2.5 TRANSCENDENTAL FUNCTIONS

The transcendental functions include the trigonometric functions sin x, cos x, tan x, the exponential function ex, and the natural logarithm function In x. These functions are developed in detail in Chapters 7 and 8. This section contains a brief discussion.

 

1  TRIGONOMETRIC  FUNCTIONS

   The Greek letters θ (theta) and Ø (phi) are often used for angles. In the calculus it is convenient to measure angles in radians instead of degrees. An angle θ in radians is defined as the length of the are of the angle on a circle of radius one (Figure 2.5.1). Since a circle of radius one has circumference 2π,

               360 degrees = 2π radians.

 

 

 

 

 

 

 

 

 

Figure 2.5.1

 

Thus a right angle is

                        90 degrees = π /2 radians.

     To define the sine and cosine functions, we consider a point P(x, y) on the unit circle x²+y² = 1. Let θ be the angle measured counterclockwise in radians from the point (1,0) to the point P(x, y) as shown in Figure 2.5.2. Both coordinates x and y depend on θ. Then value of x is called the cosine of θ , and the value of y is the sine of θ . In symbols,

 

                  x= cos θ,   y= sin θ.

 

 

 

 

 

 

 

 

 

 

 

 

 

Figure 2.5.2

 

The tangent of θ is defined by

                        tan θ = sin θ/ cos θ.

 

Negative angles and angles greater than 2π radians are also allowed.

The trigonometric functions can also be defined using the sides of a right triangle, but this method only works for θ between 0 and π/2. Let θ be one of the acute angles of a right triangle as shown in Figure 2.5.3.

 

 

 

 

 

 

 

 

Figure 2.5.3

 

Then         

 

 

 

 

 

 

The two definitions, with circles and right triangles, can be seen to be equivalent using similar triangles.

 

Table 2.5.1 gives the values of sin θ and cosθ for some important values of θ.

 

 

 

 

 

 

A useful identity which follows from the unit circle equation x2+ y2 = 1 is

                           sin2 θ + cos 2 θ =1.

Here sin2 θ means (sin θ) 2.

 

Figure 2.5.4 shows the graphs of sin θ and cosθ , which look like waves that oscillate between 1 and -1 and repeat every 2π radians.

 

The derivatives of the sine and cosine functions are:

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Figure 2.5.4

 

In both formulas θ is measured in radians. We can see intuitively why these are the derivatives in Figure 2.5.5.

 

In the triangle under the infinitesimal microscope,

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Figure 2.5.5

 

Notice that cos θ decreases, and Δ(cos θ) is negative in the figure, so the derivative of cos θ is -sinθ instead of just sin θ.

 

Using the rules of differentiation we can find other derivatives.

 

EXAMPLE 1   Differentiate y= sin² θ . Let u= sin² θ , y=u². Then

 

 

EXAMPLE  2  Differentiate y=sin² θ (1-cos θ). Let u=sinθ, v=1-cosθ . Then

               y=u · v, and

 

 

 

 

The other trigonometric functions (the secant, cosecant, and cotangent functions) and the inverse trigonometric functions are discussed in Chapter 7.

 

2  EXPONENTIAL  FUNCTIONS

   Given a positive real number b and a rational number m/n, the rational power bm/n is defined as

                         bm/n = _____,

the positive nth root of bm. The negative power b-m/n is

                          ________________.

As an example consider b=10. Several values of 10m/n are shown in Table 2.5.2.

 

 

 

 

 

If we plot all the rational powers 10m/n, we get a dotted line, with one value for each rational number m/n, as in Figure 2.5.6.

 

 

 

 

 

Figure 2.5.6

 

By connecting the dots with a smooth curve, we obtain a function y=10x, where x varies over all real numbers instead of just the rationals. 10x is called the exponential function with base 10. It is positive for all x and follows the rules

                  10a+b = 10a · 10b.     10a·b = (10a)b.

The derivative of 10x is a constant times 10x, approximately

                     

 

To see this let Δx be a nonzero infinitesimal. Then

 

 

 

The number st[(10Δx - 1) / Δx] is a constant which does not depend on x and can be shown to be approximately 2.303.

 

If we start with a given positive real number b instead of 10, we obtain the exponential function with base b, y=bx. The derivative of bx is equal to the constant st[bΔx-1)/Δ x] times bx. This constant depends on b. The derivative is computed as follows:

 

 

 

 

The most useful base for the calculus is the number e. e is defined as the real number such that the derivative of ex is ex itself.

 

 

                               

 

In other words, e is the real number such that the constant

 

 

 

(where Δx is a nonzero infinitesimal). It will be shown in Section 8.3 that there is such a number e and that e has the approximate value

                    e ~ 2.71828.

The function y= ex is called the exponential function. ex is always positive and follows the rules

                      ea+b  = ea ·eb           ea·b  = (ea )b,  e0 = 1. 

Figure 2.5.7 shows the graph of y= ex.

 

 

 

 

 

 

 

 

 

Figure 2.5.7

 

EXAMPLE  3  Find the derivative of y=x²ex. By the Product Rule,

 

 

 

3  THE NATURAL LOGARITHM

The inverse of the exponential function x=e y is the natural logarithm function, written

                          y = 1n x.

Verbally, 1n x is the number y such that e y=x. Since y= 1n x is the inverse function of x = e y, we have

                       e 1n a = a,  1n(ea ) = a.

The simplest values of y= 1n x are

                       1n(1/e) = -1,   1n(1) = 0, 1ne = 1.

Figure 2.5.8 shows the graph of y= 1n x. It is defined only for x >0.

 

 

 

 

 

Figure  2.5.8

 

The most important rules for logarithms are

                       1n (ab) = 1n a + 1n b,

                       1n (ab) = b·1n a .

The natural logarithm function is important in calculus because its derivative is simply 1/x,
                 

                      

 

This can be derived from the Inverse Function Rule.

If                     y = 1n x,

Then                  x = e y,

 

 

 

 

 

The natural logarithm is also called the logarithm to the base e and is sometimes written loge x. Logarithms to other bases are discussed in Chapter 8.

 

EXAMPLE 4  Differentiate

            

 

 

 

4  SUMMARY

   Here is a list of the new derivatives given in this section.

                

 

 

 

 

 

 

 

Tables of values for sin x, cos x, ex, and 1n x can be found at the end of the book.

 

PROBLEMS  FOR  SECTION  2.5

In Problems 1-20, find the derivative.

1   y= cos ² θ                       2   s= tan² t

3   y=2 sin x + 3 cos x                4   y= sin x ·cos x    

5   _______________                6________________

7   y= sin n θ                        8  y= tan n θ

9  s= t sin t                        10 ________________

11  y=xex                                    12  y=1/( 1+ ex )

13   y= (1n x) 2                     14   y= x 1n x

15   y= ex·1n x                     16  y= ex·sin x

17   ___________                    18   u= (1+e v ) (1 - ev )

19   y= xn·1n x                      20   y= (1n x)n 

 

 

In Problems 21-24, find the equation of the tangent line at the given point.

21  

 

23       y= x - 1n x at (e, e-1)          24    y= e - x at(0, 1)