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CLP-1 Differential Calculus
Joel Feldman, Andrew Rechnitzer, Elyse Yeager
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Front Matter
Colophon
Preface
Dedication
Acknowledgements
Using the exercises in this book
Feedback about the text
0
The basics
0.1
Numbers
0.1.1
More on Real Numbers
0.2
Sets
0.3
Other Important Sets
0.3.1
More on Sets
0.4
Functions
0.5
Parsing Formulas
0.6
Inverse Functions
1
Limits
1.1
Drawing Tangents and a First Limit
1.1.1
Drawing Tangents and a First Limit
1.1.2
Exercises
1.2
Another Limit and Computing Velocity
1.2.1
Another Limit and Computing Velocity
1.2.2
Exercises
1.3
The Limit of a Function
1.3.1
The Limit of a Function
1.3.2
Exercises
1.4
Calculating Limits with Limit Laws
1.4.1
Calculating Limits with Limit Laws
1.4.2
Exercises
1.5
Limits at Infinity
1.5.1
Limits at Infinity
1.5.2
Exercises
1.6
Continuity
1.6.1
Continuity
1.6.2
Quick Aside — One-sided Continuity
1.6.3
Back to the Main Text
1.6.4
Exercises
1.7
(Optional) — Making the Informal a Little More Formal
1.8
(Optional) — Making Infinite Limits a Little More Formal
1.9
(Optional) — Proving the Arithmetic of Limits
2
Derivatives
2.1
Revisiting Tangent Lines
2.1.1
Revisiting Tangent Lines
2.1.2
Exercises
2.2
Definition of the Derivative
2.2.1
An Important Point (and Some Notation)
2.2.2
Back to Computing Some Derivatives
2.2.3
Where is the Derivative Undefined?
2.2.4
Exercises
2.3
Interpretations of the Derivative
2.3.1
Instantaneous Rate of Change
2.3.2
Slope
2.3.3
Exercises
2.4
Arithmetic of Derivatives - a Differentiation Toolbox
2.4.1
Arithmetic of Derivatives - a Differentiation Toolbox
2.4.2
Exercises
2.5
Proofs of the Arithmetic of Derivatives
2.5.1
Proof of the Linearity of Differentiation (Theorem 2.4.2)
2.5.2
Proof of the Product Rule (Theorem 2.4.3)
2.5.3
(Optional) — Proof of the Quotient Rule (Theorem 2.4.5)
2.6
Using the Arithmetic of Derivatives – Examples
2.6.1
Using the Arithmetic of Derivatives – Examples
2.6.2
Exercises
2.7
Derivatives of Exponential Functions
2.7.1
Whirlwind Review of Logarithms
2.7.1.1
Logarithmic Functions
2.7.2
Back to that Limit
2.7.3
Exercises
2.8
Derivatives of Trigonometric Functions
2.8.1
These Proofs are Optional, the Results are Not.
2.8.2
Step 1:
d
d
x
{
sin
x
}
|
x
=
0
2.8.3
Proof that
lim
h
→
0
sin
h
h
=
1
2.8.4
Step 2:
d
d
x
{
cos
x
}
|
x
=
0
2.8.4.1
Method 1 — Multiply by the “Conjugate”
2.8.4.2
Method 2 — via the Double Angle Formula
2.8.5
Step 3:
d
d
x
{
sin
x
}
and
d
d
x
{
cos
x
}
for General
x
2.8.6
Step 4: the Remaining Trigonometric Functions
2.8.7
Summary
2.8.8
Exercises
2.9
One More Tool – the Chain Rule
2.9.1
Statement of the Chain Rule
2.9.2
(Optional) — Derivation of the Chain Rule
2.9.3
Chain Rule Examples
2.9.4
Exercises
2.10
The Natural Logarithm
2.10.1
Back to
d
d
x
a
x
2.10.2
Logarithmic Differentiation
2.10.3
Exercises
2.11
Implicit Differentiation
2.11.1
Implicit Differentiation
2.11.2
Exercises
2.12
Inverse Trigonometric Functions
2.12.1
Derivatives of Inverse Trig Functions
2.12.2
Exercises
2.13
The Mean Value Theorem
2.13.1
Rolle’s Theorem
2.13.2
Back to the MVT
2.13.3
(Optional) — Why is the MVT True
2.13.4
Be Careful with Hypotheses
2.13.5
Exercises
2.14
Higher Order Derivatives
2.14.1
Higher Order Derivatives
2.14.2
Exercises
2.15
(Optional) — Is
lim
x
→
c
f
′
(
x
)
Equal to
?
f
′
(
c
)
?
3
Applications of derivatives
3.1
Velocity and Acceleration
3.1.1
Velocity and Acceleration
3.1.2
Exercises
3.2
Related Rates
3.2.1
Related Rates
3.2.2
Exercises
3.3
Exponential Growth and Decay
3.3.1
Carbon Dating
3.3.2
Newton’s Law of Cooling
3.3.3
Population Growth
3.3.3.1
(Optional) — Logistic Population Growth
3.3.4
Exercises
3.3.4
Exercises for § 3.3.1
3.3.4
Exercises for § 3.3.2
3.3.4
Exercises for § 3.3.3
3.3.4
Further problems for § 3.3
3.4
Taylor Polynomials
3.4.1
Zeroth Approximation — the Constant Approximation
3.4.2
First Approximation — the Linear Approximation
3.4.3
Second Approximation — the Quadratic Approximation
3.4.4
Whirlwind Tour of Summation Notation
3.4.5
Still Better Approximations — Taylor Polynomials
3.4.6
Some Examples
3.4.7
Estimating Change and
,
Δ
x
,
Δ
y
Notation
3.4.8
Further Examples
3.4.9
The Error in the Taylor Polynomial Approximations
3.4.10
(Optional) — Derivation of the Error Formulae
3.4.11
Exercises
3.4.11
Exercises for § 3.4.1
3.4.11
Exercises for § 3.4.2
3.4.11
Exercises for § 3.4.3
3.4.11
Exercises for § 3.4.4
3.4.11
Exercises for § 3.4.5
3.4.11
Exercises for § 3.4.6
3.4.11
Exercises for § 3.4.7
3.4.11
Exercises for § 3.4.8
3.4.11
Further problems for § 3.4
3.5
Optimisation
3.5.1
Local and Global Maxima and Minima
3.5.2
Finding Global Maxima and Minima
3.5.3
Max/Min Examples
3.5.4
Exercises
3.5.4
Exercises for § 3.5.1
3.5.4
Exercises for § 3.5.2
3.5.4
Exercises for § 3.5.3
3.6
Sketching Graphs
3.6.1
Domain, Intercepts and Asymptotes
3.6.2
First Derivative — Increasing or Decreasing
3.6.3
Second Derivative — Concavity
3.6.4
Symmetries
3.6.5
A Checklist for Sketching
3.6.5.1
A Sketching Checklist
3.6.6
Sketching Examples
3.6.7
Exercises
3.6.7
Exercises for § 3.6.1
3.6.7
Exercises for § 3.6.2
3.6.7
Exercises for § 3.6.3
3.6.7
Exercises for § 3.6.4
3.6.7
Exercises for § 3.6.6
3.7
L’Hôpital’s Rule, Indeterminate Forms
3.7.1
L’Hôpital’s Rule and Indeterminate Forms
3.7.1.1
Optional — Proof of Part (b) of l’Hôpital’s Rule
3.7.2
Standard Examples
3.7.3
Variations
3.7.3.1
Limits at
±
∞
3.7.3.2
∞
∞
indeterminate form
3.7.3.3
Optional — Proof of l’Hôpital’s Rule for
∞
∞
3.7.3.4
0
⋅
∞
indeterminate form
3.7.3.5
∞
−
∞
indeterminate form
3.7.3.6
1
∞
indeterminate form
3.7.3.7
0
0
indeterminate form
3.7.3.8
∞
0
indeterminate form
3.7.4
Exercises
4
Towards Integral Calculus
4.1
Introduction to Antiderivatives
4.1.1
Introduction to Antiderivatives
4.1.2
Exercises
Appendices
A
High School Material
A.1
Similar Triangles
A.2
Pythagoras
A.3
Trigonometry — Definitions
A.4
Radians, Arcs and Sectors
A.5
Trigonometry — Graphs
A.6
Trigonometry — Special Triangles
A.7
Trigonometry — Simple Identities
A.8
Trigonometry — Add and Subtract Angles
A.9
Inverse Trig Functions
A.10
Areas
A.11
Volumes
A.12
Powers
A.13
Logarithms
A.14
You Should be Able to Derive
B
Origin of Trig, Area and Volume Formulas
B.1
Theorems about Triangles
B.1.1
Thales’ Theorem
B.1.2
Pythagoras
B.2
Trigonometry
B.2.1
Angles — Radians vs Degrees
B.2.2
Trig Function Definitions
B.2.3
Important Triangles
B.2.4
Some More Simple Identities
B.2.5
Identities — Adding Angles
B.2.6
Identities — Double-angle Formulas
B.2.7
Identities — Extras
B.2.7.1
Sums to Products
B.2.7.2
Products to sums
B.3
Inverse Trigonometric Functions
B.4
Cosine and Sine Laws
B.4.1
Cosine Law or Law of Cosines
B.4.2
Sine Law or Law of Sines
B.5
Circles, cones and spheres
B.5.1
Where Does the Formula for the Area of a Circle Come From?
B.5.2
Where Do These Volume Formulas Come From?
C
Root Finding
C.1
Newton’s Method
C.2
The Error Behaviour of Newton’s Method
C.3
The false position (regula falsi) method
C.4
The secant method
C.5
The Error Behaviour of the Secant Method
D
Hints for Exercises
E
Answers to Exercises
F
Solutions to Exercises
🔗
Section
A.8
Trigonometry — Add and Subtract Angles
🔗
Sine
sin
(
α
±
β
)
=
sin
(
α
)
cos
(
β
)
±
cos
(
α
)
sin
(
β
)
Cosine
cos
(
α
±
β
)
=
cos
(
α
)
cos
(
β
)
∓
sin
(
α
)
sin
(
β
)
