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92 problems have screenshot solutions; and 17 have video solutions

MATH2111 Index

Tutorial Problems

Questions List

Go to Topics

Q1

Relevant Topics: Representation of Curves | Lines

Resources: bb2 | screenshot

Q2

Relevant Topics: Representation of Curves | Projections on Coordinate Planes

Resources: geogebra | bb2 | screenshot

Q4

Relevant Topics: Lines | Tangent Vector to Parametric Curve

Resources: geogebra

Q5

Relevant Topics: Tangent Vector to Parametric Curve

Resources: geogebra | bb2/i | screenshot/i | bb2/ii | screenshot/ii | bb2/iii+iv | screenshot/iii+iv

Q6

Relevant Topics: Representation of Curves | Tangent Vector to Parametric Curve

Resources: bb2 | screenshot

Q7

Relevant Topics: Derivatives of Vector Functions

Resources: screenshot | video comments

Q8

Relevant Topics: Orthonormal Bases | Q7

Resources: bb2 | screenshot

Q9

Relevant Topics: Position, Velocity and Acceleration

Resources: geogebra | bb2 | screenshot

Q10

Relevant Topics: Q7 | Position, Velocity and Acceleration | Vector Cross Product

Resources: bb2 | screenshot

Q11

Relevant Topics: Position, Velocity and Acceleration | Q7

Q12

Relevant Topics: Representation of Surfaces

Q13

Relevant Topics: Section of Surfaces by Coordinate Planes | Section of Surfaces by Arbitrary Plane | Rotation

Resources: bb2/iii | bb2/i | geogebra | screenshot/i | screenshot/iii

Q14

Relevant Topics: Section of Surfaces by Coordinate Planes

Q15

Relevant Topics: Section of Surfaces by Coordinate Planes | Hyperbola

Resources: bb2 | screenshot

Q16

Relevant Topics:

Q17

Relevant Topics: Section of Surfaces by Coordinate Planes | Hyperbola | Ellipse | Circle

Q18

Relevant Topics: Section of Surfaces by Coordinate Planes | Projections on Coordinate Planes

Resources: geogebra | bb2/i | screenshot/i | bb2/ii | screenshot/ii | bb2/iii | screenshot/iii | video comments

Q19

Relevant Topics: Cylinder | Ellipse | Projections on Coordinate Planes | Rotation

Resources: video comments | geogebra | bb2/i+ii | screenshot/i+ii

Q20

Relevant Topics: Section of Surfaces by Coordinate Planes | Projections on Coordinate Planes

Resources: bb2/i | screenshot/i | bb2/iii | screenshot/iii | geogebra

Q21

Relevant Topics: Polar Coordinates | Projections on Coordinate Planes | Circle

Resources: bb2/i | screenshot/i

Q22

Relevant Topics: Representation of Surfaces

Q26

Relevant Topics: Open and Closed Subsets | Union, Intersection of Open, Closed Subsets

Resources: bb2/1+2 | screenshot/1+2 | part 8, Lecture notes, p.19 | video comments, part 8

Q27

Relevant Topics: Open and Closed Subsets

Resources: bb2 | screenshot

Q28

Relevant Topics: Open and Closed Subsets | Cauchy-Schwartz Inequality

Resources: bb2 | screenshot

Q29

Relevant Topics: Open and Closed Subsets | Union, Intersection of Open, Closed Subsets

Q30

Relevant Topics: Open and Closed Subsets

Resources: bb2 | screenshot | video comments

Q31

Relevant Topics: Limit of Function of Several Variables | Limits in Rn | Limits in Rn, II | Limits in Rn, III

Q32

Relevant Topics: Limit of Function of Several Variables | Limit of Vector Function | Limits in Rn | Limits in Rn, II | Limits in Rn, III

Resources: Lecture Notes, pp.25--27 | bb2 | screenshot

Q33

Relevant Topics: Limit of Function of Several Variables | Pinching Principle | Limits in Rn | Limits in Rn, II | Limits in Rn, III

Q34

Relevant Topics: Limit of Function of Several Variables | Limit of Vector Function | Limits in Rn | Limits in Rn, II | Limits in Rn, III

Q35

Relevant Topics: Limit of Function of Several Variables | Limit of Vector Function | Limits in Rn | Limits in Rn, II | Limits in Rn, III

Q36

Relevant Topics: Limit of Function of Several Variables | Pinching Principle | Limits in Rn | Limits in Rn, II | Limits in Rn, III

Resources: bb2 | screenshot

Q37

Relevant Topics: Limit of Function of Several Variables | Pinching Principle | Limits in Rn | Limits in Rn, II | Limits in Rn, III

Q38

Relevant Topics: Limit of Function of Several Variables | Pinching Principle | Limits in Rn | Limits in Rn, II | Limits in Rn, III

Resources: bb2 | screenshot

Q39

Relevant Topics: Limits and Taylor Expansions

Resources: bb2 | screenshot

Q40

Relevant Topics: Limits and Taylor Expansions

Resources: bb2 | screenshot

Q41

Relevant Topics: Limits and Taylor Expansions

Q42

Relevant Topics: Limits and Taylor Expansions

Resources: bb2 | screenshot

Q44

Relevant Topics: Continuous Functions | Limit of Function of Several Variables | Elementary Functions

Q45

Relevant Topics: Continuous Functions | Limit of Function of Several Variables | Elementary Functions

Q46

Relevant Topics: Continuous Functions | Limit of Function of Several Variables

Q47

Relevant Topics: Preimage of Subset | Open and Closed Subsets

Q48

Relevant Topics: Preimage of Subset | Open and Closed Subsets | Continuous Functions

Q49

Relevant Topics: Preimage of Subset

Q50

Relevant Topics: Preimage of Subset | Open and Closed Subsets | Continuous Functions | Preimage of Subset, II | Limits in Rn, IV

Resources: bb2 | screenshot

Q51

Relevant Topics: Preimage of Subset | 50 | Open and Closed Subsets | Union, Intersection of Open, Closed Subsets

Resources: bb2/i-iii | screenshot/i-iii | bb2/iv | screenshot/iv

Q52

Relevant Topics: Preimage of Subset | 50 | Open and Closed Subsets | Union, Intersection of Open, Closed Subsets

Resources: bb2 | screenshot

Q53

Relevant Topics: Preimage of Subset | 50 | Open and Closed Subsets | Union, Intersection of Open, Closed Subsets

Q54

Relevant Topics: Open and Closed Subsets | Open and Closed Subsets

Resources: bb2 | screenshot

Q55

Relevant Topics: Open and Closed Subsets | Open and Closed Subsets | Union, Intersection of Open, Closed Subsets | Preimage of Boundary

Q56

Relevant Topics: Bounded subset

Q57

Relevant Topics: Path-connected | Polar Map | Parellelepiped

Resources: bb2 | screenshot

Q58

Relevant Topics: Continuous map and compact and path-connected subsets

Resources: Lecture notes, pp.46--47

Q59

Relevant Topics: Continuous map and compact and path-connected subsets

Resources: bb2 | screenshot | Lecture Notes, pp.50

Q60

Relevant Topics: Continuous map and compact and path-connected subsets

Resources: bb2 | screenshot

Q61

Relevant Topics: Map of Boundary

Resources: bb2/ii | screenshot/ii | video comments/ii | bb2/iii | screenshot/iii

Q62

Relevant Topics: Continuous Functions

Resources: Lecture Notes, pp.36

Q63

Relevant Topics: Limits in Rn, II

Q64

Relevant Topics: Bounded subset | Compact subset | Path-connected subset

Resources: bb2/i | screenshot/i | bb2/iii | screenshot/iii | bb2/iv/path-conncted | screenshot/iv/path-conncted | video comments/iv/path-connected | bb2/iv/closed | screenshot/iv/closed | video comments/iv/closed

Q65

Relevant Topics: Continuous map and compact and path-connected subsets

Q66

Relevant Topics: Partial derivatives

Q67

Relevant Topics: Partial derivatives | Clairaut's Theorem

Resources: Lecture Notes, pp.13--14 | video comments

Q68

Relevant Topics: Partial derivatives

Q69

Relevant Topics: Partial derivatives

Q70

Relevant Topics: Partial derivatives | Jacobian matrix

Q71

Relevant Topics: Jacobian matrix

Resources: bb2 | screenshot

Q72

Relevant Topics: Differentiability

Resources: bb2 | screenshot

Q73

Relevant Topics: Differentiability

Resources: bb2 | screenshot

Q74

Relevant Topics: Differentiability

Resources: screenshot | video comments

Q75

Relevant Topics: Differentiability

Resources: bb2 | screenshot | video comments

Q76

Relevant Topics: Differentiability

Resources: screenshot | video comments

Q77

Relevant Topics: Differentiability

Q78

Relevant Topics: Differentiability

Resources: screenshot | video comments

Q79

Relevant Topics: Affine approximation

Q80

Relevant Topics: Affine approximation

Resources: bb2 | screenshot

Q81

Relevant Topics: Affine approximation

Q82

Relevant Topics: Chain rule

Q83

Relevant Topics: Chain rule

Q84

Relevant Topics: Chain rule

Q85

Relevant Topics: Chain rule

Q86

Relevant Topics: Directional derivative

Resources: bb2 | screenshot

Q87

Relevant Topics: Directional derivative

Q88

Relevant Topics: Directional derivative

Q89

Relevant Topics: Directional derivative

Q90

Relevant Topics: Directional derivative

Q91

Relevant Topics: Directional derivative | Directional Derivative and Gradient

Q92

Relevant Topics: Directional derivative | Directional Derivative and Gradient

Resources: bb2 | screenshot

Q93

Relevant Topics: Directional derivative | Directional Derivative and Gradient

Resources: bb2 | screenshot

Q94

Relevant Topics: Directional derivative

Q95

Relevant Topics: Directional derivative

Resources: screenshot | video comments

Q96

Relevant Topics: Directional derivative | Trace of Matrix | Orthogonal Matrix

Resources: bb2 | screenshot

Q97

Relevant Topics: Chain rule

Resources: bb2 | screenshot

Q98

Relevant Topics: Chain rule

Resources: screenshot | video comments | screenshot/alt | video comments/alt

Q99

Relevant Topics: Chain rule

Resources: screenshot | video comments

Q100

Relevant Topics: Chain rule

Resources: screenshot | video comments

Q101

Relevant Topics: Chain rule

Q102

Relevant Topics: Chain rule

Q103

Relevant Topics: Chain rule

Resources: bb2 | screenshot

Q104

Relevant Topics: Chain rule

Resources: bb2 | screenshot

Q105

Relevant Topics: Chain rule

Resources: bb2 | screenshot

Q106

Relevant Topics: Taylor Series

Q107

Relevant Topics: Taylor Series

Q108

Relevant Topics: Taylor Series

Resources: bb2 | screenshot

Q109

Relevant Topics: Tangent Plane via Gradient

Resources: bb2 | screenshot

Q110

Relevant Topics: Tangent Plane via Gradient

Resources: bb2 | screenshot

Q111

Relevant Topics: Tangent Plane via Gradient

Q112

Relevant Topics: Tangent Plane via Gradient

Resources: bb2 | screenshot

Q113

Relevant Topics: Tangent Plane via Gradient

Resources: bb2 | screenshot

Q114

Relevant Topics: Tangent Plane via Gradient

Resources: bb2 | screenshot

Q115

Relevant Topics: Absolute Max/Min

Q116

Relevant Topics: Absolute Max/Min

Q117

Relevant Topics: Absolute Max/Min

Q118

Relevant Topics: Local Max/Min/Saddle Point

Resources: bb2 | screenshot | video comments

Q119

Relevant Topics: Local Max/Min/Saddle Point

Q120

Relevant Topics: Constrained Max/Min, Lagrange Multipliers

Resources: bb2 | screenshot

Q121

Relevant Topics: Constrained Max/Min, Lagrange Multipliers

Resources: bb2 | screenshot

Q122

Relevant Topics: Constrained Max/Min, Lagrange Multipliers

Q123

Relevant Topics: Constrained Max/Min, Lagrange Multipliers

Resources: bb2 | screenshot

Q124

Relevant Topics: Constrained Max/Min, Lagrange Multipliers

Q125

Relevant Topics: Constrained Max/Min, Lagrange Multipliers

Resources: bb2 | screenshot

Q126

Relevant Topics: Constrained Max/Min, Lagrange Multipliers

Q127

Relevant Topics: Constrained Max/Min, Lagrange Multipliers

Q128

Relevant Topics: Constrained Max/Min, Lagrange Multipliers

Q129

Relevant Topics: Constrained Max/Min, Lagrange Multipliers

Resources: bb2 | screenshot

Q130

Relevant Topics: Implicit Function Theorem | Inverse Function Theorem

Q131

Relevant Topics: Implicit Function Theorem | Inverse Function Theorem

Resources: bb2 | screenshot

Q132

Relevant Topics: Implicit Function Theorem | Inverse Function Theorem

Resources: bb2 | screenshot

Q133

Relevant Topics: Implicit Function Theorem | Inverse Function Theorem

Resources: bb2/i | screenshot/i

Q134

Relevant Topics: Implicit Function Theorem | Inverse Function Theorem

Resources: bb2 | screenshot

Q135

Relevant Topics: Implicit Function Theorem | Inverse Function Theorem

Resources: bb2 | screenshot

Q136

Relevant Topics: Implicit Function Theorem | Inverse Function Theorem

Resources: bb2 | screenshot

Q137

Relevant Topics: Implicit Function Theorem | Inverse Function Theorem

Resources: bb2 | screenshot

Q138

Relevant Topics: Implicit Function Theorem | Inverse Function Theorem

Resources: bb2/i | screenshot/i

Q139

Relevant Topics: Riemann Integration

Resources: bb2 | screenshot

Q140

Relevant Topics: Fubini's Theorem in general

Resources: bb2 | screenshot

Q141

Relevant Topics: Fubini's Theorem in general

Resources: bb2/i | screenshot/i | bb2/iv | screenshot/iv

Q142

Relevant Topics: Fubini's Theorem in general

Q143

Relevant Topics: Fubini's Theorem in general

Resources: bb2 | screenshot

Q144

Relevant Topics: Fubini's Theorem in general

Resources: bb2 | screenshot

Q145

Relevant Topics: Leibniz' rule

Q146

Relevant Topics: Leibniz' rule

Resources: bb2 | screenshot

Q147

Relevant Topics: Leibniz' rule

Q148

Relevant Topics: Change of Variables

Resources: bb2 | screenshot

Q149

Relevant Topics: Change of Variables

Resources: bb2 | screenshot

Q150

Relevant Topics: Change of Variables

Resources: bb2 | screenshot

Q151

Relevant Topics: Change of Variables | Cylindrical and Spherical Coordinates

Resources: bb2 | screenshot

Q152

Relevant Topics: Cylindrical and Spherical Coordinates

Resources: bb2 | screenshot

Q153

Relevant Topics: Cylindrical and Spherical Coordinates

Resources: bb2 | screenshot

Q154

Relevant Topics: Cylindrical and Spherical Coordinates

Resources: bb2 | screenshot

Q155

Relevant Topics: Cylindrical and Spherical Coordinates

Resources: bb2 | screenshot

Q156

Relevant Topics: Cylindrical and Spherical Coordinates

Resources: bb2 | screenshot

Q157

Relevant Topics: Cylindrical and Spherical Coordinates

Resources: bb2 | screenshot

Q158

Relevant Topics: Cylindrical and Spherical Coordinates

Q159

Relevant Topics: Change of Variables

Resources: bb2 | screenshot

Q160

Relevant Topics: Change of Variables

Q161

Relevant Topics: Change of Variables

Resources: bb2 | screenshot

Q162

Relevant Topics: Change of Variables

Resources: bb2 | screenshot

Topics List (alphabetical)

Go to Questions

Absolute Max/Min

Resources: Lecture Notes, pp 63--67

What to Know:

  1. two stage method of finding absolute max/min over domain
  2. method of finding max/min on the boundary of domain

Affine approximation

Resources: Lecture Notes, pp 25--26 | in-class screen | bb2

What to Know:

  1. formula for affine approximation
  2. the role of Jacobian matrix
  3. difference from tangent plane
  4. existence of affine approximation is synonym to differentiability (see differentiability topic)

Algebra of Continuous Functions

Resources: Lecture Notes, pp 36

What to Know:

    Bounded subset

    Resources: Lecture Notes, pp 37--38 | in-class screen | bb2

    What to Know:

    1. definition of bounded subset
    2. proof that closed interval is bounded
    3. proof that an open ball (higher dimensions) is bounded
    4. proof that rectangle is bounded
    5. example of unbounded subset (with proof)

    Cauchy-Schwartz Inequality

    Resources: wiki

    What to Know:

      Chain rule

      Resources: Lecture Notes, pp 27--34

      What to Know:

      1. chain rule in matrix form
      2. chain rule in scalar form

      Change of Variables

      Resources: Lecture Notes, pp 39--51

      What to Know:

        Circle

        Resources: in-class screen | bb2 | wiki

        What to Know:

        1. implicit equation (interpretation of coefficients)
        2. parametric representation
        3. completion of squares

        Clairaut's Theorem

        Resources: Lecture Notes, pp 13--14

        What to Know:

        1. when mixed second order partial derivatives are the same
        2. example where second order mixed partial are different (Q67)

        Compact subset

        Resources: Lecture Notes, pp 43

        What to Know:

        1. definition of compact subset
        2. that compact subsets are mapped to compact subsets by continuous maps

        Constrained Max/Min, Lagrange Multipliers,

        Resources: Lecture Notes, pp 91--105

        What to Know:

        1. method of finding extremum under one constraint

        Continuous Functions

        Resources: Lecture Notes, pp 31--36 | in-class screen | bb2 | wiki

        What to Know:

        1. definition of continuous function at point
        2. definition of continuous function over domain
        3. how to find a limit of a function which is continuous at the point where the limit is needed
        4. example of discontinuous function

        Continuous map and compact and path-connected subsets

        Resources: Lecture Notes, pp 46--47 | in-class screen | bb2

        What to Know:

        1. compact, path-connected subsets are mapped to compact, path-connected subsets, respectively

        Cylindrical and Spherical Coordinates

        Resources: Lecture Notes, pp 52--53 | in-class screen | bb2

        What to Know:

          Differentiability

          Resources: Lecture Notes, pp 2--7, 17--23 | in-class screen | bb2

          What to Know:

          1. definition of differentiability
          2. result guaranteeing differentiability of C^1 functions
          3. example of non-differentiable function (e.g., Q74, Q75)
          4. direct argument of differentiability (e.g., Q73)

          Directional derivative

          Resources: Lecture Notes, pp 35--39

          What to Know:

          1. formula for the directional derivative via Gradient
          2. definition of directional derivative

          Directional Derivative and Gradient

          Resources: in-class screen | bb2 | video comments

          What to Know:

          1. formula for the directional derivative via Gradient
          2. when gradient formula works and when it does not
          3. direction of fastest change for a function
          4. direction of no-change for a function

          Elementary Functions

          Resources: in-class screen | bb2 | wiki

          What to Know:

          1. what is an elementary function
          2. continuity, differentiability of elementary function

          Ellipse

          Resources: in-class screen | bb2 | geogebra | wiki

          What to Know:

          1. implicit equation (interpretation of coefficients, intercepts)
          2. parametric representation
          3. completion of squares

          Fubini's Theorem in general

          Resources: Lecture Notes, pp 13--17,19--26

          What to Know:

          1. see 'Fubini's Theorem on Rectangles
          2. definition of x-simple and y-simple domains

          Fubini's Theorem on Rectangles

          Resources: Lecture Notes, pp 9--12

          What to Know:

          1. difference between double/triple integral and repeated integration
          2. ability to convert double integral to repeated one and vise versa

          Geometrical Interpretation of Integral

          Resources: Lecture Notes, pp 18

          What to Know:

          1. geometrical meaning of double integral when the function is constant
          2. geometrical meaning of double integral with general function
          3. geometrical meaning of triple integral when the function is constant

          Gradient

          Resources: Lecture Notes, pp 24

          What to Know:

          1. see 'Tangent Plane' for the role of gradient in tangent plane equation (graph of function)
          2. see 'Tangent plane via Gradient' the role of gradient in tangent plane equation (surface given implicitly)
          3. the role of gradient in tangent plane equation (curve given implicitly)
          4. see 'Directional derivative' for the role of gradient in Directional derivative
          5. see 'Directional Derivative via Gradient' for the direction of fastest and no change

          Hyperbola

          Resources: in-class screen | bb2 | geogebra | wiki

          What to Know:

          1. implicit equation (interpretation of coefficients, intercepts, asymptotes)
          2. parametric representation
          3. completion of squares

          Implicit Function Theorem

          Resources: Lecture Notes, pp 115--127

          What to Know:

          1. test which ensures that implicit function exists
          2. how to find derivative/Jacobian of implicit function

          Interior and Boundary Points

          Resources: Lecture Notes, pp 17 | wiki

          What to Know:

          1. definition of boundary point
          2. definition of interior point
          3. ability to proof, using definition, that a point interior
          4. ability to proof, using definition, that a point is boundary

          Inverse Function Theorem,

          Resources: Lecture Notes, pp 106--114

          What to Know:

          1. test which ensures that inverse function exists
          2. how to find derivative/Jacobian of inverse

          Jacobian matrix

          Resources: Lecture Notes, pp 15--16

          What to Know:

          1. how to find Jacobian Matrix, numerical and general
          2. the role of Jacobian matrix in Affine Approximation
          3. the role in definition of differentiability

          l'Hopital's Rule

          Resources: in-class screen | bb2 | wiki

          What to Know:

          1. why not to use it

          Leibniz's rule

          Resources: Lecture Notes, pp 28--38 | wiki

          What to Know:

          1. how to use Leibniz's rule

          Limit of Function of Several Variables

          Resources: Lecture Notes, pp 23--30 | wiki

          What to Know:

          1. see Limits in Rn I, II, III, IV

          Limit of Vector Function

          Resources: Lecture Notes, pp 7--8, 10--12 | wiki | Lecture Notes, p.9

          What to Know:

          1. see Limits in Rn I, II, III, IV

          Limits and Taylor Expansions

          Resources: wiki

          What to Know:

          1. understand the method of Taylor expansions of elementary functions for finding limits (e.g., Q39, Q42)

          Limits in Rn

          Resources: in-class screen | bb2

          What to Know:

          1. definition of limit of vector function, one variable
          2. definition of limit of vector sequence
          3. definition of limit of function of several variables

          Limits in Rn, II

          Resources: in-class screen | bb2

          What to Know:

          1. Limit of vector function and vector sequence via components
          2. Show that a limit does not exist, for function of several variables
          3. Examples of limit which does not exist

          Limits in Rn, III

          Resources: in-class screen | bb2

          What to Know:

          1. direct argument showing that limit exists, numerical function of two variables

          Limits in Rn, IV

          Resources: in-class screen | bb2

          What to Know:

          1. general statement of definition of limit, the case of vector map of several variables
          2. ability to see the definitions on slide 'Limits in Rn' as special cases of this topic

          Lines

          Resources: in-class screen | bb2 | Normal Demo | Parametrisation Demo | wiki

          What to Know:

          1. lines through parametrisation (any dimension)
          2. lines through Cartesian equation (2d)
          3. interpretation of coefficients in Cartesian equation (2d)
          4. parametrisation of straight line segment

          Local Max/Min/Saddle Point

          Resources: Lecture Notes, pp 61, 68--90

          What to Know:

          1. Q118
          2. Efficiency trick

          Map of Boundary

          Resources: in-class screen | bb2 | video comments

          What to Know:

            Mass and Centre of Mass

            Resources: Lecture Notes, pp 54--58

            What to Know:

              One-point Set Boundary

              Resources: in-class screen | bb2

              What to Know:

                Open and Closed Subsets

                Resources: Lecture Notes, pp 17--19, 21 | in-class screen | bb2 | wiki/open sets | wiki/closed sets

                What to Know:

                1. open interval is open (proof, Q26)
                2. closed interval is closed (proof, Q26)
                3. open ball in higher dimensions is open (proof)

                Orthonormal Bases

                Resources: wiki

                What to Know:

                1. definition of orthonormal basis and orthogonal basis
                2. how to find representation of a vector with respect to orthonormal basis

                Parallelepiped

                Resources: in-class screen | bb2

                What to Know:

                1. parallelepiped is path-connected (proof in one and two dimensions)

                Partial derivatives

                Resources: Lecture Notes, pp 8--14

                What to Know:

                1. compute partial derivatives via formulae of differentiation of elementary functions
                2. compute partial derivatives via definition
                3. definition of partial derivatives
                4. connection with directional derivative

                Path-connected subset

                Resources: Lecture Notes, pp 44--45 | in-class screen | bb2

                What to Know:

                1. definition of path-connected subset
                2. proof that a closed interval and closed rectangle are path-connected
                3. example of not path-connected subset (with proof)

                Pinching Principle

                Resources: in-class screen | bb2 | wiki

                What to Know:

                1. statement of the principle
                2. example of usage

                Polar Map

                Resources: in-class screen | bb2 | geogebra

                What to Know:

                1. definition of polar map
                2. images of rectangles under polar map
                3. formula for inverse of polar map
                4. seam of discontinuity of inverse polar map

                Polar Map, Global Inverse

                Resources: in-class screen | bb2

                What to Know:

                1. one example of polar inverse
                2. the seam of discontinuity of polar inverse
                3. proof that global inverse does not exist

                Position, Velocity and Acceleration

                Resources: in-class screen | bb2 | geogebra

                What to Know:

                1. how to find velocity of particle given by a position vector
                2. how to find acceleration of particle given by a position vector
                3. difference between speed and velocity

                Preimage of Boundary

                Resources: in-class screen | bb2 | video comments

                What to Know:

                  Preimage of Subset

                  Resources: in-class screen | bb2 | wiki

                  What to Know:

                  1. the statement of the preimage theorem
                  2. ability to use the preimage theorem (e.g., Q28)

                  Preimage of Subset, II

                  Resources: in-class screen | bb2

                  What to Know:

                  1. proof of preimage theorem

                  Representation of Curves

                  Resources: Lecture Notes, pp 3-7,18 | in-class screen | bb2 | Graph Demo | Parametrisation Demo (2d) | Parametrisation Demo (3d)

                  What to Know:

                  1. implicit representation (examples)
                  2. graph of functions (examples)
                  3. parametric representation (examples)

                  Representation of Surfaces

                  Resources: Lecture Notes, pp 15--17,19 | in-class screen | bb2 | Graph Demo | Parametrisation Demo

                  What to Know:

                  1. implicit representation (examples)
                  2. graph of functions (examples)
                  3. parametric representation (examples)

                  Riemann Integration

                  Resources: Lecture Notes, pp 2--8

                  What to Know:

                  1. definition of Riemann integration
                  2. how to find integral by definition (Q139)

                  Section of Surfaces by Coordinate Planes

                  Resources: Lecture Notes, pp 20--29 | in-class screen | bb2 | Paraboloid Demo | Cylinder Demo

                  What to Know:

                  1. how to construct and identify

                  Tangent Line

                  Resources: in-class screen | bb2

                  What to Know:

                  1. formula for tangent line to graph
                  2. existence of tangent line is synonym to differentiability and existence of derivative (one dimensional case)

                  Tangent Plane

                  Resources: in-class screen | bb2

                  What to Know:

                  1. formula for finding tangent plane to graph
                  2. the role of Gradient

                  Tangent Plane via Gradient

                  Resources: Lecture Notes, pp 40--49 | in-class screen | bb2

                  What to Know:

                  1. method of finding the tangent to a curve/surface given implicitly
                  2. gradient perpendicular to curve/surface given implicitly

                  Tangent Plane, II

                  Resources: in-class screen | bb2

                  What to Know:

                  1. existence of tangent plane is synonym for differentiability (see differentiability topic)
                  2. direct argument showing that tangent plane exists

                  Tangent Vector to Parametric Curve

                  Resources: Lecture Notes, pp 13--14 | in-class screen | bb2 | geogebra

                  What to Know:

                  1. how to find tangent vector to curve given parametrically

                  Taylor Series

                  Resources: Lecture Notes, pp 50--60

                  What to Know:

                  1. method of finding Taylor polynomial upto square term using partial derivatives
                  2. method of finding Taylor polynomial upto square term using know Taylor expansions of elementary functions
                  3. connection between Gradient and Taylor polynomial
                  4. connection between Hessian and Taylor polynomial

                  Union, Intersection of Open, Closed Subsets

                  Resources: in-class screen | bb2

                  What to Know:

                  1. union of open subsets is open (countable allowed)
                  2. intersection of closed subsets is closed (countable allowed)
                  3. finite union of closed subsets is closed
                  4. finite intersection of open subsets is open

                  Union, Intersection of Open, Closed Subsets, II

                  Resources: in-class screen | bb2

                  What to Know:

                  1. example showing that countable union of closed subsets is not necessarily closed
                  2. example showing that countable intersection of open subsets is not necessarily open

                  Vector Cross Product

                  Resources: wiki

                  What to Know:

                  1. algebraic formula via components for vector cross product
                  2. geometrical properties of vector cross product
                  3. formula for magnitude of vector cross product

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