Calculus
Upcoming SlideShare
Loading in...5
×
 

Calculus

on

  • 2,845 views

A tour of single variable calculus

A tour of single variable calculus

Statistics

Views

Total Views
2,845
Views on SlideShare
2,840
Embed Views
5

Actions

Likes
4
Downloads
185
Comments
0

2 Embeds 5

http://203.144.133.44 4
http://eleap.ust.edu.ph 1

Accessibility

Categories

Upload Details

Uploaded via as Microsoft PowerPoint

Usage Rights

© All Rights Reserved

Report content

Flagged as inappropriate Flag as inappropriate
Flag as inappropriate

Select your reason for flagging this presentation as inappropriate.

Cancel
  • Full Name Full Name Comment goes here.
    Are you sure you want to
    Your message goes here
    Processing…
Post Comment
Edit your comment

Calculus Calculus Presentation Transcript

  • Single Variable Calculus – A Tour
    Dr. Abdulla Al-Othman
  • Part1: Differentiation
    Motivation, Key Ideas, Theory
    Calculus 1
    2
  • Gradients
    Also called slopes
    Calculus 1
    3
  • A Line has a Single Slope
    Calculus 1
    4
    y
    ∆y
    ∆x
    x
  • A Curve has Many Slopes
    Calculus 1
    5
    y
    Many Slopes
    x
  • Example: Slopes of y=x2
    Calculus 1
    6
    y
    c
    b
    The slope increases as we move in the direction of the arrows
    a
    x
  • Measuring the Many Slopes of a Curve:
    Calculus 1
    7
    Gottfried Leibniz 1646-1717
    Isaac Newton 1642-1726
  • Calculus 1
    8
    y
    a, f(a)
    a
    x
    Objective: To find the Slope of the Curve at Point (a , f(a))
  • Be Observant
    Calculus 1
    9
    جلال‌الدین محمد رومی
    “If thou wilt be observant and vigilant, thou wilt see at every moment“
  • The Key Observation of Newton and Leibniz
    Calculus 1
    10
    Slope =l1
    y
    Slope =l2
    We make h smaller, and the limit of the slopes as it approaches 0, if it exists, will be the slope of the tangent line
    Slope=l3
    Slope =ln
    a, f(a)
    a+h3
    a+hn
    a
    x
    a+h1
    a+h2
  • Calculus 1
    11
    The key idea translated into the language of mathematics
  • Differentiability and Continuity
    How they’re related
    Calculus 1
    12
  • Differentiable => Continuous
    Calculus 1
    13
    “In Mathematics as in Design, Perfection is almost always Simplicity.” Abdulla Al-Othman
  • Proof:
    Calculus 1
    14
  • Continuous Does Not ===> Differentiable
    Calculus 1
    15
  • A Continuous non Differentiable Function
    Calculus 1
    16
    y
    x
  • Some Useful Rules
    No new ideas involved
    Calculus 1
    17
  • The List:
    Calculus 1
    18
  • Properties of Differentiable Functions cont…
    Calculus 1
    19
  • The Proofs
    Calculus 1
    20
    “What is now proven was once only imagined.”
    William Blake
  • The Sum Rule
    Calculus 1
    21
  • Extracting the Constant Rule
    Calculus 1
    22
  • The Product Rule
    Calculus 1
    23
  • The Quotient Rule
    Calculus 1
    24
  • Cont….
    Calculus 1
    25
  • The Chain Rule
    Calculus 1
    26
  • The Inverse Function Rule
    Calculus 1
    27
  • Differentiation of Polynomials
    Calculus1
    Calculus 1
    28
  • محمد بن موسى الخوارزمي
    Calculus 1
    29
  • Calculus 1
    30
    The Algorithm
  • Examples:
    Calculus 1
    31
  • Calculus 1
    32
    Example1: Calculating the Derivative of y=c
  • Calculus 1
    33
    Example2: Calculating the Derivative of y=x
  • Calculus 1
    34
    Example3: Calculating the Derivative of y=x2
  • Calculus 1
    35
    Example4: Calculating the Derivative of y=x3
  • Tabulating our results, do you see a pattern?
    Calculus 1
    36
  • Key Result1 (Our first building block)
    Plug and Play
    Calculus 1
    37
  • Polynomial Differentiation:
    Plug and Play
    Calculus 1
    38
  • Trigonometric Functions
    Review
    Calculus 1
    39
  • Basics
    Calculus 1
    40
    y
    1
    θ
    x
  • Sine
    Unit CIRCLE
    SINe
    41
    Y
    1
    1
    θ
    θ
    X
    -1
    1
    X
    0
    π/2
    π

    3π/2
    -1
    -1
    Calculus 1
  • Cosine
    Unit CIRCLE:
    Cosine:
    42
    Y
    1

    π/2
    3π/2
    π
    θ
    θ
    X
    -1
    1
    X
    0
    -1
    Calculus 1
  • Tan=Sine/Cos ine
    Calculus 1
    43
    π/2
    π

    -π/2
    0
    θ
  • Sin -1
    sINe :
    Sine-1
    Calculus 1
    44
    The Sine function is 1:1 over this range so it admits an inverse
    1
    π/2
    θ
    π/2
    0
    -π/2
    -1
    1
    -π/2
    -1
  • Cos -1
    cos:
    cos-1:
    45
    The Sine function is 1:1 over this range so it admits an inverse
    π
    1
    π/2
    θ
    π/2
    0
    -1
    1
    π
    -1
    Calculus 1
  • Tan -1
    tan :
    Tan-1
    46
    The tan function is 1:1 over this range so it admits an inverse
    θ
    π/2
    π/2
    θ
    -π/2
    0
    -π/2
    Calculus 1
  • Wikepedia Animation
    Calculus 1
    47
  • Important Formulae
    Calculus 1
    48
  • The Proofs
    Calculus 1
    49
    "Everything should be made as simple as possible, but not simpler." Einstein
  • Rule1 :
    Calculus 1
    50
    A1
    1
    1
    A2
    β
    α
  • Cont…
    Calculus 1
    51
    B3
    1
    1
    B2
    B1
    β
    α
  • Cont..
    Calculus 1
    52
  • Rule2:
    Calculus 1
    53
  • Rule3 :
    Calculus 1
    54
  • Rule 4 :
    Calculus 1
    55
  • Differentiation of Trigonometric Functions
    Calculus 1
    56
  • The Formulae
    Calculus 1
    57
  • Two Important Limits:
    Calculus 1
    58
  • d/dx (sin(x))
    Calculus 1
    59
  • d/dx (cos(x))
    Calculus 1
    60
  • d/dx tan(x))
    Calculus 1
    61
  • d/dx (sec(x))
    Calculus 1
    62
  • Derivatives of Exponentials and Logarithms
    Calculus2
    Calculus 2
    63
  • Derivatives of Exponentials:
    Calculus 2
    64
  • A Special Limit
    Calculus 2
    65
  • Geometric Proof
    Calculus 2
    66
    1
    0
  • Proofs :
    Calculus 2
    67
  • Derivatives of Logarithmic Functions:
    Calculus 2
    68
  • Proofs:
    Calculus 2
    69
  • The Mean Value Theorem
    Taylor Series Expansions
    Calculus 1
    70
  • Local Max and Min
    Calculus 1
    71
  • Local Max and Min
    Calculus 1
    72
    Local Max
    Y
    Local Min
    X
    b
    a
    Stationary Points
  • Theorem 1
    Calculus 1
    73
  • Proof:
    Calculus 1
    74
  • Stationary Points
    Calculus 1
    75
    Y
    Local Max
    Local Min
    b
    a
    X
    Point of Inflexion
  • Roll’s Theorem (Prelude to the Mean Value Theorem)
    Calculus 1
    76
  • Idea
    Calculus 1
    77
    Case 2: Max (or Min) occurs at an interior point ξ
    Case 1: Max and Min occur at end points, so function is constant.
    f(x)
    f(x)
    a
    b
    x
    a
    b
    x
  • Theorem not true if f(a) ≠ f(b)
    Calculus 1
    78
    f(x)
    f(x)
    a
    b
    x
    a
    b
    x
  • Proof:
    Calculus 1
    79
  • The Mean Value Theorem
    Calculus 1
    80
  • The Picture:
    Calculus 1
    81
    b, f(b)
    a, f(a)
    ξ
    a
    b
  • Proof:
    Calculus 1
    82
  • Taylor’s Theorem
    Calculus 1
    83
  • The Idea:
    Calculus 1
    84
  • The Idea cont:
    Calculus 1
    85
  • The Proof (optional):
    Calculus 1
    86
  • The Proof (Optional):
    Calculus 1
    87
  • The Proof (optional):
    Calculus 1
    88
  • The Proof (Optional) cont..:
    Calculus 1
    89
  • Applications: exp(x)
    Calculus 1
    90
  • Applications:
    Calculus 1
    91
  • Applications: sin(x), cos(x)
    Calculus 1
    92
  • Applications: sin(x), cos(x) cont
    Calculus 1
    93
  • Applications: sin(x), cos(x) cont
    Calculus 1
    94
  • Applications:
    Calculus 1
    95
  • Applications: Cont
    Calculus 1
    96
  • The Binomial Theorem:
    Calculus 1
    97
  • Applications: Proving the Binomial The0rem
    Calculus 1
    98
  • Applications: Binomial Cont..
    Calculus 1
    99
  • Application: L’Hopitals Rule
    Calculus 1
    100
  • Application: L’Hopitals Rule
    Calculus 1
    101
  • Exercises:
    Calculus 1
    102
  • Introduction to the Theory of Integration
    Calculus 1
    Dr. Abdulla Al-Othman
  • Part 2: Integration
    Motivation, Key Ideas, Theory
    Calculus 1
    104
  • The Ideas of Riemann
    Bernhard Riemann, 1826-66
    Calculus 1
    105
  • Objective : To Find Area S Under a Curve f
    Calculus 1
    106
  • The Key Idea
    Calculus 1
    107
  • The Key Idea Translated into the Language of Mathematics
    Calculus 1
    108
  • The Fundamental Theorems of Calculus
    Calculus 1
    109
  • The First Fundamental Theorem of Calculus (FFT)
    Calculus 1
    110
  • The First Fundamental Theorem of Calculus cont (FFT)
    Calculus 1
    111
  • The First Fundamental Theorem of Calculus (FFC) Proof (Optional)
    Calculus 1
    112
  • The Second Fundamental Theorem of Calculus
    Calculus 1
    113
  • The Second Fundamental Theorem of Calculus (SFC) Proof (Optional)
    Calculus 1
    114
  • Examples
    Calculus 1
    115
  • Example 1:
    Calculus 1
    116
  • Example 2:
    Calculus 1
    117
  • Example 3:
    Calculus 1
    118
  • Properties of the Integral
    Calculus 1
    119
  • Calculus 1
    120
  • The Anti Derivative
    Or Primitive
    Calculus 1
    121
  • Definition: Anti Derivative
    Calculus 2
    122
  • Techniques for Finding the Anti Derivative
    Method of Substitution
    Method of By Parts
    Method of Partial Fractions
    Calculus 2
    123
  • 1) Technique of Substitution
    Calculus 2
    124
  • Proof
    Calculus 2
    125
  • The Cook Book Approach Substitution (Plug and Play)
    Calculus 2
    126
  • The Cook Book Approach ( Reverse Substitution)
    Calculus 2
    127
  • Exercises:
    Calculus 2
    128
  • Partial Solutions to Selected Examples:
    Calculus 2
    129
  • Partial Solutions to Selected Class Examples cont..:
    Calculus 2
    130
  • Partial Solutions to Selected Class Exercises cont:
    Calculus 2
    131
  • Exercises: Trigonometric Integrals
    Calculus 2
    132
  • Examples cont…
    Calculus 2
    133
  • Solutions to Selected Exercises:
    Calculus 2
    134
  • Cont…
    Calculus 2
    135
  • Cont…
    Calculus 2
    136
  • Proofs…
    Calculus 2
    137
  • Trigonometric Substitution Motivating Example
    Calculus 2
    138
    y
    x
    t
  • Example Cont…
    Calculus 2
    139
  • Example Cont…
    Calculus 2
    140
    a
    x
    u
  • Further Examples: Integrals Using Trig Substitution and Trig Identities
    Calculus 2
    141
  • Further Examples cont…
    Calculus 2
    142
  • Cont…
    143
    y
    u
    2
    Calculus 2
  • 2)The Technique of “By Parts”:
    Calculus 2
    144
  • Proof:
    Calculus 2
    145
  • Cont…
    Calculus 2
    146
  • Calculus 2
    147
    The Cook Book Approach (Plug and Play)
  • Exercises:
    Calculus 2
    148
  • Exercises Cont: Solutions
    Calculus 2
    149
  • Cont..
    Calculus 2
    150
  • Cont…
    Calculus 2
    151
  • 3) Partial Fractions
    Calculus 2
    152
  • 3) The Technique of Partial Fractions..cont
    Calculus 2
    153
  • 3) The Technique of Partial Fractions..cont
    Calculus 2
    154
  • 3) The Technique of Partial Fractions..cont
    Calculus 2
    155
  • Exercises:
    Calculus 2
    156
  • Differential Equations
    An Introduction
    Calculus 1
    157
  • Definition
    A Differential Equation is an equation which contains at least one derivative of an unknown function.
    Calculus 1
    158
  • Examples
    Calculus 1
    159
  • Solution of Differential Equations
    A solution to a differentiable equation is a relation between the variables involved which is:
    Free From Derivatives
    Is Consistent with the Differential Equation
    Calculus 1
    160
  • Comments:
    Calculus 1
    161
    • Y=sin x is a solution to 1
    • 5 is called a Partial Differential Equation
    • All the others are called Ordinary Differential Equations
  • Orders and Degrees
    Calculus 1
    162
    • The Order of a Differential Equation is the order of the derivative of the highest power.
    • The Degree of a Differential Equation is the algebraic degree with which the derivative of the highest order appears in the equation
  • Examples:
    Calculus 1
    163
    • Equations 1 and 6 are first order all others are second order
    • Equation 3 is of degree 3
    • All others are of degree1
  • In this Course :
    Calculus 1
    164
    Calculus 2
  • General Solutions
    Calculus 1
    165
  • General Solution Cont…
    Calculus 1
    166
  • Boundary Conditions
    Calculus 1
    167
  • Techniques for Solving Order One Ordinary Differential Equations
    Separating Variables
    Integrating Factor
    Changing Variables
    Calculus 1
    168
  • Separating Variables Technique
    Calculus 1
    169
  • Example
    Calculus 1
    170
  • Integrating Factor Technique (Linear Equations of Order 1)
    Calculus 1
    171
  • Example1
    Calculus 1
    172
  • Example2
    Calculus 1
    173
  • Example3
    Calculus 1
    174
  • Technique of Changing the Variable
    Calculus 1
    175
    Occasionally an unpleasant looking Differential Equation can be converted into something more manageable by making a change of variable.
    Unfortunately, it is seldom easy to think of an appropriate change
  • Example1:
    Calculus 1
    176
  • Example2:
    Calculus 1
    177
  • Exercises:
    Calculus 1
    178
  • Exercises cont…
    Calculus 1
    179