第2.5节 超越函数

2.5 TRANSCENDENTAL FUNCTIONS

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

1TRIGONOMETRIC FUNCTIONS

TheGreek letters θ (theta) and Ø (phi) are often used for angles. Inthe calculus it is convenient to measure angles in radians instead ofdegrees. An angleθ in radians is defined as the length ofthe are of the angle on a circle of radius one (Figure 2.5.1). Sincea circle of radius one has circumference 2π,

360degrees= 2π radians.

Figure2.5.1

Thusa right angle is

90degrees= π /2 radians.

Todefinethe sine and cosine functions, we considera pointP(x,y) on the unit circlex²+y² = 1. Let θ be the anglemeasured counterclockwise in radians from the point (1,0) to thepointP(x, y) as shown in Figure 2.5.2. Both coordinates xand y depend on θ. Then value of x is calledthe cosine of θ , and the value ofy is the sine of θ. In symbols,

x=cosθ, y= sin θ.

Figure2.5.2

Thetangent ofθ is defined by

tanθ= sinθ/ cos θ.

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

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

Figure2.5.3

Then

Thetwo definitions, with circles and right triangles, can be seen to beequivalent using similar triangles.

Table2.5.1 gives the values of sinθ and cosθ for someimportant values of θ.

Auseful identity which follows from the unit circle equationx2+y2 = 1 is

sin2θ + cos 2θ =1.

Heresin2θ means (sin θ) 2.

Figure2.5.4 shows the graphs of sinθ and cosθ , which looklike waves that oscillate between 1 and -1 and repeat every 2πradians.

Thederivatives of the sine and cosine functions are:

Figure2.5.4

Inboth formulasθ is measured in radians. We can seeintuitively why these are the derivatives in Figure 2.5.5.

Inthe triangle under the infinitesimal microscope,

Figure2.5.5

Noticethat cosθ decreases, and Δ(cos θ) is negative inthe figure, so the derivative of cosθ is -sinθinstead of just sin θ.

Usingthe rules of differentiation we can find other derivatives.

EXAMPLE1 Differentiatey= sin² θ . Let u=sin² θ , y=u². Then

EXAMPLE2 Differentiate y=sin²θ (1-cos θ). Letu=sinθ, v=1-cosθ . Then

y=u· v, and

Theother trigonometric functions (the secant, cosecant, and cotangentfunctions) and the inverse trigonometric functions are discussed inChapter 7.

2EXPONENTIAL FUNCTIONS

Givenapositive real number b and a rational numberm/n, therational power bm/n is defined as

bm/n=_____,

thepositiventh root of bm. The negative power b-m/nis

________________.

Asan example considerb=10. Several values of 10m/n areshown in Table 2.5.2.

Ifwe plot all the rational powers 10m/n, we get a dotted line,with one value for each rational numberm/n, as in Figure2.5.6.

Figure2.5.6

Byconnecting the dots with a smooth curve, we obtain a functiony=10x,where x varies over all real numbers instead of just therationals. 10xis called the exponential function with base10. It is positive for allx and follows the rules

10a+b= 10a· 10b. 10a·b =(10a)b.

Thederivative of 10x is a constant times 10x,approximately

Tosee this let Δx be a nonzero infinitesimal. Then

Thenumberst[(10Δx - 1) / Δx] is a constant which doesnot depend on x and can be shown to be approximately 2.303.

Ifwe start with a given positive real numberb instead of 10, weobtain the exponential function with base b, y=bx. Thederivative of bx is equal to the constant st[bΔx-1)/Δx]times bx. This constant depends on b. The derivative iscomputed as follows:

Themost useful base for the calculus is the numbere. e isdefined as the real number such that the derivative of ex isex itself.

Inother words, e is the real number such that the constant

(whereΔx is a nonzero infinitesimal). It will be shown in Section8.3 that there is such a number e and that e has theapproximate value

e~ 2.71828.

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

ea+b= ea ·ebea·b = (ea )b, e0 = 1.

Figure2.5.7 shows the graph of y= ex.

Figure2.5.7

EXAMPLE3 Find the derivative of y=x²ex. By the Product Rule,

3THE NATURAL LOGARITHM

Theinverse of the exponential function x=e y is the naturallogarithm 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

e1na = a, 1n(ea ) = a.

Thesimplest values of y= 1n x are

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

Figure2.5.8 shows the graph of y= 1n x. It is defined onlyfor x >0.

Figure2.5.8

Themost important rules for logarithms are

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

1n(ab)= b·1n a .

Thenatural logarithm function is important in calculus because itsderivative is simply 1/x,

Thiscan be derived from the Inverse Function Rule.

Ify = 1n x,

Thenx = e y,

Thenatural logarithm is also called the logarithm to the base e andis sometimes written loge x. Logarithms to other bases arediscussed in Chapter 8.

EXAMPLE4 Differentiate

4SUMMARY

Hereisa list of the new derivatives given in this section.

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

PROBLEMSFOR SECTION 2.5

InProblems 1-20, find the derivative.

1y= cos ² θ2s= tan²t

3y=2 sin x + 3 cos x 4y= sin x ·cos x

5_______________6________________

7y= sin n θ 8y= tan n θ

9s= t sint 10________________

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

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

15y= ex·1n x16y= ex·sin x

17___________18 u= (1+e v )(1 - ev )

19y= xn·1n x20y= (1n x)n

InProblems 21-24, find the equation of the tangent line at the givenpoint.

21

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