Difference between revisions of "MAT2214"

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<div style="text-align: center;">4.2</div>
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<div style="text-align: center;">14.5</div>
  
 
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[[Direction Fields and Numerical Methods]]
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[[Directional Derivatives and Gradient Vectors]]
  
 
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* [[Linear Equations|Slope of a Line]]  <!-- Not Directly Mentioned (recommend 1073-Mod.R -->
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* [[Trigonometric Functions]]  <!-- 1093-2.2 -->
* [[Defining the Derivative|Equation of the tangent line]] <!-- 1214-3.1 -->
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* [[The Chain Rule for Functions of more than One Variable]]  <!-- 2214-14.4 -->
* [[Derivatives as Rates of Change|Leibnitz notation of the derivative]] <!-- 1214-3.4 -->
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* [[Vectors|Unit Vectors]] <!-- 2214-12.2 -->  <!-- 2233-1.1 -->
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* [[Partial Derivatives]] <!-- 1214-14.3 -->
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* [[The Dot Product]] <!-- 2214-12.3 -->
  
 
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* Sketch the direction field of a first-order ODE(Ordinary Differential Equation) by hand
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* Direction Derivatives in the plane
* Using direction field, find equilibria of an autonomous ODE.
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* Gradients
* Determine the stability of equilibria using a phase line diagram.
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* Properties of directional derivatives
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* Tangents to level curves
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* Directional derivatives for functions of three variables
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* The chain rule for paths
  
  
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<div style="text-align: center;">4.3</div>
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<div style="text-align: center;">14.6</div>
  
 
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[[Separable Equations]]
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[[Tangent Planes and Differentials]]
  
 
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* [[Trigonometric Integrals]] <!-- 1224-3.2 -->
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* [[Linear Approximations and Differentials]] <!-- 1214-4.2 -->
* [[Integration by Substitution]] <!-- 1224-1.5 -->
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* [[Parametric Equations]] <!-- 1224-7.1 -->
* [[Integration by Parts]] <!-- 1224-3.1 -->
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* [[Partial Derivatives]] <!-- 1214-14.3 -->
* [[Linear Approximations and Differentials]] <!-- 1224-4.2 -->
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* [[Linear Equations|Slope of a Line]] <!-- 1073-Mod R -->
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* [[Cylinders and Quadratic Surfaces]] <!-- 2214-12.5 -->
  
 
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* Recognize and solve separable differential equations
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* Tangent Planes and Normal lines
* Develop and analyze elementary mathematical models.
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* The plane tangent to a surface
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* How to linearize a function of two variables
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* Differentials for functions of two variables
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* Linearization and differentials for functions of more than two variables
  
  

Revision as of 18:49, 1 July 2020

Topics List

Date Sections Topics Prerequisite Skills Student Learning Outcomes
Week 1
1.3

Three-Dimensional Coordinate Systems


  • Three-dimensional coordinate systems
  • Distance Formula in R3
  • Standard Equation for a Sphere


Week 1 and 2
12.2


Vectors

  • Vector Algebra Operations
  • Magnitude of a vector
  • Unit Vectors
  • Midpoint of a Line Segment
  • Angle between vectors
  • Definition of Dot product
  • Orthogonal vectors
  • Vector projection
Week 3
12.3

The Dot Product


  • Find the area of plane regions bounded by the graphs of functions.
Week 3/4
12.4

The Cross Product

  • Define the cross product
  • Properties of the cross product
  • Area of a parallelogram
  • Cross product as a determinant


Week 4
12.5


Cylinders and Quadratic Surfaces

  • Find equations for cylinders that are generated by rotating lines that are parallel to a plane
  • Understand basic quadratic surfaces
  • Understand general quadratic surfaces


Week 4/5
13.1


Curves in Space and Vector Functions

  • Vector functions
  • Limits of vector functions
  • Continuity of vector functions
  • Differentiation of vector functions
  • Differentiation rules for vector functions
  • Curves and paths in space


Week 5/6
13.2

Integrals of Vector Functions

  • Indefinite integrals of vector functions
  • Definite integrals of vector functions
  • Vector and parametric equations for ideal projectile motion


Week 6/7
13.3


Arc Length

  • Length of a curve in R3
  • General arc length formula
  • Arc length for parameterized curves
  • The Unit tangent vector


Week 6
13.4


Curvature and Normal Vectors

  • Curvature in R2
  • Formula for curvature
  • Definition of Principal unit normal
  • Curvature and normal vectors for higher dimensions.


Week 7
13.5


Tangential and Normal Components of Acceleration

  • Binormal Vectors
  • Definition of tangential and normal components of acceleration
  • Torsion


Week 7/8
14.1


Functions of Several Variables


  • Domain and range of multivariable functions
  • Functions with two variables
  • Bounded regions
  • Graphs and level curves of two variable functions
  • Functions of three variables
  • Level surfaces


Week 6/7
14.2


Limits and Continuity in Higher Dimensions

  • Limits of functions of two variables
  • Properties of limits of functions of two variables
  • Continuity for functions of two variables
  • Continuity of composition
  • Functions of more than two variables
  • Extreme values on closed and bounded sets


Week 7
14.3

Partial Derivatives

  • Partial derivatives for functions of two variables
  • Partial derivatives for functions of more than two variables
  • Partial derivatives and continuity
  • Second order partial derivatives
  • Mixed derivative theorem
  • Define differentiability for functions of two variables


Week 8
14.4

The Chain Rule for Functions of more than One Variable

  • Chain rule for functions of one independent variable and two intermediate variables.
  • Chain rule for functions of one independent variable and three intermediate variables.
  • Chain rule for functions of two independent variable and two intermediate variables.
  • Additional method for implicit differentiation.
  • The general chain rule
Week 8/9
14.5

Directional Derivatives and Gradient Vectors

  • Direction Derivatives in the plane
  • Gradients
  • Properties of directional derivatives
  • Tangents to level curves
  • Directional derivatives for functions of three variables
  • The chain rule for paths


Week 9
14.6

Tangent Planes and Differentials

  • Tangent Planes and Normal lines
  • The plane tangent to a surface
  • How to linearize a function of two variables
  • Differentials for functions of two variables
  • Linearization and differentials for functions of more than two variables


Week 10/11
4.4

Exponential Growth and Decay, The Logistic Equation

  • Solve the exponential growth/decay equations and the logistic equation.
  • Describe the differences between these two models for population growth.


Week 11
5.1

Sequences

  • Find the formula for the general term of a sequence.
  • Discuss the convergence or divergence of a sequence.
  • Find the limit of a convergent sequence.
  • Determine whether a sequence is monotone.


Week 11/12
5.2

Series

  • Define the convergence or divergence of an infinite series.
  • Find the sum of a geometric or telescoping series.
Week 12
5.3

The Divergence and Integral Tests

  • Determine the convergence or divergence of a series using the Divergence or Integral Tests.
  • Estimate the sum of a series using the Remainder Estimate Theorem.


Week 13/14
5.4

Comparison Tests

  • Determine the convergence or divergence of a series using the Direct or Limit Comparison Tests.


Week 14
5.5

Alternating Series

  • Determine the convergence or divergence of alternating series.
  • Estimate the sum of an alternating series.
  • Describe the difference between conditional and absolute convergence.


Week 15
5.6

Ratio and Root Tests


  • Determine the convergence or divergence of infinite series using the ratio and root tests..


Week 15/16
6.1

Power Series and Functions

  • Recognize a power series.
  • Find its interval and radius of convergence.
  • Represent certain functions as power series.


Week 16
6.2

Properties of Power Series

  • Differentiate and integrate power series term-by-term.
  • Recognize certain continuous functions as power series on their radius of convergence.



Week 17
6.3

Taylor and Maclaurin Series

  • Find the Taylor or Maclaurin series representation of a function.
  • Find the radius of convergence of a Taylor Series.
  • Estimate the remainder in a Taylor polynomial approximation.



Week 17/18
7.1

Parametric Equations

  • Sketch the graph of a parametric curve