Inhoud
Chapter 1: Functions..............................................................................................................................6
1.1 Functions and Their Graphs..........................................................................................................6
Functions; Domain and Range........................................................................................................6
Graphs of Functions........................................................................................................................6
Representing a Function Numerically.............................................................................................6
The Vertical Line Test for a Function..............................................................................................6
Piecewise-Defined Functions..........................................................................................................6
Increasing and Decreasing Functions..............................................................................................6
Even Functions and Odd functions: Symmetry...............................................................................7
Common Functions.........................................................................................................................7
1.2 Combining Functions; Shifting and scaling Graphs.......................................................................8
Sums, Differences, Products, and Quotients...................................................................................8
Composite Functions......................................................................................................................9
Shifting a Graph of a Function........................................................................................................9
Scaling and Reflecting a Graph of a Function..................................................................................9
1.3 Trigonometric Functions.............................................................................................................10
Angels...........................................................................................................................................10
The Six Basic Trigonometric Functions..........................................................................................10
Periodicity and Graphs of the Trigonometric Functions..............................................................11
Trigonometric Identities...............................................................................................................11
Law of Cosines..............................................................................................................................11
Two Special Inequalities...............................................................................................................12
1.4 Exponential Functions.................................................................................................................12
Exponential Behaviour..................................................................................................................12
The Natural Exponential Function ex.............................................................................................13
Exponential Growth and Decay....................................................................................................13
1.5 Inverse Functions and Logarithms..............................................................................................13
One-to-one Functions...................................................................................................................13
Inverse Functions..........................................................................................................................13
Finding Inverses............................................................................................................................14
Logarithmic Functions...................................................................................................................14
Properties of Logarithms..............................................................................................................14
Chapter 2: Limits and Continuity..........................................................................................................15
2.1 Rates of Change and Tangent Lines to Curves............................................................................15
1
, Average and Instantaneous Speed...............................................................................................15
Average Rates of Change and Secant Lines..................................................................................15
Defining the Slope of a Curve.......................................................................................................15
Rates of Change and Tangent Lines..............................................................................................15
2.2 Limit of a Function and Limit Laws..............................................................................................15
Limits of Function Values..............................................................................................................15
An Informal Description of the Limit of a Function.......................................................................15
The Limit Laws..............................................................................................................................15
Evaluating Limits of Polynomials and Rational Functions.............................................................16
Eliminating Common Factors from Zero Denominators...............................................................16
Using Calculators and Computers to Estimate Limits...................................................................16
The Sandwich Theorem................................................................................................................16
2.3 The Precise Definition of a Limit.................................................................................................16
Definition of a Limit......................................................................................................................17
Examples: Testing the Definition..................................................................................................17
Finding Deltas Algebraically for Given Epsilons............................................................................17
Using the Definition to Prove Theorems.......................................................................................17
2.4 One-Sided Limits.........................................................................................................................17
Approaching a Limit from One Side..............................................................................................17
Precise Definitions of One-Sided Limits........................................................................................18
Limits Involving (sinθ)/θ...............................................................................................................18
2.5 Limits Involving Infinity; Asymptotes of Graphs..........................................................................18
Finite Limits as x→±∞...................................................................................................................18
Limits at Infinity of Rational Functions..........................................................................................18
Horizontal Asymptotes.................................................................................................................18
Oblique Asymptotes.....................................................................................................................19
Precise Definitions of Infinite Limits.............................................................................................19
Vertical Asymptotes......................................................................................................................19
2.6 Continuity...................................................................................................................................19
Continuity at a Point.....................................................................................................................19
Continuous Functions...................................................................................................................20
Inverse Functions and Continuity.................................................................................................20
Continuity of Compositions of Functions......................................................................................20
Intermediate Value Theorem for Continuous Functions...............................................................20
Chapter 3: Derivatives..........................................................................................................................20
3.1 Tangent lines and the derivative at a point................................................................................20
2
, Finding a tangent line to the graph of a function.........................................................................20
Rates of change: Derivative at a point..........................................................................................20
Summary......................................................................................................................................21
3.2 The derivative as a function........................................................................................................21
Calculating derivatives from the definition...................................................................................21
Notation........................................................................................................................................21
Graphing the derivative................................................................................................................21
Differentiable on an interval; One-sided derivatives....................................................................21
When does a function not have a derivative at a point................................................................22
Differentiable functions are continuous.......................................................................................22
3.3 Differentiation Rules...................................................................................................................22
Powers, multiples, sums, and differences.....................................................................................22
Derivatives of exponential functions............................................................................................22
Products and quotients.................................................................................................................22
Second- and higher-order derivatives...........................................................................................23
3.4 The derivative as a rate of change..............................................................................................23
Instantaneous rates of change.....................................................................................................23
Motion Along a Line: Displacement, Velocity, Speed, Acceleration, and Jerk...............................23
Derivatives in Economics..............................................................................................................23
Sensitivity to Change....................................................................................................................23
3.5 Derivatives of Trigonometric Functions......................................................................................23
Derivative of the Sine Function.....................................................................................................23
Derivative of the Cosine Function.................................................................................................23
Simple Harmonic Motion..............................................................................................................24
Derivatives of the Other Basic Trigonometric Functions...............................................................24
3.6 The Chain Rule............................................................................................................................24
Derivative of a Composite Function..............................................................................................24
‘Outside-Inside’ Rule.....................................................................................................................24
Repeated Use of the Chain Rule...................................................................................................24
The Chain Rule with Powers of a Function....................................................................................24
3.7 Implicit Differentiation................................................................................................................25
Implicitly Defined Functions.........................................................................................................25
Derivatives of Higher Order..........................................................................................................25
Lenses, Tangent Lines, and Normal Lines.....................................................................................25
3.8 Derivatives of Inverse Functions and Logarithms.......................................................................25
Derivatives of Inverses of Differentiable Functions......................................................................25
3
, Derivative of the Natural Logarithm Function..............................................................................25
The Derivatives of au and logau.....................................................................................................25
Logarithmic Differentiation...........................................................................................................26
Irrational Exponents and the Power Rule (General Version)........................................................26
The Number e Expressed as a Limit..............................................................................................26
Chapter 4: Applications of Derivatives..................................................................................................27
4.1 Extreme values of Functions on Closed Intervals........................................................................27
Local (Relative) Extreme Values....................................................................................................27
Finding Extrema............................................................................................................................27
4.2 The Mean Value Theorem..........................................................................................................27
Rolle’s Theorem............................................................................................................................27
The Mean Value Theorem............................................................................................................27
Mathematical Consequences........................................................................................................27
Finding Velocity and Position from Acceleration.........................................................................28
Law of Exponents..........................................................................................................................28
4.3 Monotonic Functions and the First Derivative Test....................................................................28
Increasing Functions and Decreasing Functions...........................................................................28
First Derivative Test for Local Extrema.........................................................................................28
4.4 Concavity and Curve Sketching...................................................................................................28
Concavity......................................................................................................................................28
Points of Inflection........................................................................................................................28
Second Derivative Test for Local Extrema.....................................................................................29
4.5 Indeterminate Forms and L’Hôpital’s Rule.................................................................................29
Indeterminate Form 0/0...............................................................................................................29
Indeterminate Powers..................................................................................................................29
Proof of L’Hôpital’s Rule...............................................................................................................29
4.6 Applied Optimization..................................................................................................................29
4.7 Newton’s Method.......................................................................................................................30
Procedure for Newton’s Method..................................................................................................30
4.8 Antiderivatives............................................................................................................................30
Finding Antiderivatives.................................................................................................................30
Indefinite Integrals........................................................................................................................31
Chapter 5: Integrals..............................................................................................................................31
5.5 Indefinite Integrals and the Substitution Method......................................................................31
Substitution: Running the Chain Rule Backwards.........................................................................31
5.6 Definite Integral Substitutions and the Area Between Curves....................................................31
4
, The Substitution Formula.............................................................................................................31
Definite Integrals of Symmetric Functions....................................................................................31
Areas Between Curves..................................................................................................................31
Integration with Respect to y.......................................................................................................32
Chapter 7: Integrals and Transcendental Functions.............................................................................32
7.1 The Logarithm Defined as an Integral.........................................................................................32
Definition of the Natural Logarithm Function...............................................................................32
The Derivative of y=ln(x)...............................................................................................................32
The Graph and Range of ln(x).......................................................................................................32
The Integral...................................................................................................................................32
The Inverse of ln(x) and the Number e.........................................................................................32
Laws of Exponents........................................................................................................................33
The General Exponential Function ax............................................................................................33
Logarithms with Base a.................................................................................................................33
Derivatives and Integrals Involving logax......................................................................................33
Chapter 8: Techniques of Integration...................................................................................................34
8.1 Using Basic Integration Formulas................................................................................................34
8.2 Integration by Parts....................................................................................................................34
Product Rule in Integral Form.......................................................................................................34
Evaluating Definite Integrals by Parts...........................................................................................35
8.3 Trigonometric Integrals..............................................................................................................35
Products of Powers of Sines and Cosines.....................................................................................35
Eliminating Square Roots..............................................................................................................35
Integrals of Powers of tan(x) and sec(x)........................................................................................35
Products of Sines and Cosines......................................................................................................35
8.4 Trigonometric Substitutions.......................................................................................................35
8.5 Integration of Rational Functions by Partial Fractions................................................................36
General Description of the Method..............................................................................................36
5
, Chapter 1: Functions
1.1 Functions and Their Graphs
A function can be represented by an equation, a graph, a numerical table, or a verbal description.
Functions; Domain and Range
The value of one variable quantity, say y, depends on the value of another variable quantity, which
we often call x. We say that ‘’y is a function of x’’ and write this symbolically as
''
y=f ( x ) (' ' y equals f of x )
The symbol f represents the function, the letter x is the independent variable representing the input
value to f, and y is the dependent variable or output value of f at x.
Definition: A function f from a set D to a set Y is a rule that assigns a unique value to f(x) in Y to each
x in D.
The domain are all possible x. The range are all possible y. Changing the domain to which we apply a
formula usually changes the range as well. If the range of a function is a set of real numbers, the
function is real-valued. The domains and ranges of most real-valued functions we consider are
intervals or combinations of intervals.
Graphs of Functions
If f is a function with domain D, its graph consists of the points in the Cartesian plane whose
coordinates are the input-output pairs for f. In set notation, the graph is
{( x , f ( x ) ) ∣ x ∊ D }
The graph of a function f is a useful picture of its behaviour. If (x,y) is a point on the graph, then y=f(x)
is the height of the graph above (or below) the point x. The height may be positive or negative,
depending on the sign of f(x).
Representing a Function Numerically
A function may be represented algebraically by a formula and visually by a graph. Another way to
represent a function is numerically, through a table of values. The graph consisting of only the points
in the table is called a scatterplot.
The Vertical Line Test for a Function
A function f can have only one value f(x) for each x in its domain, so no vertical line can intersect the
graph of a function more than once. A circle can not be the graph of one function.
Piecewise-Defined Functions
Sometimes a function is described in pieces by using different formulas on different parts of its
domain. One example is the absolute value function
{−x ,∧x <0
|x|= x ,∧x ≥ 0
Piecewise-defined functions often arise when real-world data are modelled.
Increasing and Decreasing Functions
Definitions: Let f be a function defined on an interval I and let x1 and x2 be two distinct points in I.
1. If f(x2)>f(x1) whenever x1<x2, then f is said to be increasing on I.
6
Chapter 1: Functions..............................................................................................................................6
1.1 Functions and Their Graphs..........................................................................................................6
Functions; Domain and Range........................................................................................................6
Graphs of Functions........................................................................................................................6
Representing a Function Numerically.............................................................................................6
The Vertical Line Test for a Function..............................................................................................6
Piecewise-Defined Functions..........................................................................................................6
Increasing and Decreasing Functions..............................................................................................6
Even Functions and Odd functions: Symmetry...............................................................................7
Common Functions.........................................................................................................................7
1.2 Combining Functions; Shifting and scaling Graphs.......................................................................8
Sums, Differences, Products, and Quotients...................................................................................8
Composite Functions......................................................................................................................9
Shifting a Graph of a Function........................................................................................................9
Scaling and Reflecting a Graph of a Function..................................................................................9
1.3 Trigonometric Functions.............................................................................................................10
Angels...........................................................................................................................................10
The Six Basic Trigonometric Functions..........................................................................................10
Periodicity and Graphs of the Trigonometric Functions..............................................................11
Trigonometric Identities...............................................................................................................11
Law of Cosines..............................................................................................................................11
Two Special Inequalities...............................................................................................................12
1.4 Exponential Functions.................................................................................................................12
Exponential Behaviour..................................................................................................................12
The Natural Exponential Function ex.............................................................................................13
Exponential Growth and Decay....................................................................................................13
1.5 Inverse Functions and Logarithms..............................................................................................13
One-to-one Functions...................................................................................................................13
Inverse Functions..........................................................................................................................13
Finding Inverses............................................................................................................................14
Logarithmic Functions...................................................................................................................14
Properties of Logarithms..............................................................................................................14
Chapter 2: Limits and Continuity..........................................................................................................15
2.1 Rates of Change and Tangent Lines to Curves............................................................................15
1
, Average and Instantaneous Speed...............................................................................................15
Average Rates of Change and Secant Lines..................................................................................15
Defining the Slope of a Curve.......................................................................................................15
Rates of Change and Tangent Lines..............................................................................................15
2.2 Limit of a Function and Limit Laws..............................................................................................15
Limits of Function Values..............................................................................................................15
An Informal Description of the Limit of a Function.......................................................................15
The Limit Laws..............................................................................................................................15
Evaluating Limits of Polynomials and Rational Functions.............................................................16
Eliminating Common Factors from Zero Denominators...............................................................16
Using Calculators and Computers to Estimate Limits...................................................................16
The Sandwich Theorem................................................................................................................16
2.3 The Precise Definition of a Limit.................................................................................................16
Definition of a Limit......................................................................................................................17
Examples: Testing the Definition..................................................................................................17
Finding Deltas Algebraically for Given Epsilons............................................................................17
Using the Definition to Prove Theorems.......................................................................................17
2.4 One-Sided Limits.........................................................................................................................17
Approaching a Limit from One Side..............................................................................................17
Precise Definitions of One-Sided Limits........................................................................................18
Limits Involving (sinθ)/θ...............................................................................................................18
2.5 Limits Involving Infinity; Asymptotes of Graphs..........................................................................18
Finite Limits as x→±∞...................................................................................................................18
Limits at Infinity of Rational Functions..........................................................................................18
Horizontal Asymptotes.................................................................................................................18
Oblique Asymptotes.....................................................................................................................19
Precise Definitions of Infinite Limits.............................................................................................19
Vertical Asymptotes......................................................................................................................19
2.6 Continuity...................................................................................................................................19
Continuity at a Point.....................................................................................................................19
Continuous Functions...................................................................................................................20
Inverse Functions and Continuity.................................................................................................20
Continuity of Compositions of Functions......................................................................................20
Intermediate Value Theorem for Continuous Functions...............................................................20
Chapter 3: Derivatives..........................................................................................................................20
3.1 Tangent lines and the derivative at a point................................................................................20
2
, Finding a tangent line to the graph of a function.........................................................................20
Rates of change: Derivative at a point..........................................................................................20
Summary......................................................................................................................................21
3.2 The derivative as a function........................................................................................................21
Calculating derivatives from the definition...................................................................................21
Notation........................................................................................................................................21
Graphing the derivative................................................................................................................21
Differentiable on an interval; One-sided derivatives....................................................................21
When does a function not have a derivative at a point................................................................22
Differentiable functions are continuous.......................................................................................22
3.3 Differentiation Rules...................................................................................................................22
Powers, multiples, sums, and differences.....................................................................................22
Derivatives of exponential functions............................................................................................22
Products and quotients.................................................................................................................22
Second- and higher-order derivatives...........................................................................................23
3.4 The derivative as a rate of change..............................................................................................23
Instantaneous rates of change.....................................................................................................23
Motion Along a Line: Displacement, Velocity, Speed, Acceleration, and Jerk...............................23
Derivatives in Economics..............................................................................................................23
Sensitivity to Change....................................................................................................................23
3.5 Derivatives of Trigonometric Functions......................................................................................23
Derivative of the Sine Function.....................................................................................................23
Derivative of the Cosine Function.................................................................................................23
Simple Harmonic Motion..............................................................................................................24
Derivatives of the Other Basic Trigonometric Functions...............................................................24
3.6 The Chain Rule............................................................................................................................24
Derivative of a Composite Function..............................................................................................24
‘Outside-Inside’ Rule.....................................................................................................................24
Repeated Use of the Chain Rule...................................................................................................24
The Chain Rule with Powers of a Function....................................................................................24
3.7 Implicit Differentiation................................................................................................................25
Implicitly Defined Functions.........................................................................................................25
Derivatives of Higher Order..........................................................................................................25
Lenses, Tangent Lines, and Normal Lines.....................................................................................25
3.8 Derivatives of Inverse Functions and Logarithms.......................................................................25
Derivatives of Inverses of Differentiable Functions......................................................................25
3
, Derivative of the Natural Logarithm Function..............................................................................25
The Derivatives of au and logau.....................................................................................................25
Logarithmic Differentiation...........................................................................................................26
Irrational Exponents and the Power Rule (General Version)........................................................26
The Number e Expressed as a Limit..............................................................................................26
Chapter 4: Applications of Derivatives..................................................................................................27
4.1 Extreme values of Functions on Closed Intervals........................................................................27
Local (Relative) Extreme Values....................................................................................................27
Finding Extrema............................................................................................................................27
4.2 The Mean Value Theorem..........................................................................................................27
Rolle’s Theorem............................................................................................................................27
The Mean Value Theorem............................................................................................................27
Mathematical Consequences........................................................................................................27
Finding Velocity and Position from Acceleration.........................................................................28
Law of Exponents..........................................................................................................................28
4.3 Monotonic Functions and the First Derivative Test....................................................................28
Increasing Functions and Decreasing Functions...........................................................................28
First Derivative Test for Local Extrema.........................................................................................28
4.4 Concavity and Curve Sketching...................................................................................................28
Concavity......................................................................................................................................28
Points of Inflection........................................................................................................................28
Second Derivative Test for Local Extrema.....................................................................................29
4.5 Indeterminate Forms and L’Hôpital’s Rule.................................................................................29
Indeterminate Form 0/0...............................................................................................................29
Indeterminate Powers..................................................................................................................29
Proof of L’Hôpital’s Rule...............................................................................................................29
4.6 Applied Optimization..................................................................................................................29
4.7 Newton’s Method.......................................................................................................................30
Procedure for Newton’s Method..................................................................................................30
4.8 Antiderivatives............................................................................................................................30
Finding Antiderivatives.................................................................................................................30
Indefinite Integrals........................................................................................................................31
Chapter 5: Integrals..............................................................................................................................31
5.5 Indefinite Integrals and the Substitution Method......................................................................31
Substitution: Running the Chain Rule Backwards.........................................................................31
5.6 Definite Integral Substitutions and the Area Between Curves....................................................31
4
, The Substitution Formula.............................................................................................................31
Definite Integrals of Symmetric Functions....................................................................................31
Areas Between Curves..................................................................................................................31
Integration with Respect to y.......................................................................................................32
Chapter 7: Integrals and Transcendental Functions.............................................................................32
7.1 The Logarithm Defined as an Integral.........................................................................................32
Definition of the Natural Logarithm Function...............................................................................32
The Derivative of y=ln(x)...............................................................................................................32
The Graph and Range of ln(x).......................................................................................................32
The Integral...................................................................................................................................32
The Inverse of ln(x) and the Number e.........................................................................................32
Laws of Exponents........................................................................................................................33
The General Exponential Function ax............................................................................................33
Logarithms with Base a.................................................................................................................33
Derivatives and Integrals Involving logax......................................................................................33
Chapter 8: Techniques of Integration...................................................................................................34
8.1 Using Basic Integration Formulas................................................................................................34
8.2 Integration by Parts....................................................................................................................34
Product Rule in Integral Form.......................................................................................................34
Evaluating Definite Integrals by Parts...........................................................................................35
8.3 Trigonometric Integrals..............................................................................................................35
Products of Powers of Sines and Cosines.....................................................................................35
Eliminating Square Roots..............................................................................................................35
Integrals of Powers of tan(x) and sec(x)........................................................................................35
Products of Sines and Cosines......................................................................................................35
8.4 Trigonometric Substitutions.......................................................................................................35
8.5 Integration of Rational Functions by Partial Fractions................................................................36
General Description of the Method..............................................................................................36
5
, Chapter 1: Functions
1.1 Functions and Their Graphs
A function can be represented by an equation, a graph, a numerical table, or a verbal description.
Functions; Domain and Range
The value of one variable quantity, say y, depends on the value of another variable quantity, which
we often call x. We say that ‘’y is a function of x’’ and write this symbolically as
''
y=f ( x ) (' ' y equals f of x )
The symbol f represents the function, the letter x is the independent variable representing the input
value to f, and y is the dependent variable or output value of f at x.
Definition: A function f from a set D to a set Y is a rule that assigns a unique value to f(x) in Y to each
x in D.
The domain are all possible x. The range are all possible y. Changing the domain to which we apply a
formula usually changes the range as well. If the range of a function is a set of real numbers, the
function is real-valued. The domains and ranges of most real-valued functions we consider are
intervals or combinations of intervals.
Graphs of Functions
If f is a function with domain D, its graph consists of the points in the Cartesian plane whose
coordinates are the input-output pairs for f. In set notation, the graph is
{( x , f ( x ) ) ∣ x ∊ D }
The graph of a function f is a useful picture of its behaviour. If (x,y) is a point on the graph, then y=f(x)
is the height of the graph above (or below) the point x. The height may be positive or negative,
depending on the sign of f(x).
Representing a Function Numerically
A function may be represented algebraically by a formula and visually by a graph. Another way to
represent a function is numerically, through a table of values. The graph consisting of only the points
in the table is called a scatterplot.
The Vertical Line Test for a Function
A function f can have only one value f(x) for each x in its domain, so no vertical line can intersect the
graph of a function more than once. A circle can not be the graph of one function.
Piecewise-Defined Functions
Sometimes a function is described in pieces by using different formulas on different parts of its
domain. One example is the absolute value function
{−x ,∧x <0
|x|= x ,∧x ≥ 0
Piecewise-defined functions often arise when real-world data are modelled.
Increasing and Decreasing Functions
Definitions: Let f be a function defined on an interval I and let x1 and x2 be two distinct points in I.
1. If f(x2)>f(x1) whenever x1<x2, then f is said to be increasing on I.
6
