Summary Thomas' Calculus Early Transcendentals 14th edition
53 views 0 purchase
Course
Wiskunde voor Aardwetenschappen (GEO11103)
Institution
Universiteit Utrecht (UU)
Book
Thomas\' Calculus
Mathematics summary of the chapters 1-4 , chapter 5.5 and 5.6, chapter 7.1, chapter 8.1 - 8.4 from Thomas' Calculus Early Transcendentals 14th edition.
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
The benefits of buying summaries with Stuvia:
Guaranteed quality through customer reviews
Stuvia customers have reviewed more than 700,000 summaries. This how you know that you are buying the best documents.
Quick and easy check-out
You can quickly pay through credit card or Stuvia-credit for the summaries. There is no membership needed.
Focus on what matters
Your fellow students write the study notes themselves, which is why the documents are always reliable and up-to-date. This ensures you quickly get to the core!
Frequently asked questions
What do I get when I buy this document?
You get a PDF, available immediately after your purchase. The purchased document is accessible anytime, anywhere and indefinitely through your profile.
Satisfaction guarantee: how does it work?
Our satisfaction guarantee ensures that you always find a study document that suits you well. You fill out a form, and our customer service team takes care of the rest.
Who am I buying these notes from?
Stuvia is a marketplace, so you are not buying this document from us, but from seller rianneheurneman. Stuvia facilitates payment to the seller.
Will I be stuck with a subscription?
No, you only buy these notes for $5.07. You're not tied to anything after your purchase.
