New Chapter 7 Transcendental Functions (1).docx

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M A T 1 3 3 C a l c u l u s w i t h A n a l y t i c G e o m e t r y I I Page 1

Chapter 7 Transcendental Functions

Functions can be categorized into two big groups – algebraic and non-algebraic

functions. Algebraic functions : Any function constructed from polynomials using

algebraic operations (addition, subtraction, multiplication, division and taking roots). All

rational functions are algebraic.Transcendental functions are non-algebraic functions. The following are examples of

such functions:

i. Trigonometric functions ii. Logarithmic functions

iii. Exponential functions iv. Inverse trigonometric functions

v. Hyperbolic functions vi. Inverse hyperbolic functions

In this chapter we shall study the properties, the graphs, derivatives and integrals ofeach of the transcendental function.

Many functions in the field of mathematics and science are inverses of one another. As

such, we shall briefly revise the concept of inverse functions before going on to

transcendental functions.

7.1 Inverse Functions and Their DerivativesObjectives

Determine the inverse of a function

Obtain the graph of the inverse function from the graph of the function

Find the inverse function

What exactly is a function?

Functions are a tool for describing the real world in mathematical terms. A function can

be represented by an equation, a graph, a numerical table or a verbal description. In this

section we are going to get familiar with functions and function notation.




An equation is a function if for any x in the domain of the equation, the equation yields

exactly one value of y .

The set of values that the independent variable is allowed to assume, i.e., all possible

input values, is called the domain of the function. The set of all values of f ( x ) as x varies

throughout the domain is called the range of the function.

Example 7.1.1: Given and find each of the following.

(a)

(b)

Notice that in this example . This usually does not happen.

However, when two compositions are both there is a relationship between the two

functions.

Consider the following evaluations.

In some way we can think of these two functions as undoing what the other did to a

number. Pairs of function that exhibit this behavior are called inverse functions .

We want to determine whether we can reverse f ; that is, for any given y in R we can go

back and find the x from which it came. If such function exists it is known as inverse of f

or simply f -inverse . Important inverse functions often show up in applications.

Comments

a. A function has one and only one inverse function.

b. The inverse of f is denoted by .

c. f( f -1 (y)) = y and f -1 (f(x)) = x for x D and y R.

d. f -1 (y) 1

f(y).

e. Domain is R and its range is D.




A function is called one-to-one if no two values of x produce the same y. In other words,

whenever .

Horizontal Line Test : The function f is one-to-one if and only if the graph of f is cut at

most once by any horizontal line.

Example 7.1.2 Determine whether each of the following is one-to-one function.

(b)




(c)

Given two one-to-one functions and if AND

then we say that and are inverses of each other.

is the inverse of and is denoted by or we can say that is

the inverse of and is denoted by .

Some functions are one-to-one on their entire domain. Other functions are not but byrestricting the function to a smaller domain we create a function that is one-to-one. If f

is 1 - 1 then f has an inverse function, and conversely if f has an inverse function then f is

1 - 1.

Example 7.1.3 Sketch the graphs of the following two functions.

and

(a) Determine the domain and range of each of the functions.(b) State the intervals on which the functions are increasing and decreasing.

(c) Which of the functions are one-to-one?

(d) Do both the functions have inverse functions?




If f is either an increasing function or a decreasing function on the domain of f then f has

an inverse function. Why?

Example 7.1.4 Determine whether each of the following function has an inverse

function.

(a) f(x) = 3x + 5.

(b) f(x) = x - 2x + 12 .

(c) f(x) = x + 7x + 4x +15 3 .

(d) f(x) = x - 3, x 02 .

Example 7.1.5 Using the functions from Example 7.1.4, find if it exist.

Homework

Exercise 7.1: 1, 3, 5, 7, 9, 19, 21, 23, 45a, 47

Review Questions

1. What functions have inverses? How do you know if two functions and are inversesof one another?

2. How are the domains, ranges and graphs of functions and their inverses related?

3. How can you sometimes express the inverse of a function of as a function of ?




7.2 Natural Logarithms

Objectives

The natural logarithmic function

The number e

The derivative of the natural logarithmic function

Integration of the natural logarithmic function

Even though the integrals1x

dx , tan x dx and sec x dx seem simple, but these

cannot be evaluated in terms of polynomials, rational functions or trigonometric

functions. The primary purpose of this topic is to define a new function which will

enable us to evaluate these and other important integrals.

For , y log x b is equivalent to x b y.

Special logarithms that arise frequently are

(i) common logarithm , and

(ii) natural logarithm

e is an irrational number that can be expressed as

elim

x1

1x

x

.

e 2.71828...




The following is the sketch of both the common logarithm and the natural logarithm.

lim

x ln x

lim

x 0 ln x

Properties of the Natural Logarithm

For any positive numbers a and c and any rational number r,

(a) ln 1 = 0 (b) ln ac = ln a + ln c

(c) lnac

ln a ln c (d) ln a r ln ar

(e) ln1c

ln c




Example 7.2.1: Without using a calculator or tables, solve for x.

(a) ln x 42 ans:

(b) ln x ln x 302 ans:

(c) (√ ) ans: (d) ans:

DEFINITION The natural logarithm function is given by

xdt , x

t x

10

1ln .

Derivatives

ddx

ln x1x

ddx

ln u1u

dudx

, u 0

Example 7.2.2: Find the derivative of y with respect to x .

(a) ) x( y 49ln ans:

(b) x y 1ln

(c) 33416lny x x ans:

(d) 32

2

1

1xlny

x ans:

(e) 12

45 3

x

x y ans: √

(f) | | ans: cot x




Integration

C xdx x

ln1

C uduu ln1

where u 0

Example 7.2.3: Evaluate the following integrals.

(a) tan x dx ans: | |

(b) dx

x

xln ans: ⁄

(c) x x

xd

53 2 ans: | |

(d) x) dx( x

lncos1

ans:

(e) dx ) x-( x 1

1 ans: √

Homework

Exercise 7.2: 3, 9, 11, 13, 15, 17, 21, 23, 25, 27, 31, 33, 41, 45, 49, 51, 53,

55, 65, 67

Review Questions

1. What is the domain and the range of a natural logarithm function?

2 What is logarithmic differentiation? Give an example.




7.3 The Exponential Function

Objectives

Definition of the natural exponential function

Properties of the natural exponential function

The derivative of the natural exponential function

Integrals of the natural exponential function

Differentiation and integration of other bases

In this topic we start of by introducing the inverse of the natural logarithm function, the

natural exponential function and its derivatives followed by exponential functions and

logarithm functions in general. We shall come across exponential functions frequentlythroughout this course.

y

x

y = ln x

1

y = x

y = f 1(x)

1




Given the function .

(a) Should have an inverse function?

(b) State the domain and range of .(c) State the domain and range of f 1.

(d) Find the equation of the inverse function.

The natural logarithmic and natural exponential functions are inverse functions of one

another.

Problem:

(a) Show that .(b) Simplify y, where y = e ln x, for x > 0.

(c) Simplify ln e x, for all x .

Example 7.3.1: Simplify each of the following. State conditions for , if any.

(i) e ln 2

(ii) e ln x 2

(iii) e ln(3x+5)

Derivatives

Problem: Suppose y = e x. Finddxdy

.

ddx

e e xx ddx

e edudx

u u




Example 7.3.2: Differentiate each of the following with respect to .

(a)31 xe y

(b) x xe y tan

(c) x e y cosln

(d) x

x 1ln

ee

y

(e) y xe x3 y 2 3 10

Integration

Since is its own derivative, it is also its own antiderivative.

e dx e Cx x e du e Cu u

Example 7.3.3: Solve each of the following integrals.

(a) e dx5x

(b) dxe

2

3/x

x

(c) e sec x dxt an x 2

(d)e

dx2x

xe 3

Homework

Exercise 7.3: 1, 3, 7, 9, 13, 15, 19, 21, 25, 27, 29, 37, 39, 41, 43, 45, 47, 49

Review Questions

1. What is the domain, the range and the derivative of the natural exponential function?

2. Comment on its graph.




xloganda ax

The function f defined by f(x) = a x is called the exponential function with base a, where

a is a positive number (a > 0 and ) and x is any real number.

We avoid because we get a constant function .

Sketch the graph of y = a x, a > 0.

From the graph we can state some of the properties of the exponential function.

Problem:

(a) If y = a x, finddydx

.

(b) Solve a dxx .

dxdu

alnaadxd uu

Caln

adua

uu

y

xy = a x for 0 < a < 1

y = a x for a > 1

1




Example 7.3.4: Find the derivative of with respect to .

(a) x y 3

(b) 12102 101 x x y

(c) x y sin2


(a) dxx2

(b) dx

x2

1/x5

(c) x dx x sin2 cos

If is any positive number other than 1, the function a x is one-to-one and has an inverse

function. We call the inverse the logarithm of x with base a and is denoted by log ax.

Two basic properties that are very useful in this topic are as follows.

For any number a > 0,

(i) xa xalog for

(ii) xa xalog for x all real number

Example 7.3.5: Solve for log ax: xa xalog .

Example 7.3.6: Show that a x x

dxd a ln

1log .




Example 7.3.7: Solve each of the following.

(a) ddx

xlog 1023 2 5

(b)dxx

xlog

4

1

2

(c) ddx

x 2

(d) ddx

xx where x > 0

(e) ddx

xsintan x

Homework

Exercise 7.3: 57, 59, 61, 63, 65, 67, 71, 73, 77, 81, 87, 89, 91, 95, 97, 99, 101,

105, 111, 115, 117

Review Question

1. For the functions and , are there any restrictions on ? Explain.






By restricting the domain to

2 2, , we obtain a one-to-one function so that it has

an inverse function.

Note:

1. Inverse sine function is denoted by sin 1.

2. D : 1, 1f 1 R

f 1 2 2

: ,

3. sin(sin x) x , if 1 x 11 .

sin (sin y) y , if 1 2 2

y .

4. x y 1sin if and only if x = sin y for 11 x and22

y .

Example 7.6.1: Evaluate each of the following and give your answer in terms of .

(a) sin

1 12

ans:

(b) sin

1 1

2 ans:

(c) sin sin

1

6 ans:

(d) sin sin

1 3

2 ans:

Evaluating an inverse trigonometric function is the same as asking what angle did we

substitute into the sine function to get x . The restrictions on y given above are there to

make sure that we get a consistent answer out of the inverse sine. We know that thereare in fact an infinite number of angles that will work and we want a consistent value

when we work with inverse sine. When using the range of angles above gives all

possible values of the sine function exactly once.




Example 7.6.2: Simplify the expression cos(sin x)1 .

Inverse tangent function

Note:

1. Inverse tangent function is denoted by tan 1

.

2. D f : , R f

1 2 2

: ,

3. tan(tan 1 x) = x , if x .

tan 1(tan y) = y , if

2 2y .

4. y = tan 1 x if and only if x = tan y for x and

2 2y .

5.

y

x

2

2

y = tan x

2

y

x

2

y = tan 1 x




Example 7.6.3: Calculate tan ( )1 3 .

Example 7.6.4: Simplify sec (tan x)2 1 .

Other Inverse Trigonometric Functionsi. y = cos

1 x if and only if x = cos y if 0 y and 1 1x .

ii. y = cot 1 x if and only if x = cot y if 0 y and x .

iii. y = csc 1 x if and only if x = csc y if y

20

2, 0 , and

x , 1 1, .

iv. y = sec 1 x if and only if x = sec y if y

0

2 2, ,

and

, ,1 1 .

Example 7.6.5: Without using a calculator, calculate

(a) sin cos223

1

(b) cos tan2 1 x

Derivatives

Problem: Suppose x y 1sin . Finddxdy

.




Derivatives of Inverse Trigonometric Functions

(1) 11

1sin 2

1u ,dx

du

u udx

d

(2) 11

1cos

2

1 u ,dxdu

u u

dxd

(3) dxdu

uu

dxd

21

1

1tan

(4) dxdu

uu

dxd

21

11

cot

(5) 11

1sec

2

1 u ,dxdu

uu u

dxd

(6) 11

1csc2

1 u ,dxdu

uu u

dxd

Example 7.6.6: Find the derivative of y with respect to x. Simplify your answer where

possible.

(a) )(sin 31 x y

(b) )(sin 1 x x y

(c) y ex sec ( )1




(d) x x y 11 tan10cos4

(e) yxx

tan 1 1

1

Identities

1.

2.

3.

Integrals of Inverse Trigonometric Functions

(1) 1sin1

1 1

2 uc,udu

u

(2) cuduu

12 tan

1

1

(3) 1sec1

1 1

2

uc,uduuu


(a)

dx

e

e x

x

2

1

(b)19 2 x x

dx

(c) dx x 249

1




(d) dx x x x 34)2(

12

(e) dx x x 1

1

Homework

Exercise 7.6: 21, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41, 47, 49, 55, 57, 59, 65, 67,

71, 75, 77, 79, 81, 87, 89




Graphs of trigonometric functions

1. y = tan x

2. y = cot x

2

2

2 2

3

1

1

22

3

2

2

2

23

2

4

2

4

25

2

5

x

y

y

x




3. y = csc x

4. y = sec x

1

1

y = sec x y = cos x

y = sec x

2

23

2

2

23

2

23

2

2

23

2

1

1

y = csc x y = sin x

2

y = csc x

x

y

y

x




7.8 Hyperbolic Functions

Objectives

Define Hyperbolic functions

Differentiation and integration of hyperbolic functions

Inverse hyperbolic functions

Differentiation and integration of inverse hyperbolic functions

Many of the advanced application of calculus involve certain combinations of e x and e x.

These combinations are called the hyperbolic functions. The hyperbolic functions and

their inverses are used to solve a variety of problems in the physical sciences and

engineering.

Definition

The hyperbolic sine function, denoted by sinh, and the hyperbolic cosine function,

denoted by cosh, are defined by

2sinh

x x ee

x and 2cosh

x x ee

x

for every real number x.

Note: We pronounce sinh x and cosh x as sinch x and kosh x, respectively.




Other hyperbolic functions

hyperbolic tangent : tanh xsinh xcosh x

e ee e

x x

x x

hyperbolic cotangent: coth xcosh xsinh x

e ee e , x 0

x x

x x

hyperbolic secant : sech x1

cosh x2

e ex x

hyperbolic cosecant : csch x1

sinh x2

e e , x 0x x

Note: We pronounce the four functions as tansh x, cotansh x, setch x.

The following are graphs of three main hyperbolic functions.




The graphs of the remaining three hyperbolic functions are given at the end of this

topic.

Many identities for the hyperbolic functions are similar to the identities for

trigonometric functions. Any differences which occur usually involve signs of the terms.

(1) 1sinhcosh 22 x x

(2) xh x 22 sectanh1

(3) xh x 22 csc1coth

(4) x x x coshsinh22sinh

(5) cosh 2x x x 22 sinhcosh

1sinh2 2 x

1cosh2 2 x

(6) x x)( coshcosh

(7) x x)( sinhsinh

In each case it is sufficient to express the hyperbolic functions in terms of exponential

functions and show that one side of the equation can be transformed into the other.

Derivatives

Because the hyperbolic functions are defined in terms of exponential functions finding their

derivatives is fairly simple.




Problem : Suppose y = sinh x. Finddydx

.

To find the derivatives of and we use the definition and quotient rule. For

the remaining functions we can use the results obtained together with the quotient

rule.

Derivatives of Hyperbolic Functions

(1) dxdu

uudxd

coshsinh

(2) dxduuu

dxd sinhcosh

(3) dxdu

uhudxd 2sectanh

(4) dxdu

uhud d 2csccoth

(5) dxdu

uh uh udxd

tanhsecsec

(6) dxdu

uh uh udxd

cothcsccsc

Example 7.7.1: Find the derivative of y with respect to x for each of the following.

(a) y x cosh 2 1

(b) y tanh(sin x)

(c) y x sinh ( )2 3 1




Integral formulas for hyperbolic functions

(1) sinh u du

(2) cosh u du

(3) sech u du2

(4) csch u du2

(5) sech u tanh u du

(6) csch u coth u du


(a) x dx x coshsinh 5

(b) x dxtanh

(c) dxh x

e x

sec

sinh

HomeworkExercise 7.7: 13, 15, 19, 21, 43, 45, 47, 49, 53, 55, 59




Inverse Hyperbolic Functions

The inverse of the six hyperbolic functions are very useful in integration.

Inverse hyperbolic sine function

Since the hyperbolic sine function is continuous and increasing for all x, it has an

inverse function.

(a) Inverse hyperbolic sine is denoted by sinh 1 .

(b) Df 1 : R

f 1 :

(c) x y 1sinh if any only if x = sinh y.

(d) Since sinh x is defined in terms of e , sinh xx 1 can be expressed in terms of the

natural logarithmic function.

(e) Find xdxd 1sinh .

Results

(a) x,1lnsinh 21 x x x

(b) 1x,1lncosh21

x x x

(c) tanh ln 1 1

211

1xxx

x,

(d) sec lnh xx

xx

121 1

0 1,

(e) csc lnh xx

xx

1

21 10, x

(f) coth ln 1 1

211

1xxx

x,




Derivatives of Inverse Hyperbolic Functions

(a) dxdu

uu

dxd

2

1

1

1sinh

(b) 1,1

1cosh

2

1 udxdu

uu

dxd

(c) 1,dxdu

11

tanh 21 u

uu

dxd

(d) 1,dxdu

11

coth 21 u

uu

dxd

(e) 10,dxdu

1

1sec

2

1 uuu

uhdxd

(f) 0,dxdu

11 csc

2

1 uuu

uhdxd

Example 7.7.3: Find the derivative of y with respect to x for each of the following.

(a) y sinh 1 (tan x)

(b) y x x

sinh

1 1

(c) y x tanh ( )1 2 1



Integrals leading to inverse hyperbolic functions

(a)1

1 2

1

uu cdu sinh

(b)1

12

1

udu u c cosh

(c)1

1 2

u

dutanh u c if u 1

coth u c if u 1

1

1

(d)1

10 1

2

1

u udu h u c u sec ,

(e)1

10

2

1

u udu h u c csc , u


(a) dxx1x

1

(b)dx

x1 9 2

(c)

dx

x x 2ln1

1

(d)1

x 1 x dx

6

Homework

Exercise 7.7 25, 27, 31, 33, 67, 71, 73

Chapter 7 Practice Exercises

13, 21, 25, 27, 29, 31, 35, 41, 49, 55, 57, 63, 69, 73, 75, 77, 79, 81

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New Chapter 7 Transcendental Functions (1).docx