PLANNED COURSE STATEMENT
Department: Mathematics
Course Title: Advanced Calculus
State no. 3178 (RCPS – 3210N)
Grade level(s) 12
Length of Course: Full Year
Credits: 1 weighted
Prerequisite(s); AP Calculus AB
Check one:
Required
Selective
x
Elective
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Course Description:
Advanced Calculus is a course designed to build on the concepts learned in
AP Calculus AB and to prepare students for the higher levels of mathematics. The
diversity of topics and versatility of the knowledge gained in this course will enable
students to solve problems in many fields of study, including physics, engineering,
biology, and chemistry. The course includes vectors and dynamical systems, series
and approximation, advanced integration techniques, parametric and polar
functions, multiple variable functions, differential equations, and real-world
modeling examples. The students will extend the applications of calculus to a wider
arena of physical phenomena through the use of graphing calculators and computer
software.
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Course Objectives: Attached
Course Objectives:
1.
The student will extend topics from AP Calculus AB as preparation for
future study.
1.1 Find the derivatives of algebraic, trigonometric, exponential, and
inverse functions.
1.2 Compare the value of the differential, dy, to the actual change in y.
1.3 Find the differential of a function using differentiation formulas.
1.4 Use Newton’s Method to find the zeros of a function, manually and
using appropriate technology.
1.5 Evaluate definite and indefinite integrals of algebraic, trigonometric,
exponential, and inverse functions.
1.6 Calculate volumes of solids of revolution using the methods of disks,
washers, shells, and cross sections.
1.7 Calculate the arc length and area of a surface of revolution.
2.
The student will explore advanced integration techniques for evaluating
definite and indefinite integrals.
2.1 Find the antiderivatives of algebraic, trigonometric, exponential, and
inverse functions.
2.2 Use a table appropriately to evaluate integrals.
2.3 Use integration by parts to evaluate definite and indefinite integrals.
2.4 Integrate rational functions using linear and quadratic factors to find a
partial fraction composition.
2.5 Use identities and reduction formulas to integrate higher order
trigonometric functions.
2.6 Use trigonometric substitutions to evaluate integrands involving square
roots of quadratic functions.
2.7 Complete the square to integrate quadratic functions.
2.8 Identify improper integrals with infinite discontinuities and infinite
limits of integration.
2.9 Define convergence and divergence of improper integrals.
2.10 Evaluate improper integrals.
2.11 Use Riemann Sums, Trapezoidal Rule and Simpson’s Rule to numerically
evaluate integrals, both manually and with appropriate technology.
2.12 Recognize limits that produce indeterminate forms.
2.13 Apply L’Hopital’s Rule to evaluate limits.
3.
The student will use the properties of sequences and series to determine
their convergence of divergence and to calculate the sums of series.
3.1 Define monotonic and bounded sequences.
3.2 Apply the Sandwich Theorem and the Absolute Value Theorem for
sequences.
3.3 Use limits to determine the convergence or divergence of sequences.
3.4 Identify the properties of infinite series.
3.5 Use sequences of partial sums to determine the sums of infinite series.
3.6 Identify the following types of series and their properties:
Infinite series
Geometric series, including formula for the nth partial sum
Alternating series
Telescoping series
Harmonic and p-series
3.7 Determine the convergence or divergence of series using:
Absolute convergence
nth-term test
Geometric series test
Telescoping series test
p-series test
Alternating series test
Integral test
Root test
Ratio test
Direct comparison test
Limit comparison test
4.
The student will use Taylor and MacLaurin polynomials and power series to
approximate functions.
4.1 Identify Taylor and MacLaurin polynomials and the general formula for
each.
4.2 Find Taylor and MacLaurin polynomials of degree n and centered at c to
approximate functions.
4.3 Use appropriate graphing software to explore Taylor and MacLaurin
approximations.
4.4 Use Taylor and MacLaurin polynomials to approximate function values at
a point.
4.5 Calculate the error of Taylor and MacLaurin approximations.
4.6 Identify a power series and determine its convergence.
4.7
4.8
4.9
4.10
Determine the radii and intervals of convergence for power series.
Represent functions as sums of power series.
Perform differentiation and integration on power series.
Express Taylor and MacLaurin polynomials as power series.
5.
The student will explore the basic foundations of differential equations in
order to solve them.
5.1 Use separation of variables to derive the exponential formula for
growth and decay.
5.2 Explore appropriate applications of exponential growth, including
Newton’s Law of Cooling
Population Growth
Radioactive Decay
5.3 Identify the type and order of differential equations.
5.4 Use separation of variables to find the general solutions for
differential equations.
5.5 Use initial values to find specific solutions to differential equations.
5.6 Verify solutions to differential equations.
5.7 Examine direction fields to explore graphically the solutions to
differential equations, using technology as appropriate.
5.8 Find partial derivatives of functions.
5.9 Define homogeneous differential functions.
5.10 Use y = vx substitution and Use separation of variables to to solve
differential equations.
5.11 Find orthogonal trajectories for a family of curves.
6. The student will define a function in parametric mode.
6.1 Draw graphs manually to explore the relationship between rectangular
and parametric coordinates.
6.2 Sketch a plane curve and label its orientation from its parametric
equations.
6.3 Find parametric equations to describe a graph.
6.4 Convert equations between rectangular and parametric coordinates.
6.5 Identify smooth and piecewise smooth curves.
6.6 Find the parametric form of the derivative for a plane curve.
6.7 Identify the points of horizontal and vertical tangency using
derivatives.
6.8 Find the arc length of a plane curve.
6.9 Calculate the area of a surface of revolution generated from a plane
curve.
7.
The student will study the polar coordinate system in the context of
calculus.
7.1 Identify and describe the polar coordinate system.
7.2 Plot a point given its polar coordinates.
7.3 Find the four sets of polar coordinates describing a point.
7.4 Convert coordinates between rectangular and polar and vice versa.
7.5 Convert equations between rectangular and polar and vice versa.
7.6 Graph equations in polar coordinates.
7.7 Know the rules for symmetry of equations.
7.8 Find the relative extrema of r.
7.9 Find the slope of a polar graph.
7.10 Identify horizontal and vertical tangents to polar curves.
7.11 Find the tangents at the pole.
7.12 Identify and sketch the basic types of polar curves from their
equations.
7.13 Use integration to calculate area of regions bounded by polar curves.
7.14 Find arc length and area of surface of revolution for polar curves.
8.
The student will solve application problems using vectors and vector-valued
functions.
8.1 Identify a scalar and a vector.
8.2 Find the component form of a vector and translate any vector to its
standard position.
8.3 Find the magnitude of a vector.
8.4 Perform operations on vectors.
8.5 Find unit vectors tangent and normal to a given curve at a point.
8.6 Use the formula for finding angle measures between vectors.
8.7 Solve appropriate application problems, including navigation problems.
8.8 Define a vector-valued function and find its domain.
8.9 Find the limit of a vector-valued function.
8.10 Calculate the dot-product and cross-product of two vectors.
8.11 Find the derivative of a vector-valued function at a point.
8.12 Integrate vector-valued functions.
8.13 Use vector-valued functions to describe velocity and acceleration.
8.14 Solve problems involving the position function for a projectile.
9.
The student will calculate area and volume using iterated integrals.
9.1 Explain the relationship between single integrals and equivalent double
integrals.
9.2 Use double integrals to calculate the area of a plane region.
9.3 Define vertically and horizontally simple regions.
9.4 Use double integrals to calculate volume.
9.5 Use iterated integrals to find area and volume in polar coordinates.
9.6 Convert iterated integrals between rectangular and polar coordinates
and evaluate.
10.
The student will identify function of several variables and differentiate
them.
10.1 Define a function of two independent variables and find its domain and
range.
10.2 Sketch functions of two independent variables.
10.3 Find the curves describing level surfaces for a function and sketch.
10.4 Identify the level curve sketch equivalent to a given function and its
graph.
10.5 Find partial derivatives for functions of several variables.
10.6 Define and use the Chain Rule for functions of one and two
independent variables.