Microsoft Word - instr_calc9e_0501.doc - WebStartswhscalculus.webstarts.com/uploads/notetaking_guide... · Web viewo. ns. 98Chapter 5Logarithmic, Exponential, and Other Transcendental

Chapter 5 Logarithmic, Exponential, and Other Transcendental Functions

Section 5.1 The Natural Logarithmic Function: Differentiation

Objective: In this lesson you learned the properties of the natural logarithmic function and how to find the derivative of the natural logarithmic function.

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Number

Instructor

Date

Important Vocabulary Define each term or concept.

Natural logarithmic function The natural logarithmic function is defined by ln x =x∫ 1/t dt, x > 0. 1

e The letter e denotes the positive real number such thate

ln e = ∫ 1/t dt = 1.1

I. The Natural Logarithmic Function (Pages 324−326)

The domain of the natural logarithmic function is the set

of all positive real nu m bers .

The value of ln x is positive for x > 1 and negative

for 0 < x < 1 . Moreover, ln (1) = 0 ,

because the upper and lower limits of integration are equal

when x = 1 .

The natural logarithmic function has the following properties:

1. The domain is (0, ∞) and the range is (−∞, ∞).

2. The function is continuous, increasing, and one-to-one.

3. The graph is concave downward.

If a and b are positive numbers and n is rational, then the following properties are true:

1. ln (1) = 0 .

2. ln (ab) = ln a + ln b .

3. ln(an) = n ln a .

What you should learn How to develop and use properties of the natural logarithmic function

Larson/Edwards Calculus 9e Notetaking Guide IAECopyright © Cengage Learning. All rights reserved. 95

96 Chapter 5 Logarithmic, Exponential, and Other Transcendental Functions

Larson/Edwards Calculus 9e Notetaking Guide IAE Copyright © Cengage Learning. All rights

b⎜4. ln ⎛ a ⎞ = l n a − ln b .

⎝ ⎠

Example 1: Expand the logarithmic expression ln

ln x + 4 ln y − ln 2

xy 4.

2

II. The Number e (Page 327)

The base for the natural logarithm is defined using the fact

that the natural logarithmic function is continuous, is one-to-

one, and has a range of (−∞, ∞). So, there must a unique real

number x such that ln x = 1 . This

number is denoted by the letter e ,

which has the decimal approximation

2. 71 828 1 82 8 46 .

III. The Derivative of the Natural Logarithmic Function(Pages 328−330)

Let u be a differentiable function of x. Complete the

following rules of differentiation for the natural logarithmic

function:d [ln x] = 1/x , x > 0dxd [ln u] = 1/u [du/dx] = u′ /u , u > 0dx

What you should learn How to understand the definition of the number e

What you should learn How to find derivatives of functions involving the natural logarithmic function

Example 2: Find the derivative ofx + 2x lnx

f (x) = x2 ln x .

If u is a differentiable function of x such that u ≠ 0 , thend

[ln u ] = u ′ / u . In other words, functions ofdxthe form

y = ln u

can be differentiated as if the absolu t e

value signs were not present .

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98 Chapter 5 Logarithmic, Exponential, and Other Transcendental Functions


Section 5.2 The Natural Logarithmic Function: Integration 97

Section 5.2 The Natural Logarithmic Function: Integration

Objective: In this lesson you learned how to find the antiderivative of the natural logarithmic function.

Course Number

Instructor

Date

I. Log Rule for Integration (Pages

334−337) Let u be a differentiable function of

x.

1 dx = ln| x | + C

x

u′ 1

u dx =

u du = ln| u | + C

What you should learn How to use the Log Rule for Integration to integrate a rational function

Example 1: Find ⎛1 − 1 ⎞ dx .

⎝ ⎠⎜ x ⎟

x − ln| x | + C

Example 2: Find2

∫ 3

− x3

dx .

− (1/3) ln|3 − x3| + C

Example 3: Find

∫x2 − 4x + 1

x

dx .

(1/2)x2 − 4x + ln| x | + C


If a rational function has a numerator of degree greater than

or equal t o that of the d e no m inator ,

division may reveal a form to which you can apply the Log Rule.

9 Chapter Logarithmic, Exponential, and Other Transcendental


Guidelines for Integration

1. Learn a basic list of integration formulas.

2. Find an integration formula that resembles all or part of the integrand, and, by trial and error, find a choice of u that will make the integrand conform to the formula.

3. If you cannot find a u-substitution that works, try altering the integrand. You might try a trigonometric identity, multiplication and division by the same quantity, addition and subtraction of the same quantity, or long division. Be creative.

4. If you have access to computer software that will find antiderivatives symbolically, use it.

II. Integrals of Trigonometric Functions (Pages 338−339) What you should learn

How to integrate trigonometric functions

∫sin u du

=

- cos u + C

∫cos u du = sin u + C

∫ tan u du =

- ln|cos u | + C

∫cot u du = ln|sin u | + C

∫sec u du = ln | s ec u + tan u | + C

∫cscu du =

- l n |csc u + cot u | + C

Example 4: Find ∫csc5x dx

−(1/5)ln|csc 5x + cot 5x| + C

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9Section Inverse


Section 5.3 Inverse Functions

Objective: In this lesson you learned how to determine whether a function has an inverse function.

Course

Number

Instructor

Date

Important Vocabulary Define each term or concept.

Inverse function A function g is the inverse function of the function f if f(g(x)) = x for each x in the domain of g and g(f(x)) = x for each x in the domain of f. The function g is denoted by f −1.Horizontal Line Test A test that states that a function f has an inverse function if and only if every horizontal line intersects the graph of f at most once.

I. Inverse Functions (Pages 343−344)

For a function f that is represented by a set of ordered pairs, you

can form the inverse function of f by interchang i ng the first

and second coordinates of each ordered pairs .

For a function f and its inverse f −1, the domain of f is equal

to the range of f −1 , and the range of f is equal to

the domain of f −1 .

State three important observations about inverse functions.

1. If g is the inverse function of f, then f is the inverse function of g.

2. The domain of f −1 is equal to the range of f, and the range off −1 is equal to the domain of f.

3. A function need not have an inverse function, but if it does, the inverse function is unique.

To verify that two functions, f and g, are inverse functions of

each other, . . . find f(g(x)) and g(f(x)). If both of these

compositions are equal to the identity function x for every x in

the domain of the inner function, then the functions are

inverses of each other.

What you should learn How to verify that one function is the inverse function of another function



Example 1: Verify that the functions f (x) = 2x − 3 and

g(x) = x +

3

2

are inverse functions of each other.

The graph of f −1 is a reflection of the graph of f in the line

y = x .

The Reflective Property of Inverse Functions states that the

graph of f contains the point (a, b) if and only if the

graph of f −1 contains the point (b, a) .

II. Existence of an Inverse Function (Pages

345−347) State two reasons why the horizontal line test

is valid.

1. A function has an inverse function if and only if it is one-to- one.

2. If f is strictly monotonic on its entire domain, then it is one- to-one and therefore has an inverse function.

Example 2: Does the graph of the function shown below have an inverse function? Explain.No, it doesn’t pass the Horizontal Line Test.

What you should learn How to determine whether a function has an inverse function

y5

3

-5 -3

1

x-1 1 3 5

-1

-3

-5

Complete the following guidelines for finding an inverse

function.

1Section Inverse


1) Determine whether the function given by y = f(x) has an

inverse function.



2) Solve for x as a function of y: x = g(y) = f −1(y).

3) Interchange x and y. The resulting equation is y = f −1(x).

4) Define the domain of f −1 to be the range of f.

5) Verify that f(f −1(x)) = x and f −1(f(x)) = x.

Example 3: Find the inverse (if it exists) of

f −1(x) = 0.25x + 1.25

f (x) = 4x − 5 .

III. Derivative of an Inverse Function (Pages 347−348)

Let f be a function whose domain is an interval I. If f has an inverse function, then the following statements are true.

1. If f is continuous on its domain, then f −1 is continuous on its domain.

2. If f is increasing on its domain, then f −1 is increasing on its domain.

3. If f is decreasing on its domain, then f −1 is decreasing on its domain.

4. If f is differentiable on an interval containing c and f ′(c) ≠ 0, then f −1 is differentiable at f(c).

Let f be a function that is differentiable on an interval I. If f has

an inverse function g, then g is dif f erentiable

at any x f or w h ich f ′ ( g ( x )) ≠ 0 . Moreover,

What you should learn How to find the derivative of an inverse function

g′(x) =

1,f ′(g(x))

f ′(g(x)) ≠ 0 .

This last theorem can be interpreted to mean that graphs

of inverse functions have reciprocal slopes .



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Section 5.4 Exponential Functions: Differentiation and Integration 103

Section 5.4 Exponential Functions: Differentiation and Integration

Objective: In this lesson you learned about the properties of the natural exponential function and how to find the derivative and antiderivative of the natural exponential function.

I. The Natural Exponential Function (Pages 352−353)

The inverse function of the natural logarithmic function

f (x) = ln x is called the natural exponential

Course Number

Instructor

Date

What you should learn How to develop properties of the natural exponential function

function and is denoted by f −1 (x) = ex . That is,

y = ex if and only if x = ln y .

Example 1: Solve e x−2 − 7 = 59 for x. Round to three decimal places.x ≈ 6.190

Example 2: Solve 4 ln 5x = 28 for x. Round to three decimal places.x ≈ 219.327

Complete each of the following operations with exponential functions.

1. ea eb = ea+b .

2.e

= ea−b .eb

List four properties of the natural exponential function.

1. The domain of f(x) = ex is (−∞, ∞), and the range is (0, ∞).

2. The function f(x) = ex is continuous, increasing, and one-to- one on its entire domain.

3. The graphs of f(x) = ex is concave upward on its entire domain.



4. lim ex = 0 and lim ex = ∞x→ − ∞ x→ ∞

II. Derivatives of Exponential Functions (Pages 354−355)

Let u be a differentiable function of x. Complete the

following rules of differentiation for the natural exponential

function:

What you should learn How to differentiate natural exponential functions

d ⎡ex ⎤ =

ex .

dx ⎣ ⎦d ⎡eu ⎤ =

eu [du/dx] .

dx ⎣ ⎦

Example 3: Find the derivative of2xex + x2ex

f (x) = x2ex .

III. Integrals of Exponential Functions (Pages

356−357) Let u be a differentiable function of x.What you should learn How to integrate natural exponential functions

∫ex dx

=

∫eu du

=

ex + C

eu + C

Example 4: Find ∫e2 x dx .(1/2) e2x + C

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Section 5.5 Bases Other Than e and Applications

Objective: In this lesson you learned about the properties, derivatives, and antiderivatives of logarithmic and exponential functions that have bases other than e.

I. Bases Other than e (Pages 362−363)

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Date

What you should learnHow to define

If a is a positive real number (a ≠ 1)

and x is any real number, exponential functions that

then the exponential function to the base a is denoted by a x

and is defined by ax = e(ln a)x . If a = 1, then

y = 1x = 1 is a constant function .

In a situation of radioactive decay, half-life is the nu m ber

of y ears required for half of the atoms in a sa m ple of

radioactive mater i al to decay .

have bases other than e

If a is a positive real number (a ≠ 1)

and x is any positive real

number, then the logarithmic function to the base a is denoted

by loga x and is defined by loga x = 1/(ln a ) ln x .

Complete the following properties of logarithmic functions to the base a.

1) loga 1 = 0

2) loga (xy) = l og a x + lo g a y

3. log

xn = n loga x

x4. loga y = l og a x − log a y

State the Properties of Inverse Functions

y = ax if and only if x = loga y

aloga x = x for x > 0

loga ax = x, for all x



The logarithmic function to the base 10 is called the com m on

logarith m ic function .

Example 1: (a) Solve log8 x = 1

3

for x.

(b) Solve 5 x = 0.04 for x.(a) x = 2 (b) x = − 2

II. Differentiation and Integration (Pages 364−365)

To differentiate exponential and logarithmic functions to

other bases, you have three options:

1. Use the definitions of ax and loga x and differentiate using the rules for the natural exponential and logarithmic functions.

2. Use logarithmic differentiation, or

3. Use the differentiation rules for bases other than e.

What you should learn How to differentiate and integrate exponential functions that have bases other than e

Let a be a positive real number (a ≠ 1)

and let u be a

differentiable function of x. Complete the following formulas

for the derivatives for bases other than e.d ⎡a x ⎤ =

(ln a) ax .

dx ⎣ ⎦d ⎡au ⎤ =

(ln a) au [du/dx] .

dx ⎣ ⎦d

[log x] = 1 / [ (ln a ) x ] .dx a

d [log u] = 1 / [ (ln a ) u ] [ du / dx ] .

dx a

Occasionally, an integrand involves an exponential function to

a base other than e. When this occurs, there are two options:

(1) convert to base e using the formula ax = e(ln a)x and

then integrate or (2) integrate directly using the



integration formula ∫a x dx =

(1/ln a) ax + C .



Let n be any real number and let u be a differentiable function ofx. The Power Rule for Real Exponents gives.

d [xn ] = nxn−1 .

dxd

[un ] = nun−1 [du/dx] .dx

III. Applications of Exponential Functions (Pages

366−367) Complete the following limit statement:

x x

What you should learn How to use exponential functions to model compound interest and

lim ⎛1 + 1 ⎞ = lim ⎛ x + 1 ⎞ = e . exponential growth

x→∞ ⎜ x ⎟ x→∞ ⎜ x ⎟⎝ ⎠ ⎝ ⎠

Let P be the amount deposited, t the number of years, A the balance after t years, and r the annual interest rate (in decimal form), and n the number of compounding per year. Complete the following compound interest formulas:

Compounded n times per year: A = P(1 + r/n)nt

Compounded continuously: A = Pert

Example 2: Find the amount in an account after 10 years if$6000 is invested at an interest rate of 7%,(a) compounded monthly.(b) compounded continuously.

(a) $12,057.97 (b) $12,082.52



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Section 5.6 Inverse Trigonometric Functions: Differentiation 109

Section 5.6 Inverse Trigonometric Functions: Differentiation

Objective: In this lesson you learned about the properties of inverse trigonometric functions and how to find derivatives of inverse trigonometric functions.

Course Number

Instructor

Date

I. Inverse Trigonometric Functions (Pages 373−375)

None of the six basic trigonometric functions has an

inverse function . This is true because all six

trigonometric functions are p e riodic and th e refore are

not one-to-one . However, their domains can be

redefined in such a way that they will have inverse functions on

th e restricted domains .

For each of the following definitions of inverse trigonometric functions, supply the required restricted domains and ranges.

What you should learn How to develop properties of the six inverse trigonometric functions

y = arcsin x iff sin y = x

D o m a in

− 1 ≤ x ≤ 1

Range

− π /2 ≤ y ≤ π /2

y = arccos x iff cos y = x − 1 ≤ x ≤ 1 0 ≤ y ≤ π

y = arctan x iff tan y = x − ∞ < x < ∞ − π /2 < y < π /2

y = arccot x iff cot y = x − ∞ < x < ∞ 0 < y < π

y = arcsec x iff sec y = x | x | ≥ 1 0 ≤ y ≤ π , y ≠ π /2

y = arccsc x iff csc y = x | x | ≥ 1 − π /2 ≤ y ≤ π /2, y ≠ 0

An alternative notation for the inverse sine function is

sin−1 x .

Example 1: Evaluate the function:− π/2

Example 2: Evaluate the function:

π/3

arcsin (−1) .

arccos 1

.2

Example 3: Evaluate the function: arcos (0.85).0.5548



State the Inverse Property for the Sine function.

If − 1 ≤ x ≤ 1 and − π/2 ≤ y ≤ π/2, then sin(arcsin x) = x and arcsin(sin y) = y.

State the Inverse Property for the Cosine function.

If − 1 ≤ x ≤ 1 and 0 ≤ y ≤ π, then cos(arccos x) = x and arccos(cos y) = y.

State the Inverse Property for the Tangent function.

If x is a real number and − π/2 < y < π/2, then tan(arctan x) = x and arctan(tan y) = y.

II. Derivatives of Inverse Trigonometric Functions(Pages 376−377)

Let u be a differentiable function of x.

What you should learn How to differentiate an inverse trigonometric function

d [arcsin u] =dx

d [arccos u] =dx

d [arctan u] =dx

d [arc cot u] =dxd [arc secu] =

u′

1 − u2

− u′

1 − u2

u′

1 + u2

− u′

1 + u2

u′

dx

d [arc cscu] =dx

| u |

| u |

u2 − 1

− u′

u2 − 1


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Section 5.7 Inverse Trigonometric Functions: Integration 111

Section 5.7 Inverse Trigonometric Functions: Integration

Objective: In this lesson you learned how to find antiderivatives of inverse trigonometric functions.

Course Number

Instructor

Date

I. Integrals Involving Inverse Trigonometric Functions

(Pages 382−383)

Let u be a differentiable function of x, and let a > 0.

du

What you should learn How to integrate functions whose antiderivatives involve inverse trigonometricfunctions

∫ a2 −

u2

= arcsin ( u / a ) + C .

du a2 + u2

= ( 1/ a ) arctan ( u / a ) + C .

du

u u2 − a2

Example 1:

= ( 1/ a ) arcs e c (| u | / a ) + C .

6x dx4 + 9x4

½ arctan(3x2/2) + C

II. Completing the Square (Pages 383−384)

Completing the square helps when quadratic

functions are involved in t h e integrand .

Example 2: Complete the square for the polynomial:x2 + 6x + 3 .(x + 3)2 − 6

Example 3: Complete the square for the polynomial: 2x2 + 16x .

2[(x + 4)2 − 16]


What you should learn How to use the method of completing the square to integrate a function



∫ ∫

∫

III. Review of Basic Integration Rules (Pages

385−386) Complete the following selected basic

integration rules.

u′ 1

u dx =

u du = ln| u | + C

What you should learn How to review the basic integration rules involving elementary functions

∫du = u + C

∫cot u du = ln|sin u | + C

du a2 + u2

= ( 1/ a ) arctan ( u / a ) + C

∫cos u du = sin u + C

∫sec2 u du = t a n x + C

Homework Assignment

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Exercises

sinh x = (ex − e−x) / 2 .

cosh x = (ex + e−x) / 2 .

11Section Hyperbolic


Section 5.8 Hyperbolic Functions

Objective: In this lesson you learned about the properties of hyperbolic functions and how to find derivatives and antiderivatives of hyperbolic functions.

I. Hyperbolic Functions (Pages 390−392)

Complete the following definitions of the hyperbolic functions.

Course Number

Instructor

Date

What you should learn How to develop properties of hyperbolic functions

tanh x = (sinh x ) / (cosh x ) .

csch x = 1 / (sinh x ), x ≠ 0 .

sech x = 1 / (cosh x ) .

coth x = 1 / (tanh x ), x ≠ 0 .

Complete the following hyperbolic identities.

cosh2 x − sinh2 x = 1 .

tanh2 x + sech2 x = 1 .

coth2 x − csch2 x = 1 .

− 1 + c o sh 2 x 2

= sinh2 x .

1 + co s h 2 x 2 = cosh2 x .

2 sinh x cosh x = sinh 2 x .

cosh2 x + sinh2 x = cosh 2 x .

sinh (x + y)= sinh x c o s h y + cosh x sinh y .

sinh (x − y)= sinh x c o s h y − cosh x sinh y .

cosh (x + y)= cosh x cosh y + sinh x sinh y .

cosh (x − y)= cosh x cosh y − sinh x sinh y .



II. Differentiation and Integration of Hyperbolic Functions(Pages 392−394)

Let u be a differentiable function of x. Complete each of the following rules of differentiation and integration.

d [sinh u] = (cosh u ) u ′ .

dx

d [cosh u] = (sinh u ) u ′ .

dx

d [tanh u] = (sech2 u) u′ .

dx

d [coth u] = − (csch2 u) u′ .

dx

d [sech u] = − (sech u tanh u ) u ′ .

dx

d [csch u] = − (csch u coth u ) u ′ .

dx

What you should learn How to differentiate and integrate hyperbolic functions

∫cosh u

du

∫sinh u

du

= sinh u + C .

= cosh u + C .

∫sech2 u

du

∫csch2 u

du

= tanh u + C .

= − coth u + C .

∫sech u tanh u

du

= − sech u + C .

∫csch u coth u du = − csch u + C .



III. Inverse Hyperbolic Functions (Pages 394−396)

State the inverse hyperbolic function given by each of the following definitions and give the domain for each.

D o m a in

What you should learn How to develop properties of inverse hyperbolic functions

ln ( x +

ln ( x +

x2 + 1) = sinh−1 x , ( −∞ , ∞ ) .

x2 −1) = cosh−1 x , [ 1, ∞ ) .

1 ln 1 +

x2 1 −

x

= tanh−1 x , ( − 1, 1) .

1 ln

x + 1 = coth−1 x , ( −∞ , − 1) ∪ (1, ∞ ) .

2 x −1

ln 1

+1 − x2

x

= sech−1 x , (0, 1] .

⎛ 1 1 + x2 ⎞ln ⎜+

⎟ = csch−1 x , ( −∞ , 0) ∪ (0, ∞ ) .

⎜ x x ⎟⎝ ⎠

IV. Differentiation and Integration of Inverse Hyperbolic Functions (Pages 396−397)

Let u be a differentiable function of x. Complete each of the following rules of differentiation and integration.

What you should learn How to differentiate and integrate functions involving inverse hyperbolic functions

d [ sinh−1

u] = u′

dx u2 + 1

d [ cosh−1

u ] = u′

dx u2 −1

d [ tanh −1 u ] =dx

u′1 − u2



d [ coth −1 u ] =dx

u′1 − u2

d [ sech−1

u] = − u ′

dx u 1 − u2

∫

∫



d [ csch−1

u] = − u ′

dx u 1 + u2

du

u2 ±

a2

= ln(u + √u2 ± a2) + C .

du a2 − u2

du

= 1 / (2 a ) ln| ( a + u ) / ( a − u ) | + C .

−(1/a) ln( (a + √a2 ± u2) / |u|) + C .∫ u a2 ± u2 =

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Microsoft Word - instr_calc9e_0501.doc - WebStartswhscalculus.webstarts.com/uploads/notetaking_guide... · Web viewo. ns. 98Chapter 5Logarithmic, Exponential, and Other Transcendental