Springfield Technical Community College
Division of Mathematics, Sciences, and Engineering Transfer
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I.
Mathematics
Calculus III
MATH 355
Spring, 2015
Richard Burns
burns@stcc.edu
http://faculty.stcc.edu/burns
17/412
TBA
Calculus with Early Transcendentals, Fifth Edition
Larson and Edwards, Brooks/Cole pub.
Course Descriptive Guide
A.
Organization of the Course
This is a lecture course where all the homework and Mathematica labs are
given online.
B.
Course Description
Topics include conic sections, parametric equations and polar coordinates,
vectors in the plane and space, solid analytic geometry, vector-valued functions
in the plane and in space, applications of vector-valued functions, limits and
continuity of functions of several variables, partial derivatives, chain rule,
directional derivatives, the gradient, tangent plane and normal lines, extrema of
functions of several variables and applications, multiple integration and
applications, cylindrical and spherical coordinates, and topics in vector calculus.
C.
Course Competencies and Objectives
Competency: To solve problems involving conic sections.
Objectives:
To recognize a conic section by its equation.
To find vertex, focus, directrix of parabola.
To find center, minor and major axes of ellipse.
To find center and orientation of hyperbola.
Competency: To solve problems involving parametric equations.
Objectives:
To state domain for a set of parametric equations.
To determine the orientation of the graph of a set of parametric equations
To sketch the graph of a set of parametric equations.
To eliminate the parameter from a set of parametric equations.
To find a set of parametric equations for a given equation.
To compute the first and second derivative for a set of parametric
equations.
Competency: To solve problems involving parametric equation.
Objectives:
To compute the slope of line tangent to graph of a set of parametric
equations.
To find an equation of the line tangent to the graph of a set of parametric
equations.
To compute the length of an arc determined by a set of parametric
equations.
Competency: To solve problems involving polar coordinates.
Objectives:
To plot points in polar coordinates.
To convert equations from rectangular form to polar form.
To convert equations from polar form to rectangular form.
To test for symmetry in polar coordinates.
To sketch the graph of a polar function.
To identify special polar forms and sketch their graph.
To compute all vertical and horizontal tangents to polar graphs.
To compute points of intersection of polar graphs.
To compute the area of regions bounded by one or more polar functions.
To compute the length of an arc determine by a polar function.
Competency: To solve problems involving vectors in the plane.
Objectives:
To state the definition of the special unit vectors.
To express a vector in component form using the special unit vectors.
To plot a vector given in rectangular form.
To plot a vector given in polar form.
To convert a vector from rectangular form to polar form.
To convert a vector from polar form to rectangular form.
To add two vectors mathematically and graphically.
To perform scalar multiplication mathematically and graphically.
To compute the length of a vector.
To compute the unit vector in the direction of a given vector.
To compute the scalar (dot) product of two vectors.
To state and prove the properties of the dot product.
To determine if two vectors are orthogonal.
To compute the projection of one vector onto another.
To evaluate and plot vector-valued functions.
To compute the limit of vector-valued functions.
To discuss the continuity of a vector-valued function.
To apply the definition of the derivative to a vector-valued function.
To explain graphically the meaning of the derivative of a vector-valued
function.
To compute the derivative of vector-valued functions.
To state and derive the properties of the derivative of a vector-valued
function.
To compute the vector tangent to the path of a vector-valued function.
To compute the integral of vector-valued functions.
To compute the velocity and acceleration of a particle given its position.
To compute the velocity and position of a particle given its acceleration.
To apply the calculus of vector-valued functions to physics and engineering
problems.
To define and compute the unit tangent vector for a given vector-valued
function.
To define and compute the unit normal vector for a given vector-valued
function..
To compute the tangential and normal components of an acceleration
vector for a given vector-valued function.
To compute curvature for a given vector-valued function.
To compute radius and center of the circle of curvature for a given vectorvalued function.
To plot the path, velocity vector, acceleration vector and its components,
and circle of curvature associated with a given vector-valued function at a given
point.
Competency: To solve problems involving vectors in space.
Objectives:
To derive and apply the distance formula.
To compute the direction cosines for a line or vector in space.
To decompose a vector in space into components.
To define and apply the cross product for two vectors.
To state and derive the properties of the cross product.
To compute the scalar triple product of three vectors.
To compute areas using the cross product.
To compute volumes using the scalar triple product.
To write the parametric forms for the line passing through two points.
To write the symmetric forms for the line passing through two points.
To write an equation of the line satisfying specified conditions.
To determine if two lines are parallel or skew.
To determine the point of intersection of two lines.
To write an equation of the plane passing through three non-collinear
points.
To write an equation of the plane satisfying specified conditions.
To compute the distance between a point and a plane.
To compute the distance between a point and a line.
To compute the distance between a line and a plane.
To compute the distance between two parallel lines.
To compute the distance between skew lines.
To identify and sketch cylindrical surfaces.
To identify and sketch certain quadratic surfaces.
To define and sketch space curves.
To evaluate and sketch certain vector-valued functions in space.
To compute the length of an arc of a space curve.
To compute the velocity and acceleration given the position function for a
particle in space.
To compute the tangent and normal vectors for a position function.
To compute the tangential and normal components of acceleration for a
particle in space.
To compute the curvature for a space curve.
Competency: To solve problems involving functions of several variables.
Objectives:
To define and evaluate a function of several variables.
To state the domain and range of a function of several variables.
To sketch the graph for a function of two variables.
To sketch the level curves associated with a function of two variables.
To sketch the level surfaces associated with a function of three variables.
To compute the limit of a function of two variables.
To discuss the continuity of a function of two variables.
To state and apply the definition of partial derivative.
To compute the partial derivatives of a function of several variables.
To compute higher order derivatives of functions of several variables.
To state and apply the definition of total differential.
To apply the concept of differentials to compute relative error.
To apply the chain rule to compute the derivative of a function.
To apply the concept of implicit differentiation to compute the derivatives of
functions.
To define and compute the directional derivative of a function.
To define and compute the gradient of a function.
To apply the concept of gradient to solve problems involving rates of
change of a function.
To find an equation for the plane tangent to a surface at a given point.
To find an equation for a line normal to a surface at a given point.
To compute the angle of inclination of a tangent plane.
To state and apply the Extreme Value Theorem for a function of two
variables.
Competency: To solve problems involving functions of several variables.
Objectives:
To compute and classify critical points for a function of two variables.
To compute the absolute extrema for a function of two variables.
To solve applied extrema problems.
Competency: To solve problems involving multiple integration.
Objectives:
To evaluate an iterated integral.
To find areas using double integration in two ways.
To switch the order of integration of a double integral.
To compute volumes using double integration.
To areas in polar coordinates using double integration.
To change an integral from rectangular form to polar form.
To change an integral from polar form to rectangular form.
To apply double integration to compute mass, center of mass, and moment
of inertia.
To apply triple integration to compute volume, mass, center of mass, and
moment of inertia.
To convert triple integrals between systems (rectangular, cylindrical, and
spherical) and evaluate.
Competency: To solve problems involving vector calculus.
Objectives:
To define and sketch a vector field.
To compute the curl of a vector field.
To determine if a vector field is conservative.
To compute the potential function for a conservative field.
To compute the divergence of a vector field.
To evaluate line integrals.
To apply line integrals to compute work.
To state and apply the Fundamental Theorem of Line Integrals.
To list and apply the fundamental properties of a conservative field.
To state and apply Green's Theorem.
D.
Evaluation Criteria
If at any time during the semester there is evidence of a student using
unauthorized material or submitting answers which are not of their own original
composition on examinations, then that student will be dismissed from the
course, receive a grade of F for the semester, and be referred to the Dean of
Students for further action.
There will be six one-hour examinations and a cumulative final examination,
which will be valued as two hour examinations. Homework is online on the
webassign.net site and is counted as 10% of the grade. There are six
Mathematica labs that are given on webassign.net that count as one hour exam.
Your grade will be calculated as follows:
g = .9*( (t1 + t2 + t3 + t4 + t5 + t6 + lab + 2*final)/9 ) + .1*hw
II.
Course Outline
Chapter 10: Plane Curves, Parametric Equations, and Polar Coordinates
10.1 Conic Sections
10.2 Plane Curves and Parametric Equations
10.3 Parametric Equations and Calculus
10.4 Polar Coordinates and Polar Graphs
10.5 Area and Arc Length in Polar Coordinates
Examination 1
Chapter 11: Vectors and the Geometry of Space
11.1 Vectors in the Plane
11.2 Space Coordinates and Vectors in Space
11.3 The Dot Product of Two Vectors
11.4 The Cross Product of Two Vectors in Space
11.5 Lines and Planes in Space
11.6 Surfaces in Space
11.7 Cylindrical and Spherical Coordinates
Examination 2
Chapter 12: Vector-valued Functions
12.1 Vector-valued Functions
12.2 Derivatives of Vectors in Space
12.3 Velocity and Acceleration
12.4 Tangent Vectors, Normal Vectors
12.5 Arc Length and Curvature
Examination 3
Chapter 13: Functions of Several Variables
13.1 Introduction to Functions of Several Variables
13.2 Limits and Continuity
13.3 Partial Derivatives
13.4 Differentials
13.5 Chain Rules for Functions of Several Variables
13.6 Directional Derivatives and gradients
13.7 Tangent Planes and Normal Lines
13.8 Extrema of Functions of Several Variables
13.9 Applications of Extrema of Functions of Two Variables
Examination 4
Chapter 14: Multiple Integration
14.1
14.2
14.3
14.4
14.6
14.7
Iterated Integrals and Area in the Plane
Double Integrals and Volume
Change of Variables: Polar Coordinates
Center of Mass and Moments of Inertia
Triple Integrals and Applications
Triple Integrals in Cylindrical and Spherical Coordinates
Examination 5
Chapter 15: Vector Analysis
15.1 Vector Fields
15.2 Line Integrals
15.3 Conservative Vector Fields and Independence of Path
15.4 Green's Theorem
Examination 6
Final Examination