function itself divided by the natural logarithm of the base.
In the particular case of the base e, the integral of the exponential function is the exponential function itself.
Table of derivatives and integrals of transcendent functions
Below is a summary table of the main transcendent functions, their derivatives and indefinite integrals (antiderivatives):
Table of derivatives and indefinite integrals for some transcendent functions. (Own elaboration)
Examples
Example 1
Find the function resulting from the composition of the function f (x) = x ^ 3 with the function g (x) = cos (x):
(fog) (x) = f (g (x)) = cos 3 (x)
Its derivative and its indefinite integral is:
Example 2
Find the composition of the function g with the function f, where g and f are the functions defined in the previous example:
(gof) (x) = g (f (x)) = cos (x 3)
It should be noted that the composition of functions is not a commutative operation.
The derivative and the indefinite integral for this function are respectively:
The integral was left indicated because it is not possible to write the result as a combination of elementary functions exactly.
References
- Calculus of a Single Variable. Ron Larson, Bruce H. Edwards. Cengage Learning, Nov 10 2008
- The Implicit Function Theorem: History, Theory, and Applications. Steven G. Krantz, Harold R. Parks. Springer Science & Business Media, Nov 9. 2012
- Multivariable Analysis. Satish Shirali, Harkrishan Lal Vasudeva. Springer Science & Business Media, Dec 13. 2010
- System Dynamics: Modeling, Simulation, and Control of Mechatronic Systems. Dean C. Karnopp, Donald L. Margolis, Ronald C. Rosenberg. John Wiley & Sons, Mar 7 2012
- Calculus: Mathematics and Modeling. William Bauldry, Joseph R. Fiedler, Frank R. Giordano, Ed Lodi, Rick Vitray. Addison Wesley Longman, Jan 1 1999
- wikipedia. Transcendent function. Recovered from: es.wikipedia.com