Advanced Calculus & Number Theory...

Advanced Calculus & Number Theory #322

Instructional Unit Elementary Number Theory Advanced Calculus & Number Theory #322 Unit Content Objective Performance Indicator Performance Task State Standards Code:

Elementary Number Theory The students will be - Understand the Archimedean - in-class problems 2.5.11.A; - Mathematical Induction able to understand and property of number theory - in-class discussions 2.5.11.B; - Binomial Theorem apply some preliminary - Apply the principal of finite - small group work 2.5.11.C; 2.5.11.D - Early Number Theory considerations relating induction - take home evaluation - Division Algorithm to number theory. - Understand and apply the problems - Greatest Common Divisor concepts involved in the - in-class evaluation - Euclidean Algorithm binomial theorem problems - Diophantine Equation - Explore methods and the - Fundamental Theorem of development of early number Arithmetic theory - Sieve of Eratosthenes - Goldbach conjecture - Gauss - Basic properties of congruence - Special divisibility tests - Linear congruencies - Fermat - Fermat's factorization method - The little theorem - Wilson's theorem


Elementary Number Theory The students will be - Understand and apply the - in-class problems 2.5.11.A; - Mathematical Induction able to apply the division algorithm - in-class discussions 2.5.11.B; - Binomial Theorem divisibility theory to the - Explore the special casein - small group work 2.5.11.C; 2.5.11.D - Early Number Theory set of integers. which the remainder in the - take home evaluation - Division Algorithm division algorithm is zero problems - Greatest Common Divisor - Understand and apply the - in-class evaluation - Euclidean Algorithm concepts involved in the problems - Diophantine Equation Euclidean algorithm - Fundamental Theorem of - Explore the Diophantine Arithmetic equation - Sieve of Eratosthenes - Goldbach conjecture - Gauss - Basic properties of congruence - Special divisibility tests - Linear congruencies - Fermat - Fermat's factorization method - The little theorem - Wilson's theorem


Elementary Number Theory The students will be - Apply the Fundamental - in-class problems 2.5.11.A; - Mathematical Induction able to understand Theorem of Arithmetic and its - in-class discussions 2.5.11.B; - Binomial Theorem primes and explore relationship to prime numbers - small group work 2.5.11.C; 2.5.11.D - Early Number Theory their distribution. - Use the technique of the - take home evaluation - Division Algorithm Sieve of Eratosthenes to find all problems - Greatest Common Divisor primes below a given values - in-class evaluation - Euclidean Algorithm - Use Euclid's technique to problems - Diophantine Equation prove there is an infinite - Fundamental Theorem of number of primes Arithmetic - Understand and apply the - Sieve of Eratosthenes Goldbach Conjecture - Goldbach conjecture - Gauss - Basic properties of congruence - Special divisibility tests - Linear congruencies - Fermat - Fermat's factorization method - The little theorem - Wilson's theorem


Elementary Number Theory The students will be - Review the work of Gauss - in-class problems 2.5.11.A; - Mathematical Induction able to understand the relating to number theory - in-class discussions 2.5.11.B; - Binomial Theorem theory of congruencies. - Explore basic properties of - small group work 2.5.11.C; 2.5.11.D - Early Number Theory congruence - take home evaluation - Division Algorithm - Review special divisibility problems - Greatest Common Divisor tests and their applications to - in-class evaluation - Euclidean Algorithm congruence problems - Diophantine Equation - Understand and apply the - Fundamental Theorem of concept of linear congruence Arithmetic - Sieve of Eratosthenes - Goldbach conjecture - Gauss - Basic properties of congruence - Special divisibility tests - Linear congruencies - Fermat - Fermat's factorization method - The little theorem - Wilson's theorem


Elementary Number Theory The students will be - Review the work of Fermat - in-class problems 2.5.11.A; - Mathematical Induction able to gain an - Understand Fermat's - in-class discussions 2.5.11.B; - Binomial Theorem understanding and factorization method - small group work 2.5.11.C; 2.5.11.D - Early Number Theory appreciation of - Review the proof of Fermat's - take home evaluation - Division Algorithm Fermat's Theorem. theorem problems - Greatest Common Divisor - Understand the significance - in-class evaluation - Euclidean Algorithm of Fermat's theorem problems - Diophantine Equation - Understand the concept of - Fundamental Theorem of Wilson's Theorem Arithmetic - Sieve of Eratosthenes - Goldbach conjecture - Gauss - Basic properties of congruence - Special divisibility tests - Linear congruencies - Fermat - Fermat's factorization method - The little theorem - Wilson's theorem

Instructional Unit Functions of Several Variables Advanced Calculus & Number Theory #322 Unit Content Objective Performance Indicator Performance Task State Standards Code:

Functions of Several Variables The students will be - Describe the level surface of - in-class problems 2.5.11.A; - Notation for functions of several able to find the a function of several variables - in-class discussions 2.5.11.B; variables equations of tangent - small group work 2.5.11.C; 2.5.11.D - Graphs of functions of two planes and normal lines - Find the equation of the - take home evaluation variables to surfaces. tangent plane to a function of problems - Graphs of level curves for a several variables - in-class evaluation function of two variables - Find the equation of a normal problems - Graphs of level surfaces for a line to a surface described by a function of three variables function of several variables - Definition of the limit of a function - Find the equation of a tangent to two variables line to a surface described by - Continuity of functions of two a function of several variables and three variables - Partial derivatives of functions of - Find the angle of inclination of two and three variables a tangent plane - Higher-order derivatives of function of two and three variables - Increments and differentials - The chain rule for a function of several variables - Implicit partial derivatives - Directional derivatives of a function of two and three variables - The gradient of a function of two and three variables - Tangent planes and normal lines to surfaces - Angle on inclination - Absolute and relative extreme of a function of two variables - Second partials test


- Optimization problems on functions of several variables - Method of least squares - Method of Lagrange multipliers


Functions of Several Variables The students will be - Verify a limit of a function of - in-class problems 2.5.11.A; - Notation for functions of several able to determine the two variables by using the - in-class discussions 2.5.11.B; variables limit and describe the definition of a limit - small group work 2.5.11.C; 2.5.11.D - Graphs of functions of two continuity of a function - Evaluate the limit of a function - take home evaluation variables of several variables. of several variables problems - Graphs of level curves for a - Understand examples where - in-class evaluation function of two variables the limit of a function of several problems - Graphs of level surfaces for a variables does not exist function of three variables - Discuss the continuity of a - Definition of the limit of a function function of several variables to two variables - Test the continuity of a - Continuity of functions of two function of three variables and three variables - Partial derivatives of functions of two and three variables - Higher-order derivatives of function of two and three variables - Increments and differentials - The chain rule for a function of several variables - Implicit partial derivatives - Directional derivatives of a function of two and three variables - The gradient of a function of two and three variables - Tangent planes and normal lines to surfaces - Angle on inclination - Absolute and relative extreme of a function of two variables - Second partials test


Functions of Several Variables The students will be - Find and evaluate the partial - in-class problems 2.5.11.A; - Notation for functions of several able to find and use derivative of a function of two - in-class discussions 2.5.11.B; variables partial derivative of a variables - small group work 2.5.11.C; 2.5.11.D - Graphs of functions of two function of two or - Find the slopes of a surface - take home evaluation variables three variables. of a function of two variables problems - Graphs of level curves for a - Use partial derivatives to find - in-class evaluation function of two variables the rate of change of a function problems - Graphs of level surfaces for a of two variables function of three variables - Find the partial derivative of a - Definition of the limit of a function function of three variables to two variables - Find the second partial - Continuity of functions of two derivative of a function of two and three variables variables - Partial derivatives of functions of - Find higher order partial two and three variables derivatives of functions of two - Higher-order derivatives of or three variables function of two and three variables - Increments and differentials - The chain rule for a function of several variables - Implicit partial derivatives - Directional derivatives of a function of two and three variables - The gradient of a function of two and three variables - Tangent planes and normal lines to surfaces - Angle on inclination - Absolute and relative extreme of a function of two variables - Second partials test


Functions of Several Variables The students will be - Determine the domain of a - in-class problems 2.5.11.A; - Notation for functions of several able to understand the function of several variables - in-class discussions 2.5.11.B; variables notation and sketch the - Determine the range and - small group work 2.5.11.C; 2.5.11.D - Graphs of functions of two graphs for functions describe the graph of a - take home evaluation variables of several variables. function of two variables problems - Graphs of level curves for a - Sketch a contour map using - in-class evaluation function of two variables level curves for a function of problems - Graphs of level surfaces for a two variables function of three variables - Describe the level surfaces - Definition of the limit of a function of a function of several to two variables variables - Continuity of functions of two and three variables - Partial derivatives of functions of two and three variables - Higher-order derivatives of function of two and three variables - Increments and differentials - The chain rule for a function of several variables - Implicit partial derivatives - Directional derivatives of a function of two and three variables - The gradient of a function of two and three variables - Tangent planes and normal lines to surfaces - Angle on inclination - Absolute and relative extreme of a function of two variables - Second partials test


Functions of Several Variables The students will be - Find the directional derivative - in-class problems 2.5.11.A; - Notation for functions of several able to find and use of a function of two variables - in-class discussions 2.5.11.B; variables directional derivative of in the direction of a given - small group work 2.5.11.C; 2.5.11.D - Graphs of functions of two a function of two vector - take home evaluation variables variables and use the - Find the gradient of a problems - Graphs of level curves for a gradient of a function function of two variables - in-class evaluation function of two variables of two variables in - Apply the gradient of a problems - Graphs of level surfaces for a applications. function to determine the function of three variables direction of maximum or - Definition of the limit of a function minimum increase to two variables - Use the gradient to determine - Continuity of functions of two the normal vector to a level and three variables curve at several points on the - Partial derivatives of functions of curve two and three variables - Find the gradient for a - Higher-order derivatives of function of three variables function of two and three variables - Increments and differentials - The chain rule for a function of several variables - Implicit partial derivatives - Directional derivatives of a function of two and three variables - The gradient of a function of two and three variables - Tangent planes and normal lines to surfaces - Angle on inclination - Absolute and relative extreme of a function of two variables - Second partials test


Functions of Several Variables The students will be - Find the relative extrema of a - in-class problems 2.5.11.A; - Notation for functions of several able to find absolute function of two variables - in-class discussions 2.5.11.B; variables and relative extreme of - Use the second partials test - small group work 2.5.11.C; 2.5.11.D - Graphs of functions of two a function of two to find the relative extreme of a - take home evaluation variables variables. function of two variables problems - Graphs of level curves for a - Explore the failure of the - in-class evaluation function of two variables second partials test problems - Graphs of level surfaces for a - Find the absolute extreme of function of three variables a function of two variables on - Definition of the limit of a function a closed interval; to two variables - Continuity of functions of two and three variables - Partial derivatives of functions of two and three variables - Higher-order derivatives of function of two and three variables - Increments and differentials - The chain rule for a function of several variables - Implicit partial derivatives - Directional derivatives of a function of two and three variables - The gradient of a function of two and three variables - Tangent planes and normal lines to surfaces - Angle on inclination - Absolute and relative extreme of a function of two variables - Second partials test


Functions of Several Variables The students will be - Find the maximum volume of - in-class problems 2.5.11.A; - Notation for functions of several able to solve a rectangular solid resting on - in-class discussions 2.5.11.B; variables optimization problems the xy plane with one vertex at - small group work 2.5.11.C; 2.5.11.D - Graphs of functions of two involving functions of the origin - take home evaluation variables several variables. - Find the production level that problems - Graphs of level curves for a will produce a maximum profit - in-class evaluation function of two variables - Determine the least squares problems - Graphs of level surfaces for a regression line given several function of three variables points - Definition of the limit of a function to two variables - Continuity of functions of two and three variables - Partial derivatives of functions of two and three variables - Higher-order derivatives of function of two and three variables - Increments and differentials - The chain rule for a function of several variables - Implicit partial derivatives - Directional derivatives of a function of two and three variables - The gradient of a function of two and three variables - Tangent planes and normal lines to surfaces - Angle on inclination - Absolute and relative extreme of a function of two variables - Second partials test


Functions of Several Variables The students will be - Use a Lagrange multiplier - in-class problems 2.5.11.A; - Notation for functions of several able to understand the with one constraint - in-class discussions 2.5.11.B; variables method of Lagrange - Use a Lagrange multiplier to - small group work 2.5.11.C; 2.5.11.D - Graphs of functions of two multipliers. find the minimum value of a - take home evaluation variables function of several variables problems - Graphs of level curves for a with one constraint - in-class evaluation function of two variables - Find the extreme values of a problems - Graphs of level surfaces for a function of several variables function of three variables subject to one constraint - Definition of the limit of a function - Use a Lagrange multiplier to to two variables maximize a function subject to - Continuity of functions of two two constraints and three variables - Partial derivatives of functions of two and three variables - Higher-order derivatives of function of two and three variables - Increments and differentials - The chain rule for a function of several variables - Implicit partial derivatives - Directional derivatives of a function of two and three variables - The gradient of a function of two and three variables - Tangent planes and normal lines to surfaces - Angle on inclination - Absolute and relative extreme of a function of two variables - Second partials test


Functions of Several Variables The students will be - Find the total differential of a - in-class problems 2.5.11.A; - Notation for functions of several able to understand the function of several variables - in-class discussions 2.5.11.B; variables concepts of - Evaluate the differentiability - small group work 2.5.11.C; 2.5.11.D - Graphs of functions of two increments and of a function of several - take home evaluation variables differentials. variables at every point in a problems - Graphs of level curves for a plane. - in-class evaluation function of two variables - Use a differential as method problems - Graphs of level surfaces for a of approximating the value of a function of three variables function of two variables - Definition of the limit of a function - Show that the value of a to two variables function of two variables exists - Continuity of functions of two at a point but is not and three variables differentiable at the point - Partial derivatives of functions of two and three variables - Higher-order derivatives of function of two and three variables - Increments and differentials - The chain rule for a function of several variables - Implicit partial derivatives - Directional derivatives of a function of two and three variables - The gradient of a function of two and three variables - Tangent planes and normal lines to surfaces - Angle on inclination - Absolute and relative extreme of a function of two variables - Second partials test


Functions of Several Variables The students will be - Use the chain rule to evaluate - in-class problems 2.5.11.A; - Notation for functions of several able to apply the chain the derivative of a function - in-class discussions 2.5.11.B; variables rule to a function of with one independent variable - small group work 2.5.11.C; 2.5.11.D - Graphs of functions of two several variables. - Apply the use of the chain - take home evaluation variables rule of the function of several problems - Graphs of level curves for a variables to a related rates - in-class evaluation function of two variables situation problems - Graphs of level surfaces for a - Find the partial derivative of a function of three variables function of several variables - Definition of the limit of a function using substitution to two variables - Use the chain rule to evaluate - Continuity of functions of two the derivative of a function and three variables with two independent variables - Partial derivatives of functions of two and three variables - Apply the chain rule to a - Higher-order derivatives of function of three variables function of two and three variables - Find a partial derivative of a function of several variables - Increments and differentials implicitly - The chain rule for a function of several variables - Implicit partial derivatives - Directional derivatives of a function of two and three variables - The gradient of a function of two and three variables - Tangent planes and normal lines to surfaces - Angle on inclination - Absolute and relative extreme of a function of two variables - Second partials test

Instructional Unit Infinite Series Advanced Calculus & Number Theory #322 Unit Content Objective Performance Indicator Performance Task State Standards Code:

Infinite Series The students will be -List the terms of a sequence - in-class problems 2.5.11.A; - Sequences able to identify and - Find the limit of a sequence - in-class discussions 2.5.11.B; - Convergence and divergence of establish notation for a - Determine the convergence - small group work 2.5.11.C; a sequence sequence. or divergence of a sequence - take home evaluation 2.5.11.D; - Formulas for the nth term of a - Apply the squeeze theorem problems 2.11.11.A, sequence and absolute value theorem to - in-class evaluation 2.11.11.D - Monotonic and bounded find the limit of a sequence problems sequences - Find the nth term of a - Convergent infinite series sequence - infinite geometric series - Determine if a sequence is - n-th term test for divergence monotonic and/or bounded - Integral test - p-series - harmonic series - Direct comparison test - Limit comparison test - Alternating series test - Alternating series remainder - Absolute and conditional convergence - Ratio test - Root test - Taylor and Maclaurin approximations of elementary functions - Power series - Radius and interval of convergence - Endpoint convergence of a power series - Derivatives and integrals of a power series - Geometric power series


- Taylor and Maclaurin series - Binomial series


Infinite Series The students will be - Determine the convergence - in-class problems 2.5.11.A; - Sequences able to identify series or divergence of a series - in-class discussions 2.5.11.B; - Convergence and divergence of and establish if the - Write a series in telescoping - small group work 2.5.11.C; a sequence series converges. form - take home evaluation 2.5.11.D; - Formulas for the nth term of a - Determine the convergence problems 2.11.11.A, sequence or divergence of a geometric - in-class evaluation 2.11.11.D - Monotonic and bounded series problems sequences - Utilize the nth term test for - Convergent infinite series divergence with a series - infinite geometric series - n-th term test for divergence - Integral test - p-series - harmonic series - Direct comparison test - Limit comparison test - Alternating series test - Alternating series remainder - Absolute and conditional convergence - Ratio test - Root test - Taylor and Maclaurin approximations of elementary functions - Power series - Radius and interval of convergence - Endpoint convergence of a power series - Derivatives and integrals of a power series - Geometric power series


Infinite Series The students will be - Utilize the integral test to - in-class problems 2.5.11.A; - Sequences able to apply the establish the convergence or - in-class discussions 2.5.11.B; - Convergence and divergence of integral test to a series divergence of a series - small group work 2.5.11.C; a sequence and understand the - Determine the convergence - take home evaluation 2.5.11.D; - Formulas for the nth term of a properties of a p- or divergence of p-series problems 2.11.11.A, sequence series and harmonic - in-class evaluation 2.11.11.D - Monotonic and bounded series. problems sequences - Convergent infinite series - infinite geometric series - n-th term test for divergence - Integral test - p-series - harmonic series - Direct comparison test - Limit comparison test - Alternating series test - Alternating series remainder - Absolute and conditional convergence - Ratio test - Root test - Taylor and Maclaurin approximations of elementary functions - Power series - Radius and interval of convergence - Endpoint convergence of a power series - Derivatives and integrals of a power series - Geometric power series


Infinite Series The students will be - Determine the convergence - in-class problems 2.5.11.A; - Sequences able to apply both the or divergence of a series using - in-class discussions 2.5.11.B; - Convergence and divergence of direct comparison and the direct comparison test - small group work 2.5.11.C; a sequence limit comparison tests - Determine the convergence - take home evaluation 2.5.11.D; - Formulas for the nth term of a to establish the or divergence of a series using problems 2.11.11.A, sequence convergence or the limit comparison test - in-class evaluation 2.11.11.D - Monotonic and bounded divergence of a series. problems sequences - Convergent infinite series - infinite geometric series - n-th term test for divergence - Integral test - p-series - harmonic series - Direct comparison test - Limit comparison test - Alternating series test - Alternating series remainder - Absolute and conditional convergence - Ratio test - Root test - Taylor and Maclaurin approximations of elementary functions - Power series - Radius and interval of convergence - Endpoint convergence of a power series - Derivatives and integrals of a power series - Geometric power series


Infinite Series The students will be - Determine the convergence - in-class problems 2.5.11.A; - Sequences able to assess the or divergence of an alternating - in-class discussions 2.5.11.B; - Convergence and divergence of convergence of an series - small group work 2.5.11.C; a sequence alternating series and - Explore the effects of using - take home evaluation 2.5.11.D; - Formulas for the nth term of a classify a series as the alternating series test when problems 2.11.11.A, sequence absolutely or the test fails - in-class evaluation 2.11.11.D - Monotonic and bounded conditionally - Approximate the sum of an problems sequences convergent. alternating series - Convergent infinite series - Determine if a series is - infinite geometric series absolutely or conditionally - n-th term test for divergence convergent - Integral test - p-series - harmonic series - Direct comparison test - Limit comparison test - Alternating series test - Alternating series remainder - Absolute and conditional convergence - Ratio test - Root test - Taylor and Maclaurin approximations of elementary functions - Power series - Radius and interval of convergence - Endpoint convergence of a power series - Derivatives and integrals of a power series - Geometric power series


Infinite Series The students will be - Determine the convergence - in-class problems 2.5.11.A; - Sequences able to apply the ratio or divergence of a series using - in-class discussions 2.5.11.B; - Convergence and divergence of and root tests to the ratio test - small group work 2.5.11.C; a sequence determine the - Explore a series example - take home evaluation 2.5.11.D; - Formulas for the nth term of a convergence or where the ration test fails problems 2.11.11.A, sequence divergence of a series. - Determine the convergence - in-class evaluation 2.11.11.D - Monotonic and bounded or divergence of a series using problems sequences the root test - Convergent infinite series - Establish guidelines for - infinite geometric series determine the convergence or - n-th term test for divergence divergence of any series - Integral test - Create a flow chart of steps - p-series for establishing the - harmonic series convergence or divergence of - Direct comparison test a series - Limit comparison test - Alternating series test - Alternating series remainder - Absolute and conditional convergence - Ratio test - Root test - Taylor and Maclaurin approximations of elementary functions - Power series - Radius and interval of convergence - Endpoint convergence of a power series - Derivatives and integrals of a power series - Geometric power series


Infinite Series The students will be - Write a power series term by - in-class problems 2.5.11.A; - Sequences able to understand and term and express the power - in-class discussions 2.5.11.B; - Convergence and divergence of work with power series using summation - small group work 2.5.11.C; a sequence series. notation - take home evaluation 2.5.11.D; - Formulas for the nth term of a - Determine the radius of problems 2.11.11.A, sequence convergence of a power - in-class evaluation 2.11.11.D - Monotonic and bounded series problems sequences - Determine the interval of - Convergent infinite series convergence of a power - infinite geometric series series - n-th term test for divergence - Evaluate the derivative of a - Integral test power series and find the - p-series interval of convergence - harmonic series - Evaluate the integral of a - Direct comparison test power series and find the - Limit comparison test interval of convergence - Alternating series test - Alternating series remainder - Absolute and conditional convergence - Ratio test - Root test - Taylor and Maclaurin approximations of elementary functions - Power series - Radius and interval of convergence - Endpoint convergence of a power series - Derivatives and integrals of a power series - Geometric power series


Infinite Series The students will be - Determine a first degree - in-class problems 2.5.11.A; - Sequences able to find Taylor and Taylor polynomial for e to the x - in-class discussions 2.5.11.B; - Convergence and divergence of Maclaurin polynomial power - small group work 2.5.11.C; a sequence approximations of - Determine higher order - take home evaluation 2.5.11.D; - Formulas for the nth term of a elementary functions. degree Taylor polynomials for e problems 2.11.11.A, sequence to the x power - in-class evaluation 2.11.11.D - Monotonic and bounded - Determine Taylor and problems sequences Maclaurin polynomials for - Convergent infinite series exponential and logarithmic - infinite geometric series functions - n-th term test for divergence - Determine Taylor and - Integral test Maclaurin polynomials for - p-series trigonometric functions - harmonic series - Approximate the value of a - Direct comparison test function using a Maclaurin - Limit comparison test polynomial - Alternating series test - Utilize Taylor's theorem to - Alternating series remainder approximate the value of a - Absolute and conditional function and determine the convergence accuracy - Ratio test - Determine the degree of a - Root test Taylor polynomial required to - Taylor and Maclaurin limit the error within a specified approximations of elementary bound functions - Power series - Radius and interval of convergence - Endpoint convergence of a power series - Derivatives and integrals of a power series - Geometric power series


Infinite Series The students will be - Form a Taylor or Maclaurin - in-class problems 2.5.11.A; - Sequences able to find Taylor and series for a given function - in-class discussions 2.5.11.B; - Convergence and divergence of Maclaurin series for a - Determine a Maclaurin series - small group work 2.5.11.C; a sequence function. for a composite function - take home evaluation 2.5.11.D; - Formulas for the nth term of a - Find a Maclaurin series for a problems 2.11.11.A, sequence binomial function and determine - in-class evaluation 2.11.11.D - Monotonic and bounded the radius of convergence problems sequences - Find a Maclaurin series for a - Convergent infinite series function that can be treated as - infinite geometric series a binomial function - n-th term test for divergence - Derive a power series from a - Integral test given list of elementary - p-series functions - harmonic series - Utilize rewriting techniques to - Direct comparison test determine a power series - Limit comparison test representation of a more - Alternating series test complex function - Alternating series remainder - Absolute and conditional convergence - Ratio test - Root test - Taylor and Maclaurin approximations of elementary functions - Power series - Radius and interval of convergence - Endpoint convergence of a power series - Derivatives and integrals of a power series - Geometric power series


Infinite Series The students will be - Determine a geometric power - in-class problems 2.5.11.A; - Sequences able to represent series centered at zero - in-class discussions 2.5.11.B; - Convergence and divergence of functions by using - Determine a geometric power - small group work 2.5.11.C; a sequence power series. series centered at a value - take home evaluation 2.5.11.D; - Formulas for the nth term of a other than zero problems 2.11.11.A, sequence - Utilize operations with power - in-class evaluation 2.11.11.D - Monotonic and bounded series to combine two power problems sequences series - Convergent infinite series - Determine a power series - infinite geometric series through integration techniques - n-th term test for divergence - Integral test - p-series - harmonic series - Direct comparison test - Limit comparison test - Alternating series test - Alternating series remainder - Absolute and conditional convergence - Ratio test - Root test - Taylor and Maclaurin approximations of elementary functions - Power series - Radius and interval of convergence - Endpoint convergence of a power series - Derivatives and integrals of a power series - Geometric power series

Instructional Unit Parametric and Polar Equations Advanced Calculus & Number Theory #322 Unit Content Objective Performance Indicator Performance Task State Standards Code:

Parametric and Polar Equations The students will be - Sketch a curve in parametric - in-class problems 2.5.11.A; - Define and graph parametric able to work with form - in-class discussions 2.5.11.B; equations parametric equations. - Determine the domain and - small group work 2.5.11.C; 2.5.11.D - Graph functions and their range of a parametric equation - take home evaluation inverses in parametric mode problems - Use the Chain Rule to find first - Eliminate the parameter from - in-class evaluation and second derivatives for a set of parametric equations problems parametric equations - Determine a set of parametric - Use the Derivative feature of the equations for a given function TI-89 to find derivatives of parametric equations - Use the Tangent feature of the TI-89 to find tangent lines and their slopes - Find lengths of parametric curves using the definition of arc lengthy - Find arc lengths with the Arc feature of the TI-89 graph math menu - Convert coordinates from rectangular to polar and from polar to rectangular - Graph polar functions and find their intersections - Use the Tangent and Arc features for polar graphs on the TI- 89 - Find areas bounded by polar functions


Parametric and Polar Equations The students will be - Sketch a curve in polar form - in-class problems 2.5.11.A; - Define and graph parametric able to work with polar - in-class discussions 2.5.11.B; equations equations. - Determine the domain and - small group work 2.5.11.C; 2.5.11.D - Graph functions and their range of a polar equation - take home evaluation inverses in parametric mode - Determine a polar equation problems - Use the Chain Rule to find first - in-class evaluation and second derivatives for problems parametric equations - Use the Derivative feature of the TI-89 to find derivatives of parametric equations - Use the Tangent feature of the TI-89 to find tangent lines and their slopes - Find lengths of parametric curves using the definition of arc lengthy - Find arc lengths with the Arc feature of the TI-89 graph math menu - Convert coordinates from rectangular to polar and from polar to rectangular - Graph polar functions and find their intersections - Use the Tangent and Arc features for polar graphs on the TI- 89 - Find areas bounded by polar functions


Parametric and Polar Equations The students will be - Define and graph parametric - in-class problems 2.5.11.A; - Define and graph parametric able to graph and equations using a TI-89 - in-class discussions 2.5.11.B; equations analyze parametric - Graph functions and their - small group work 2.5.11.C; 2.5.11.D - Graph functions and their equations using the TI- inverses in parametric mode - take home evaluation inverses in parametric mode 89 and find slopes and - Use the Chain Rule to find problems - Use the Chain Rule to find first arc lengths of first and second derivative for - in-class evaluation and second derivatives for parametric curves. parametric equations problems parametric equations - Use the Derivative feature of - Use the Derivative feature of the the TI-89 to find derivative of TI-89 to find derivatives of parametric equations parametric equations - Use the Tangent feature of - Use the Tangent feature of the the TI-89 to find tangent lines TI-89 to find tangent lines and their and their slopes slopes - Find lengths of parametric - Find lengths of parametric curves using the definition of curves using the definition of arc arc length lengthy - Find arc lengths with the Arc - Find arc lengths with the Arc feature of the TI-89 Graph Math feature of the TI-89 graph math menu menu - Convert coordinates from rectangular to polar and from polar to rectangular - Graph polar functions and find their intersections - Use the Tangent and Arc features for polar graphs on the TI- 89 - Find areas bounded by polar functions


Parametric and Polar Equations The students will be - Use the TI-89 to convert - in-class problems 2.5.11.A; - Define and graph parametric able to use the TI-89 to coordinates form rectangular to - in-class discussions 2.5.11.B; equations graph polar functions polar and from polar to - small group work 2.5.11.C; 2.5.11.D - Graph functions and their and perform calculus rectangular - take home evaluation inverses in parametric mode operations on polar - Graph polar functions and problems - Use the Chain Rule to find first functions to find values find their intersections - in-class evaluation and second derivatives for such as slope, arc - Us the TI-89 Tangent and Arc problems parametric equations length and area. features for polar functions - Use the Derivative feature of the - Find areas bounded by polar TI-89 to find derivatives of functions parametric equations - Use the Tangent feature of the TI-89 to find tangent lines and their slopes - Find lengths of parametric curves using the definition of arc lengthy - Find arc lengths with the Arc feature of the TI-89 graph math menu - Convert coordinates from rectangular to polar and from polar to rectangular - Graph polar functions and find their intersections - Use the Tangent and Arc features for polar graphs on the TI- 89 - Find areas bounded by polar functions

Instructional Unit Review, Application, and Extension Problems from First Year Calculus Advanced Calculus & Number Theory #322 Unit Content Objective Performance Indicator Performance Task State Standards Code:

Review, Application, and Extension The students will be - Determine areas over which - in-class problems 2.5.11.A; Problems from First Year Calculus able to determine a function is increasing and - in-class discussions 2.5.11.B; extreme values of a decreasing - small group work 2.5.11.C; - Functions and graphs function and apply - Determine all relative extreme - take home evaluation 2.5.11.D; - The derivative these values to of a function problems 2.11.11.A, - Extreme values problems involving - Use the second derivative - in-class evaluation 2.11.11.B; - Antiderivatives and differential optimization. test to determine the concavity problems 2.11.11.C equations of a function - The definite integral - Apply the concepts of - Revisiting the definite integral extreme values to problems - Sequences and series of involving maximums and numbers minimums - Sequences and series of functions

Review, Application, and Extension The students will be - Determine the antiderivative - in-class problems 2.5.11.A; Problems from First Year Calculus able to determine the of a function - in-class discussions 2.5.11.B; antiderivative of a - Graph the slope field of a - small group work 2.5.11.C; - Functions and graphs function and evaluate differential equation - take home evaluation 2.5.11.D; - The derivative differential equations. - Find the general and specific problems 2.11.11.A, - Extreme values solution to a differential equation - in-class evaluation 2.11.11.B; - Antiderivatives and differential problems 2.11.11.C equations - The definite integral - Revisiting the definite integral - Sequences and series of numbers - Sequences and series of functions


Review, Application, and Extension The students will be - Determine an average rate of - in-class problems 2.5.11.A; Problems from First Year Calculus able to find the change - in-class discussions 2.5.11.B; derivative of a function - Evaluate a derivative using - small group work 2.5.11.C; - Functions and graphs and apply the concept the definition of a derivative - take home evaluation 2.5.11.D; - The derivative of a derivative to - Determine a function from the problems 2.11.11.A, - Extreme values related rates. graph of it's derivative - in-class evaluation 2.11.11.B; - Antiderivatives and differential - Evaluate the lime of an problems 2.11.11.C equations elementary function and a - The definite integral piece wise function - Revisiting the definite integral - Apply the power, sum, and - Sequences and series of product rules for differentiation numbers - Sequences and series of - Apply the chain rule for functions differentiation - Evaluate the derivative of trigonometric, logarithmic, and exponential functions - Review Rolle's theorem, the Intermediate Value Theorem and Newton's method - Solve problems involving related rates


Review, Application, and Extension The students will be - Express a definite integral as - in-class problems 2.5.11.A; Problems from First Year Calculus able to evaluate the a Riemann sum - in-class discussions 2.5.11.B; definite integral. - Apply basic properties of an - small group work 2.5.11.C; - Functions and graphs integral - take home evaluation 2.5.11.D; - The derivative - Evaluate an integral using the problems 2.11.11.A, - Extreme values geometry of the graph - in-class evaluation 2.11.11.B; - Antiderivatives and differential - Apply the Fundamental problems 2.11.11.C equations Theorem of Calculus - The definite integral - Evaluate a function defined - Revisiting the definite integral by an integral - Sequences and series of numbers - Sequences and series of functions

Review, Application, and Extension The students will be - Determine domain and range - in-class problems 2.10.11.A; Problems from First Year Calculus able to analyze of elementary functions - in-class discussions 2.5.11.A; elementary functions - Evaluate and graph - small group work 2.5.11.B; - Functions and graphs including trigonometric, trigonometric functions - take home evaluation 2.5.11.C; 2.5.11.D - The derivative exponential, - Evaluate composite functions problems - Extreme values logarithmic, composite, - in-class evaluation - Antiderivatives and differential parametric, and polar - Describe and graph functions problems equations functions. given in parametric form - The definite integral - Describe and graph functions - Revisiting the definite integral given in polar form - Sequences and series of numbers - Sequences and series of functions


Review, Application, and Extension The students will be - Find the exact value of an - in-class problems 2.5.11.A; Problems from First Year Calculus able to find both exact integral using the fundamental - in-class discussions 2.5.11.B; and approximations of theorem of calculus - small group work 2.5.11.C; - Functions and graphs a definite integral. - Apply techniques of - take home evaluation 2.5.11.D; - The derivative integration problems 2.11.11.A, - Extreme values - Utilize approximation - in-class evaluation 2.11.11.B; - Antiderivatives and differential techniques and error analysis problems 2.11.11.C equations for definite integrals - The definite integral - Revisiting the definite integral - Sequences and series of numbers - Sequences and series of functions

Instructional Unit Vectors and the Geometry of Space Advanced Calculus & Number Theory #322 Unit Content Objective Performance Indicator Performance Task State Standards Code:

Vectors and the Geometry of The students will be - Sketch the surface given by - in-class problems 2.5.11.A; Space able to recognize and an equation in three-space - in-class discussions 2.5.11.B; - Component form of a vector write equations for - Sketch a quadric surface in - small group work 2.5.11.C; 2.5.11.D - Vector operations cylindrical surfaces, space - take home evaluation - Standard unit vectors quadratic surfaces, - Classify quadric surfaces problems - Applications of vectors and surfaces of given their equations in space - in-class evaluation - Three-dimensional rectangular revolution. - Determine an equation for a problems coordinate system surface of revolution - Vectors in space - Determine a generating curve - Applications of three-dimensional for a surface of revolution vectors - Dot product of two vectors - Angle between two vectors - Direction cosines - Projection of a vector - Cross product of two vectors - Triple scalar product - Parametric equations for a line in space - Distance between points, lines and planes in space - Cylindrical surfaces - Quadratic surfaces - Surfaces of revolution - Cylindrical and spherical coordinates


Vectors and the Geometry of The students will be - Determine parametric and - in-class problems 2.5.11.A; Space able to understand symmetric equations of a line - in-class discussions 2.5.11.B; - Component form of a vector lines and planes in that passes through a given - small group work 2.5.11.C; 2.5.11.D - Vector operations space and evaluate point and is parallel to a given - take home evaluation - Standard unit vectors distances between vector problems - Applications of vectors lines and/or planes. - Find a set of parametric - in-class evaluation - Three-dimensional rectangular equations of a line that passes problems coordinate system through two given points - Vectors in space - Determine the equation of a - Applications of three-dimensional plane in three-space vectors - Find the parametric equations - Dot product of two vectors of he line of intersection of - Angle between two vectors two planes - Direction cosines - Determine the distance - Projection of a vector between a point and a plane - Cross product of two vectors - Determine the distance - Triple scalar product between two parallel planes - Parametric equations for a line in - Determine the distance space between a point and a plane in - Distance between points, lines three-space and planes in space - Cylindrical surfaces - Quadratic surfaces - Surfaces of revolution - Cylindrical and spherical coordinates


Vectors and the Geometry of The students will be - Represent a vector using - in-class problems 2.5.11.A; Space able to understand directed line segments - in-class discussions 2.5.11.B; - Component form of a vector apply operations for - Find the component form and - small group work 2.5.11.C; 2.5.11.D - Vector operations components of a length of a vector given the - take home evaluation - Standard unit vectors vector in a plane. two endpoints problems - Applications of vectors - Apply basic vector - in-class evaluation - Three-dimensional rectangular operations of additions and problems coordinate system multiplication - Vectors in space - Find a unit vector in a given - Applications of three-dimensional direction vectors - Write a vector as a linear - Dot product of two vectors combination of unit vectors - Angle between two vectors - Write a vector using unit - Direction cosines vectors given a length and - Projection of a vector direction - Cross product of two vectors - Find the resultant force given - Triple scalar product two applied forces - Parametric equations for a line in space - Distance between points, lines and planes in space - Cylindrical surfaces - Quadratic surfaces - Surfaces of revolution - Cylindrical and spherical coordinates


Vectors and the Geometry of The students will be - Determine the cross product - in-class problems 2.5.11.A; Space able to determine the of two vectors in space - in-class discussions 2.5.11.B; - Component form of a vector cross product of two - Use the cross product to find - small group work 2.5.11.C; 2.5.11.D - Vector operations vectors in space. a unit vector that is orthogonal - take home evaluation - Standard unit vectors to two given vectors problems - Applications of vectors - Utilize the cross product in - in-class evaluation - Three-dimensional rectangular geometric applications problems coordinate system - Determine the volume of a - Vectors in space parallelepiped using the triple - Applications of three-dimensional scalar product vectors - Dot product of two vectors - Angle between two vectors - Direction cosines - Projection of a vector - Cross product of two vectors - Triple scalar product - Parametric equations for a line in space - Distance between points, lines and planes in space - Cylindrical surfaces - Quadratic surfaces - Surfaces of revolution - Cylindrical and spherical coordinates


Vectors and the Geometry of The students will be - Convert from cylindrical to - in-class problems 2.5.11.A; Space able to use cylindrical rectangular coordinates - in-class discussions 2.5.11.B; - Component form of a vector and spherical - Convert from rectangular to - small group work 2.5.11.C; 2.5.11.D - Vector operations coordinates to cylindrical coordinates - take home evaluation - Standard unit vectors represent surfaces in - Determine equations in problems - Applications of vectors space. cylindrical coordinates for - in-class evaluation - Three-dimensional rectangular surfaces given in rectangular problems coordinate system equations - Vectors in space - Determine a rectangular - Applications of three-dimensional equation for a graph vectors represented by a cylindrical - Dot product of two vectors equation - Angle between two vectors - Determine an equation is - Direction cosines spherical coordinates for a - Projection of a vector surface represented by a - Cross product of two vectors rectangular equation - Triple scalar product - Parametric equations for a line in space - Distance between points, lines and planes in space - Cylindrical surfaces - Quadratic surfaces - Surfaces of revolution - Cylindrical and spherical coordinates


Vectors and the Geometry of The students will be - Find the dot product of two - in-class problems 2.5.11.A; Space able to evaluate the dot vectors - in-class discussions 2.5.11.B; - Component form of a vector product of two - Determine the angle between - small group work 2.5.11.C; 2.5.11.D - Vector operations vectors and apply the two vectors - take home evaluation - Standard unit vectors concept of a dot - Determine the directions problems - Applications of vectors product to real life cosines and angles for a - in-class evaluation - Three-dimensional rectangular applications. vector problems coordinate system - Find a vect5or component of - Vectors in space a given vector that is - Applications of three-dimensional orthogonal to a second vector vectors - Decompose a vector into - Dot product of two vectors vector components - Angle between two vectors - Find a force using vector - Direction cosines projections and dot products - Projection of a vector - Find the work done in moving - Cross product of two vectors an object using vector - Triple scalar product projections and dot products - Parametric equations for a line in space - Distance between points, lines and planes in space - Cylindrical surfaces - Quadratic surfaces - Surfaces of revolution - Cylindrical and spherical coordinates


Vectors and the Geometry of The students will be - Find the distance between - in-class problems 2.5.11.A; Space able to understand two points in space - in-class discussions 2.5.11.B; - Component form of a vector space coordinates and - Find the standard equation of - small group work 2.5.11.C; 2.5.11.D - Vector operations work with vectors in a sphere given the endpoints - take home evaluation - Standard unit vectors space. of a diameter problems - Applications of vectors - Determine the component - in-class evaluation - Three-dimensional rectangular form of a vector in space problems coordinate system - Determine if two space - Vectors in space vectors are parallel - Applications of three-dimensional - Find the resultant force given vectors three applies forces in space - Dot product of two vectors - Angle between two vectors - Direction cosines - Projection of a vector - Cross product of two vectors - Triple scalar product - Parametric equations for a line in space - Distance between points, lines and planes in space - Cylindrical surfaces - Quadratic surfaces - Surfaces of revolution - Cylindrical and spherical coordinates

Instructional Unit Vector-Valued Functions Advanced Calculus & Number Theory #322 Unit Content Objective Performance Indicator Performance Task State Standards Code:

Vector-Valued Functions The students will be - Sketch a plane curve - in-class problems 2.5.11.A; - Space curves given by a vector- able to analyze and represented by a vector-valued - in-class discussions 2.5.11.B; valued function sketch a space curve function - small group work 2.5.11.C; 2.5.11.D - Limits and continuity related to given by a vector- - Sketch a space curve - take home evaluation vector-valued functions valued function. represented by a vector-valued problems - Derivative of a vector-valued function - in-class evaluation function - Represent a graph and find a problems - Integral of a vector-valued vector-valued function to function represent the graph - Velocity and acceleration related - Discuss the continuity of a to vector-valued functions vector-valued function through - Unit tangent vector limits - Tangential and normal components of acceleration - Arc length of a space curve - Curvature of a curve at a point on a curve


Vector-Valued Functions The students will be - Differentiate a vector-valued - in-class problems 2.5.11.A; - Space curves given by a vector- able to differentiate function - in-class discussions 2.5.11.B; valued function and integrate a vector- - Evaluate higher order - small group work 2.5.11.C; 2.5.11.D - Limits and continuity related to valued function. derivatives of a vector-valued - take home evaluation vector-valued functions function problems - Derivative of a vector-valued - Determine intervals over - in-class evaluation function which a curve is smooth problems - Integral of a vector-valued - Apply properties of a function derivative to vector-valued - Velocity and acceleration related functions to vector-valued functions - Integrate a vector-valued - Unit tangent vector function - Tangential and normal - Evaluate a definite integral of components of acceleration a vector-valued function - Arc length of a space curve - Determine the antiderivative - Curvature of a curve at a point of a vector-valued function that on a curve satisfies given conditions -


Vector-Valued Functions The students will be - Find the velocity vector, - in-class problems 2.5.11.A; - Space curves given by a vector- able to describe the speed, and acceleration vector - in-class discussions 2.5.11.B; valued function velocity and of a curve given by a vector- - small group work 2.5.11.C; 2.5.11.D - Limits and continuity related to acceleration valued function - take home evaluation vector-valued functions associated with a - Sketch the velocity and problems - Derivative of a vector-valued vector-valued function acceleration vectors along the - in-class evaluation function and analyze projectile plane curve problems - Integral of a vector-valued motion. - Use integration techniques to function find a position function given - Velocity and acceleration related the acceleration vector to vector-valued functions - Use vector-valued functions - Unit tangent vector to describe the path of a - Tangential and normal projectile in space components of acceleration - Arc length of a space curve - Curvature of a curve at a point on a curve


Vector-Valued Functions The students will be - Find the unit tangent vector to - in-class problems 2.5.11.A; - Space curves given by a vector- able to find a unit a curve given by a vector- - in-class discussions 2.5.11.B; valued function tangent vector at a valued function - small group work 2.5.11.C; 2.5.11.D - Limits and continuity related to point on a space curve - Determine the tangent line at - take home evaluation vector-valued functions and find the tangential a point on a curve given by a problems - Derivative of a vector-valued and normal vector-valued function - in-class evaluation function components of - Determine the principal unit problems - Integral of a vector-valued acceleration. normal vector of a curve given function by a vector-valued function - Velocity and acceleration related - Determine the tangential and to vector-valued functions normal components of - Unit tangent vector acceleration of a position - Tangential and normal vector components of acceleration - Find the tangential component - Arc length of a space curve of acceleration for the position - Curvature of a curve at a point vector for projectile motion on a curve


Vector-Valued Functions The students will be - Determine the arc length of a - in-class problems 2.5.11.A; - Space curves given by a vector- able to find the arc curve in space - in-class discussions 2.5.11.B; valued function length of a space - Determine the arc length of a - small group work 2.5.11.C; 2.5.11.D - Limits and continuity related to curve and find the helix - take home evaluation vector-valued functions curvature of a curve at - Find the arc length function problems - Derivative of a vector-valued a point on the curve. for a line segment given by a - in-class evaluation function vector-valued function problems - Integral of a vector-valued - Determine the curvature of a function circle - Velocity and acceleration related - Determine the curvature of a to vector-valued functions space curve - Unit tangent vector - Determine the curvature of a - Tangential and normal function given in rectangular components of acceleration coordinates - Arc length of a space curve - Find the tangential and normal - Curvature of a curve at a point components of acceleration on a curve - Determine the frictional force required to keep an object from sliding off a curved surface

Advanced Calculus & Number Theory...

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