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A tour of single variable calculus

A tour of single variable calculus

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    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