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Goals:
General: Above the topical material covered
in this course, the student will enhance their mathematical problem
solving abilities:
- Technical reading/comprehension,
- Extracting useful information from the content of a problem's description,
- Pairing information contained in the problem's description to related
mathematical truths (definitions/theorems/corollaries),
- Producing a rough sketch outlining the steps leading to the solution.
Topical: The topics included in this
course will be covered to an extent that will meet the following goals:
- The ability to work with inverse functions using the definition
of one-to-one (1:1) in a setting in which the function is not necessarily
given explicitly:
- the calculus of inverse functions (finding the derivative of
an inverse function at a point),
- exponential functions, hyperbolic functions, their inverses,
domain of definition and derivatives,
- inverse trigonometric functions, their domain of definition
and derivatives,
- The ability to evaluate limits of expressions that are classified
as indeterminate as well as the ability to understand the proper use
and application of L'Hospital's rule,
- A command of the techniques necessary for finding antiderivatives
(that is, techniques for finding indefinite integrals). The student
will learn to read the integrand and choose the most appropriate integration
technique:
- Integration by parts,
- Trigonometric integration,
- Trigonometric substitution,
- Integration of rational functions by partial fractions,
- Integration using tables (if time permits),
- The ability to apply certain techniques for approximating definite
integrals and estimating their respective approximation errors:
- Trapezoidal Rule,
- Midpoint Rule,
- Simpson's Rule,
- An understanding of common techniques for identifying and, if convergent,
evaluating improper integrals,
- The ability to apply appropriate formulae for the determination
of the length of a curve (arc length),
- The ability to apply mathematical theory and formulae required to
determine the area of a surface that is constructed by rotating a
curve about either the horizontal or vertical axis,
- An ability to apply the above topical material to applications
in the fields of physics, engineering, economics, biology and probability
(as time permits),
- The ability to solve simple first order differential equations involving
exponential growth and decay,
- The ability to use techniques of parametric equations and polar
coordinates to transform curves (that do not satisfy the definition
of a function) into expressions that are functions by learning
- parametric and polar graphing techniques,
- to apply calculus to parametric and polar representations of
curves to determine,
- tangent lines,
- arc length,
- area under a curve,
- surface area (parametric equations only),
- The ability to use the definitions of conics (parabolas, ellipses,
and hyperbolas) and to derive their standard equations,
- The ability to analyse sequences and series to include establishing
the value of the limit if convergent,
- The ability to apply certain tests to determine the convergence
properties of series,
- Integral test (also used to estimate the limit of a convergent
series),
- Comparison test,
- Alternating series test,
- Ratio test (absolute convergence),
- Root test (absolute convergence),
- The ability to analyse power series to include the determination
of the radius and interval of convergence as well as the ability to
represent a given function as a power series,
- An understanding of the Taylor series, a generalization of the Maclaurin
series, to include the ability to present a function as either a Taylor
or Maclaurin series,
- The ability to determine the remainder term of the Taylor or Maclaurin
series,
- The ability to derive the Taylor or Maclaurin series for the main
function categories (such as trigonometric, rational, exponential,
logarithmic, etc.) as well as determining power series for products
of functions as a product of power series (here, a product includes
division),
- The understanding of the binomial series and one of its uses to
determine power series representations.
In this course, the student will demonstrate
an understanding of the topics outlined in the Goal section, `Topical'.
The student is cautioned not to memorize solution steps but rather
to learn general thinking procedures that can be applied to any problem
in a specific category. For example, to learn integration techniques
is simple, but to apply these techniques is considered by many students
difficult. One explanation is the following: When an integration technique
is introduced, it can be described as a recipe or procedure. However,
if exercises that cover many integration techniques are scrambled
together, and the student is not told which procedure to use, troubles
may surface. To avoid this, it is important to bring home the understanding
that to "think" about a strategy for solving a problem in
calculus (or, more generally, in mathematics), one must be able to
associate definitions/theorems/etc. (that might be useful) to the
information given in the problem description. The beginning point
for achieving this is through memorization. That is, to become acquainted
with the facts, it is helpful to first memorize them before expecting
to be able to use them freely. Later, if the details of the facts
are forgotten, they can be looked up - but what is not forgotten is
their existence and their usefulness. Below, there are many references
to "recite on demand", or to "memorize". These
are to encourage you, the student, to stay on track by making the
best use of your study time: First learn the facts (memorization),
then learn how to use them (working exercises). The more important
definitions/theorems (facts) may appear on quizzes/exams/etc. by asking
that they be stated in detail, but for the majority the memorization
is only a necessary step before understanding: To do well, you will
need to understand the topical material. Here, the objectives for
each topic will be given in reference to what defines "understanding":
- Inverse functions:
- Recite on demand all definitions/theorems pertaining to inverse
functions (e.g., one-one, domain/range, continuity theorem,
differentiability theorem),
- Read questions pertaining to inverse functions, write out
all information contained in the question to then be followed
by writing all information that might be helpful in answering
the question (i.e., listing all definitions/theorems, related/similar
examples, etc. that appear to be related to the question),
- To be able to recite on demand the general behavior of exponential
functions, their properties, graphs, derivatives, their inverses
(i.e., be able to identify the appropriate log function for
any exponential function) and the derivatives of the log functions,
- To be able to recite on demand the general behavior of hyperbolic
functions, their inverses, their derivatives (both for hyperbolic
as well as their inverses), to write from memory the hyperbolic
inverse functions in terms of logarithmic functions, their domain
of definition and range,
- For each inverse trigonometric function, to give from memory
their domain of definition and derivative,
- For each of the above, to demonstrate an ability to apply
algebra, and any differentiation technique as applied to inverse
functions,
- For each of the above, to demonstrate an ability to apply
part 1 and part 2 of the Fundamental Theorem of Calculus as
they pertain to inverse functions,
- To demonstrate an ability to apply the above in solving word
problems.
- L'Hospital's Rule:
- To recite on demand L'Hospital's Rule,
- To recite on demand all indeterminate forms of a limit (0/0,
infinity/infinity, mixed product, indeterminate differences,
indeterminate powers,
- To demonstrate an ability to read a question and determine
the most appropriate technique for applying L'Hospital's Rule
or any other technique covered in Calculus I for analyzing limits.
- Antiderivatives (that is, techniques for finding indefinite integrals),
- The student will demonstrate a working knowledge of the procedures
for integration (by working knowledge, it is meant that the
student will be able to describe in procedural form how each
technique is applied and be able to list the most appropriate
structures to use for a given integration technique),
- The student will demonstrate the ability to read the integrand
and choose the most appropriate integration technique:
- Integration by parts,
- Trigonometric integration,
- Trigonometric substitution,
- Integration of rational functions by partial fractions,
- If time permits, the student will demonstrate how integration
tables are applied,
- Approximating definite integrals and estimating their respective
approximation errors:
- To recite on demand the definition of each approximating integration
method, the error produced from the approximation:
- Trapezoidal Rule,
- Midpoint Rule,
- Simpson's Rule,
- To apply the definitions and error formulae for each approximating
integration method as needed to answer questions,
- Improper integrals,
- To know via memorization the definitions and theorems that
pertain to improper integrals (e.g., definition of a convergent/divergent
improper integral, the definition for all types of improper
integrals (type 1 and type 2), the Comparison Theorem)
- To demonstrate the ability for properly applying the above
by answering questions pertaining to improper integrals.
- Length of a curve (arc length),
- To recite on demand the arc length formula for curves given
as y=f(x) or as x=g(y) for an appropriate interval,
- To know what is meant by the term `arc length function' and
how to use it to derive the differential for arc length and
the change in arc length,
- To apply the above for the solution of any question involving
arc length, to include word problems.
- Area of a surface of revolution.
- To be able to answer questions involving the derivation of
the various formulae for surface area found by rotating a curve
about either the x- or y-axis,
- To demonstrate a working ability to choose the proper surface
area of rotation form for a particular question (exercise),
to include word problems.
- Applications,
- If time permits, the student will demonstrate their ability
to apply the above topical material to applications in the fields
of physics, engineering, economics, biology and probability
as demonstrated via questions (exercises). As examples, the
applications may include hydrostatic pressure and force, moments
and centers of mass, the theorem of Pappus and its application,
- For any of the applications covered, any formulae, definitions
and theorems will be memorized for recitation on demand.
- First order differential equations involving exponential growth
and decay,
- To recite on demand the general formulae that describes exponential
growth and decay,
- To recite on demand any and all terminology such as `law of
natural growth', `relative growth rate', `half-life',
- To recite on demand certain applications in which the function
value is proportional to its rate of change (examples may include
Newton's law of cooling, and continuously compounded interest),
- To demonstrate the knowledge required to solve exercises related
to the above (to include word problems).
- Parametric equations,
- To recite on demand any terminology needed to describe parametric
equations (e.g., parameter, parametric equations, parametric
curve, initial point, terminal point,
- To be able to recognize the curve of a cycloid,
- To demonstrate the ability to analyze families of parametric
equations,
- To demonstrate the ability to recognize the graph of a parametric
equation,
- To demonstrate the ability to graph parametric equations,
to include sketching the direction the curve takes with increasing
values of the parameter,
- To demonstrate the ability to eliminate the parameter to express
the curve in the form y=F(x), if possible,
- To demonstrate the ability to apply calculus to parametric
curves that includes,
- tangents
- areas,
- arc length,
- surface area.
- Polar coordinates,
- To recite on demand any terminology needed to describe polar
coordinates (e.g., polar coordinate system, pole, polar axis,
polar coordinates, polar equation, positive and negative angles),
- To demonstrate the ability to plot polar coordinates, to switch
from polar coordinates to Cartesian coordinates and from Cartesian
coordinates to polar coordinates,
- To demonstrate the ability to graph polar curves,
- To recognize whether it is better to represent a described
curve in Cartesian or polar coordinates, and then to write the
equation out in detail,
- To demonstrate the ability to match polar equations with its
graph,
- To demonstrate a working ability with polar coordinates by
solving questions (to include word problems),
- To demonstrate a working ability with the application of calculus
to polar equations that includes,
- areas,
- arc lengths
- tangent lines.
- Conics,
- To recite on demand the definitions of conics (parabolas,
ellipses, and hyperbolas),
- To recite on demand any terminology that describes conics
(e.g., parabola, directrix, axis of the conic, vertex, foci,
focus, major and minor axes, ellipse, hyperbola, asymptotes,
branches, standard equations),
- To demonstrate the ability to put a conic into its standard
form given any of a variety of data,
- To graph (by hand) conics given general information pertaining
to it (in either Cartesian or polar form)
- To demonstrate the ability to write a polar equation of a
conic given sufficient information,
- To demonstrate the ability to describe any defining element
of the conic (given information in either polar or Cartesian
form),
- Sequences,
- To recite on demand any terminology (and notation) that describes
sequences to include definitions/theorems (e.g., sequence, Fibonacci
sequence, limit, convergent, divergent, increasing, decreasing,
monotonic, bounded (above/below/sequence), Monotonic Sequence
theorem)
- To demonstrate the ability to describe sequences in various
ways (notation),
- To demonstrate the ability to analyse sequences by applying
the definitions, theorems and definitions that define the main
properties (convergence/divergence/limit),
- To recite on demand the main arithmetic properties for limits
of converging sequences (Limit laws for sequences),
- To demonstrate the ability to use the Limit laws when determining
properties of sequences,
- To demonstrate the ability to write clear, concise arguments
establishing the properties of sequences,
- To demonstrate the ability to apply the above for answering
word problems.
- Series,
- To recite on demand any terminology (and notation) that describes
series to include definitions/theorems (e.g., infinite series,
series, partial sum, convergent, divergent, geometric series,
harmonic series, necessary condition for convergence (theorem),
test for divergence, p-series, remainder, alternating series,
alternating series test, alternating series estimation theorem,
absolutely convergent, conditionally convergent, ratio test,
root test, rearrangement, power series about a, power
series centered at a, power series in (x-a), coefficients,
Bessel function, radius of convergence, interval of convergence,
term-by-term differentiation or integration, Taylor series,
Maclaurin series, binomial series),
- To demonstrate the ability to analyze series to determine
their properties (convergence/divergence) that include the following
skills,
- the application of the laws governing the sum and product
of two or more converging series,
- the application of the various tests for convergence
- Integral test (also used to estimate the limit of
a convergent series),
- Comparison test,
- Alternating series test,
- Ratio test (absolute convergence),
- Root test (absolute convergence),
- To demonstrate the ability to analyze power series to include
the determination of the radius and interval of convergence
as well as the ability to represent a given function as a power
series,
- To demonstrate the ability needed to determine the Taylor
series of a given function, or as a special case, to determine
the Maclaurin series,
- To demonstrate the ability needed to determine the remainder
term of the Taylor or Maclaurin series,
- The ability to derive the Taylor or Maclaurin series for the
main function categories (such as trigonometric, rational, exponential,
logarithmic, etc.) as well as determining power series for products
of functions as a product of power series (here, a product includes
division),
- The understanding of the binomial series and one of its uses
to determine power series representation
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