Chapter 7: The Transcendental Functions

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Section 7.1 One-to-One Functions; Inverses

a. Definition; One-to-One

b. Theorem; Inverse Functions

c. One-to-One Functions; Properties and Graphs

d. Graphs of f and f ^-1

e. Continuity and Differentiability of Inverses

f. Theorem; Inverses

Section 7.2 The Logarithm, Part I

a. The Logarithm Function

b. Properties

c. The Number e

d. Natural Log Function

e. Graph of the Logarithm Function

Section 7.3 The Logarithm Function, Part II

a. Differentiation and Graphing

b. Integration

c. Integration of Trigonometric Functions

Section 7.4 The Exponential Function

a. Definition

b. Properties 1-3

c. Properties 4-6

d. Theorem

e. The Derivative and Integral

Section 7.5 Arbitrary Powers; Other Bases

a. Properties

b. Base p: The Function f(x) = p^x

c. Base p: The Function f(x) = log_px

Chapter 7: The Transcendental Functions

Section 7.6 Exponential Growth and Decay

a. Theorem

b. Population Growth

c. Radioactive Decay

d. Compound Interest

e. Rule of 72

Section 7.7 Inverse Trigonometric functions

a. Arc Sine

b. Arc Sine; Inverses

c. Arc Sine; Integrals

d. Arc Tangent

e. Arc Tangent: Inverses

f. Arc Tangent; Derivatives

g. Arc Cosine, Arc Cotangent, Arc Secant, Arc Cosecant

h. Relations to 1/2π and Derivatives

Section 7.8 The Hyperbolic Sine and Cosine

a. Hyperbolic Sine and Hyperbolic Cosine

b. The Graphs

c. Identities

Section 7.9 The Other Hyperbolic Functions

a. Hyperbolic Tangent, Hyperbolic Cotangent, Hyperbolic Secant, Hyperbolic Cosecant

b. Hyperbolic Inverses

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There is a simple geometric test, called the horizontal line test, which can be used to determine whether a function is one-to-one. Look at the graph of the function. If some horizontal line intersects the graph more than once, then the function is not one-to-one.

(Figure 7.1.1) If, on the other hand, no horizontal line intersects the graph more than once, then the function is one-to-one (Figure 7.1.2).

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One-to-One Functions: Inverses

Inverses

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One-to-One Functions: Inverses

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One-to-One Functions: Inverses

The Graphs of f and f ⁻¹

The graph of f consists of points (x, f (x)). Since f ⁻¹ takes on the value x at f (x), the graph of f ⁻¹ consists of points ( f (x), x). If, as usual, we use the same scale on the y-axis as we do on the x-axis, then the points (x, f (x)) and ( f (x), x) are symmetric with respect to the line y = x.

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One-to-One Functions: Inverses

Continuity and Differentiability of Inverses

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One-to-One Functions: Inverses

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From Theorem 5.3.5 we know that L is differentiable and

This function has a remarkable property that we’ll get to in a moment. First some preliminary observations:

(1) L is defined for all x > 0.

(2) Since

(3) L(x) is negative if 0 < x < 1, L(1) = 0, L(x) is positive for x > 1.

L increases on (0,∞).

( )

1 x1

L x ′ = ∫

t

dt

( )

¹ ^{for all} ⁰

L x x

′ = x >

( )

¹ ^{for all} ⁰

L x x

′ = x >

The Logarithm Function, Part I

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The Logarithm Function, Part I

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The Logarithm Function, Part I

Since the range of L is (−∞,∞) and L is an increasing function (and therefore one-to one), we know that L takes on as a value every real number and it does so only

once. In particular, there is one and only one real number at which L takes on the value 1. This unique number is denoted throughout the world by the letter e.

The area under the curve y = 1/t from t = 1 to t = e is exactly 1.

The Number e

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The Logarithm Function, Part I

The Logarithm Function Since

( )

1

e

L e = ∫

t

dt =

we can show that

The function that we have labeled L is known as the natural logarithm function, or more simply as the logarithm function, and from now on L(x) will be written ln x.

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The Logarithm Function, Part I

The Graph of the Logarithm Function You know that the logarithm function

1

ln

^x1

x = ∫

t

dt

has domain (0,∞), range (−∞,∞), and derivative

( )

^ln ¹

d x

dx = x

The second derivative,

is negative on (0,∞). (The graph is concave down throughout.)

2

( )

2 2

ln 1

d x

dx = − x

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The Logarithm Function, Part II

Differentiating and Graphing

For x > 0,

For x < 0, we have |x| = −x > 0, and therefore

(

^ln

)

¹ ^{for all} ⁰

d x x

dx = x ≠

(

^ln

)

^ln

^{( )}

¹

^{( )}

¹

^{( )}

¹ ¹

d d d

x x x

dx dx x dx x x

 

=  −  = − − = −  − =

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The Logarithm Function, Part II

Integration

The relation is valid on every interval that does not include 0.

Integrals of the form

by setting u = u(x), du = u´(x) dx.

( ) ( )

^with

( )

0 can be written ¹

u x dx u x du

u x u

′ ≠

∫ ∫

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The Logarithm Function, Part II

Integration of the Trigonometric Functions

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The Exponential Function

Recall the following property:

The following definitions are patterned after this relation.

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The Exponential Function

Some properties of the exponential function:

(1) In the first place,

This says that the exponential function is the inverse of the logarithm function.

(3) Since the graph of the logarithm lies to the right of the y- axis, the graph of the exponential function lies above the x- axis:

(2) The graph of the exponential function can be obtained from the graph of the logarithm by reflection in the line y = x.

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The Exponential Function

(4) Since the graph of the logarithm crosses the x-axis at (1, 0), the graph of the exponential function crosses the y-axis at (0, 1):

ln 1 = 0 gives e⁰ = 1.

(5) Since the y-axis is a vertical asymptote for the graph of the logarithm function, the x-axis is a horizontal asymptote for the graph of the exponential function:

as x → −∞, e^x → 0.

(6) Since the exponential function is the inverse of the logarithm function, the logarithm function is the inverse of the exponential function; thus

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The Exponential Function

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The Exponential Function

The integral counterpart of Theorem 7.4.9 takes the form

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Arbitrary Powers; Other Bases

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Arbitrary Powers; Other Bases

Base p: The Function f (x) = p

^x

Functions of the form f (x) = p^x are called exponential functions with base p.

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Arbitrary Powers; Other Bases

Base p: The Function f (x) = log

_p

x

Functions of the form are called the logarithm of x to the base p.^ln

ln x p

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Exponential Growth and Decay

Let y = y(t) be a function of time t.

If y is a linear function, a function of the form

y(t) = kt + C, (k, C constants)

then y changes by the same additive amount during all periods of the same duration:

y(t +Δt) = k(t +Δt) + C = (kt + C) + kΔt = y(t) + kΔt.

During every period of length t, y changes by the same amount kt.

If y is a function of the form

y(t) = Ce^kt , (k, C constants)

then y changes by the same multiplicative factor during all periods of the same duration:

y(t +Δt) = Ce^k(t+^Δt) = Ce^kte^k^Δt = e^k^Δt y(t).

During every period of length t, y changes by the factor e^kt .

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Exponential Growth and Decay

Population Growth

Under ideal conditions (unlimited space, adequate food supply, immunity to disease, and so on), the rate of increase of a population P at time t is proportional to the size of the population at time t. That is,

P´(t) = kP(t)

where k > 0 is a constant, called the growth constant. Thus, by our theorem, the size of the population at any time t is given by

P(t) = P(0)e^kt ,

and the population is said to grow exponentially. This is a model of uninhibited growth.

In reality, the rate of increase of a population does not continue to be proportional to the size of the population. After some time has passed, factors such as limitations on space or food supply, diseases, and so forth set in and affect the growth rate of the population.

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Exponential Growth and Decay

Radioactive Decay

Although different radioactive substances decay at different rates, each radioactive substance decays at a rate proportional to the amount of the substance present: if A(t) is the amount present at time t, then

A´(t) = kA(t) for some constant k.

Since A decreases, the constant k, called the decay constant, is a negative number.

From general considerations already explained, we know that A(t) = A(0) e^kt

where A(0) is the amount present at time t = 0.

The half-life of a radioactive substance is the time T it takes for half of the substance to decay. The decay constant k and the half-life T are related by the equation

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Exponential Growth and Decay

Compound Interest

Consider money invested at annual interest rate r. If the accumulated interest is credited once a year, then the interest is said to be compounded annually; if twice a year, then semiannually; if four times a year, then quarterly. The idea can be pursued further. Interest can be credited every day, every hour, every second, every half- second, and so on. In the limiting case, interest is credited instantaneously.

Economists call this continuous compounding.

The economists’ formula for continuous compounding is a simple exponential:

Here t is measured in years,

A₀ = A(0) = the initial investment,

r = the annual interest rate expressed as a decimal, A(t) = the principal at time t.

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Exponential Growth and Decay

A popular estimate for the doubling time at an interest rate α% is the rule of 72:

According to this rule, the doubling time at 8% is approximately 72/8 = 9 years.

Here is how the rule originated:

For rough calculations 72 is preferred to 69 because 72 has more divisors.

doubling time 72

≅ α

0.69 69 72

α

100 ⁼

α

^≅

α

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Inverse Trigonometric Functions

Arc Sine

Clearly the sine function is not one-to-one: it takes on every value from −1 to 1 an infinite number of times. However, on the interval [−½π, ½π] it takes on every value from −1 to 1, but only once. Thus the function

y = sin x, x ∈ [− ½_π,½_π]

maps the interval [− ½_π,½π] onto [−1, 1] in a one-to-one manner and has an inverse that maps [−1, 1] back to [− ½π, ½π], also in a one-to-one manner.

The inverse is called the arc sine function:

y = arcsin x, x ∈ [−1, 1]

is the inverse of the function

y = sin x, x ∈[− ½ _π,½_π].

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Inverse Trigonometric Functions

Since the sin and arcsin functions are inverses,

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Inverse Trigonometric Functions

The integral counterpart of (7.7.3) reads

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Inverse Trigonometric Functions

Arc Tangent

Although not one-to-one on its full domain, the tangent function is one-to-one on the open interval (−½π, ½π) and on that interval the function takes on as a value every real number.

Thus the function

y = tan x, x ∈ − ½π, ½π

maps the interval (− ½π, ½π) onto (−∞,∞) in a one-to-one manner and has an inverse that maps (−∞,∞) back to (− ½π, ½π), also in a one-to-one manner. This inverse is called the arc tangent: the arc tangent function

y = arctan x, x ∈ (−∞,∞)

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Inverse Trigonometric Functions

Since the tan and arctan functions are inverses,

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Inverse Trigonometric Functions

The integral counterpart of (7.7.8) reads

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Inverse Trigonometric Functions

Arc Cosine

While the cosine function is not one-to-one, it is one-to-one on [0, π] and maps that interval onto [−1, 1]. The arc cosine function

y = arccos x, x ∈ [−1, 1]

is the inverse of the function

y = cos x, x ∈ [0, π].

Arc Cotangent

The cotangent function is one-to-one on (0, π) and maps that interval onto (−∞,∞). The arc cotangent function

y = arccot x, x ∈ (−∞,∞) is the inverse of the function

y = cot x, x ∈ (0, π).

Arc Secant, Arc Cosecant

These functions can be defined explicitly in terms of the arc cosine and the arc sine.

For |x| ≥ 1, we set

arcsec x = arccos (1/x), arccsc x = arcsin (1/x).

It can be shown that for all |x| ≥ 1

sec (arcsec x) = x and csc (arccsc x) = x.

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Inverse Trigonometric Functions

Relations to ½π

Derivatives

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The Hyberbolic Sine and Cosine

The hyperbolic sine (sinh) and hyperbolic cosine (cosh) are the functions defined as follows:

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The Hyberbolic Sine and Cosine

The Graphs

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The Hyberbolic Sine and Cosine

Identities

The hyperbolic sine and cosine functions satisfy identities similar to those satisfied by the “circular” sine and cosine.

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The Other Hyberbolic Functions

The hyperbolic tangent is defined by setting

There is also a hyperbolic cotangent, a hyperbolic secant, and a hyperbolic cosecant:

tanh sinh

cosh

x x

x e e

x x e e

−

= = −

+

cosh 1 1

coth , sech , csch

sinh cosh sinh

x x x x

x x x

= = =

The derivatives are as follows:

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The Other Hyberbolic Functions

The Hyperbolic Inverses

These functions,

y = sinh⁻¹ x, y = cosh⁻¹ x, y = tanh⁻¹x, are the inverses of

y = sinh x, y = cosh x (x ≥ 0), y = tanh x