In this page you will find a collection of old exams and solutions. It will be available about 10 days before the exam date.
· Quiz 1 S2006 ( PDF, solutions) NB: For problem 3, please use this solution: http://web.mit.edu/8.022/www/quiz/cap_solutn.pdf
Readings marked with an asterisk (*) are required in the sense that they contain content necessary for the online quiz (available only via WebCT).
Introduction
Practice Quiz (not graded)
Interactive Concept-Visualization Tools
Parametrization of a Line
Matrices and determinants
The cross product
The component formula for the cross product
The scalar triple product
The relationship between determinants and area or volume
Parametrization of a line examples
Cross product examples
Triple scalar product example
Forming planes*
Forming Planes quiz*
A line, a point, or a plane?
Distance from point to plane
Forming plane examples
Distance from point to plane example
Intersecting planes example
Multiplying matrices and vectors*
Matrix-vector multiplication quiz*
Dot product in matrix notation
Matrices and linear functions
Examples of n-dimensional vectors
Matrix and vector multiplication examples
Level sets*
Level sets quiz*
Function notation
Translation, rescaling, and reflection
Quadric surfaces
Level set examples
Introduction to partial derivatives*
Partial derivatives intro quiz*
Partial derivatives by limit definition
Partial derivative examples
Differentiability in higher dimensions*
Differentiability quiz*
More on Differentiability
Differentiability examples
The chain rule*
Chain rule quiz*
Chain rule examples
Parametrized curves and their derivatives
Tangent lines to parametrized curves
Tangent line to parametrized curve examples
The directional derivative and the gradient*
Gradient and directional derivative quiz*
Directional derivative and gradient examples
Double integrals*
Double integral quiz*
Double integrals as iterated integrals
Double integrals examples
Gradient and directional derivative review quiz*
Initial readings and quizzes survey
Double integrals as volume or area
Change order of integration examples
Area of region example
Changing order of integration is not always feasible
Triple Integrals*
Triple integral quiz*
Triple Integral examples
The length of a path*
Length of path quiz*
Physical interpretation of a parametrization and its derivative
Length of a path examples
The idea of divergence and curl*
Divergence and curl quiz*
Vector fields
The components of the curl
More details on the components of the curl
Divergence and curl example
Path integral of a scalar-valued function*
Path integral quiz*
Path integral examples
Line integral of a vector field
Alternate notation for line integrals
Line integral examples
The idea behind Green's theorem*
Green's theorem quiz*
Other ways of writing Green's theorem
Using Green's theorem to find area
Sketch of proof for circulation per unit area
Green's theorem examples
Path-independent or conservative vector fields*
Path-independence quiz*
An example of a conservative vector field
Path-independence implies no circulation
Understanding the conditions for path-independence
Finding the potential function for path-independent vector fields
Path-independence example in three dimensions
Change of variables in double integrals*
Change of variables 2D quiz*
Change of variables examples
Change of variables in triple integrals*
Change of variables 3D quiz*
Spherical Coordinates
Change of variables examples
Parametrized surfaces*
Parametrized surface quiz*
Parametrized surface examples
Surface area of parametrized surfaces*
Surface area quiz*
Surface area calculation
Normal vector of parametrized surfaces
Surface area example
Surface Integrals*
Surface integral quiz*
Orienting surfaces
A surface that is not orientable
Parametrization of a Plane
Plane Parametrization Example
Surface integral examples
The idea of Stokes' theorem*
Stokes' theorem quiz*
Stokes theorem examples
The idea of the divergence theorem*
Divergence theorem quiz*
Divergence theorem examples
Taylor's Theorem*
Taylor theorem quiz*
Taylor polynomial example
Local Extrema*
Extrema quiz*
Extrema examples
Review quiz*
The integrals
The fundamental theorems
Length, area, and volume factors
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| The views and opinions expressed in this page are strictly those of the page author. The contents of this page have not been reviewed or approved by the University of Minnesota. nykamp@math.umn.edu |
