Difference between revisions of "ModelCourses/Differential Calculus"
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==Problem sets== |
==Problem sets== |
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− | * '''Set 01 Functions and Change''' Students will be able to |
+ | * '''Set 01 Functions and Change''' <br> Students will be able to |
** Find equations of lines |
** Find equations of lines |
||
** Find equations of perpendicular lines |
** Find equations of perpendicular lines |
||
Line 25: | Line 25: | ||
** Find the domain and range of functions |
** Find the domain and range of functions |
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− | * '''Set 02 Exponential Functions''' Students will be able to |
+ | * '''Set 02 Exponential Functions''' <br> Students will be able to |
** Construct exponential functions based on given numerical data |
** Construct exponential functions based on given numerical data |
||
** Construct exponential functions based on given graphical data |
** Construct exponential functions based on given graphical data |
||
** Find the concavity of a function based on graphical data |
** Find the concavity of a function based on graphical data |
||
− | * '''Set 03 New Functions from Old''' Students will be able to |
+ | * '''Set 03 New Functions from Old''' <br> Students will be able to |
** Evaluate compositions of functions |
** Evaluate compositions of functions |
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** Determine if a function is invertible or not |
** Determine if a function is invertible or not |
||
Line 36: | Line 36: | ||
** Evaluate an inverse function |
** Evaluate an inverse function |
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− | * '''Set 04 Logarithmic Functions''' Students will be able to |
+ | * '''Set 04 Logarithmic Functions''' <br> Students will be able to |
** Solve exponential equation using logarithms |
** Solve exponential equation using logarithms |
||
** Find doubling times |
** Find doubling times |
||
** Identify the growth rate of an exponential function |
** Identify the growth rate of an exponential function |
||
− | * '''Set 05 Trigonometric Functions''' Students will be able to |
+ | * '''Set 05 Trigonometric Functions''' <br> Students will be able to |
** Find the period and amplitude of trigonometric functions |
** Find the period and amplitude of trigonometric functions |
||
** Find the equation of a function based on the graph |
** Find the equation of a function based on the graph |
||
** Apply concepts to problems in an applied setting |
** Apply concepts to problems in an applied setting |
||
− | * '''Set 06 Powers, Polynomials, and Rational Functions''' Students will be able to |
+ | * '''Set 06 Powers, Polynomials, and Rational Functions''' <br> Students will be able to |
** Find horizontal asymptotes |
** Find horizontal asymptotes |
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** Find vertical asymptotes |
** Find vertical asymptotes |
||
Line 52: | Line 52: | ||
** Apply concepts to problems in an applied setting |
** Apply concepts to problems in an applied setting |
||
− | * '''Set 07 Introduction to Continuity''' Students will be able to |
+ | * '''Set 07 Introduction to Continuity''' <br> Students will be able to |
** Apply the Intermediate Value Theorem |
** Apply the Intermediate Value Theorem |
||
** Determine how to find parameters so that a piece-wise defined function is continuous |
** Determine how to find parameters so that a piece-wise defined function is continuous |
||
** Determine where a function is continuous |
** Determine where a function is continuous |
||
− | * '''Set 08 Limits ''' Students will be able to |
+ | * '''Set 08 Limits ''' <br> Students will be able to |
** Use a graph to estimate limits |
** Use a graph to estimate limits |
||
** Use a table to estimate limits |
** Use a table to estimate limits |
||
** Determine the limit of sums, differences, products and quotients of two functions if the limits of the individual functions are known |
** Determine the limit of sums, differences, products and quotients of two functions if the limits of the individual functions are known |
||
− | * '''Set 09 Introduction to the derivative''' Students will be able to |
+ | * '''Set 09 Introduction to the derivative''' <br> Students will be able to |
** |
** |
||
− | * '''Set 10 The Derivative at a Point''' Students will be able to |
+ | * '''Set 10 The Derivative at a Point''' <br> Students will be able to |
** |
** |
||
− | * '''Set 11 The Derivative Function''' Students will be able to |
+ | * '''Set 11 The Derivative Function''' <br> Students will be able to |
** |
** |
||
− | * '''Set 12 Interpretations of the Derivative''' Students will be able to |
+ | * '''Set 12 Interpretations of the Derivative''' <br> Students will be able to |
** |
** |
||
− | * '''Set 13 The Second Derivative''' Students will be able to |
+ | * '''Set 13 The Second Derivative''' <br> Students will be able to |
** |
** |
||
− | * '''Set 14 Differentiability''' Students will be able to |
+ | * '''Set 14 Differentiability'''<br> Students will be able to |
** |
** |
||
− | * '''Set 15 Derivatives of Powers and Polynomials''' Students will be able to |
+ | * '''Set 15 Derivatives of Powers and Polynomials''' <br> Students will be able to |
** |
** |
||
− | * '''Set 16 Derivative of the Exponential Function''' Students will be able to |
+ | * '''Set 16 Derivative of the Exponential Function''' <br> Students will be able to |
** |
** |
||
− | * '''Set 17 The Product and Quotient Rules''' Students will be able to |
+ | * '''Set 17 The Product and Quotient Rules''' <br> Students will be able to |
** |
** |
||
− | * '''Set 18 The Chain Rule''' Students will be able to |
+ | * '''Set 18 The Chain Rule''' <br> Students will be able to |
** |
** |
||
− | * '''Set 19 The Trigonometric Functions''' Students will be able to |
+ | * '''Set 19 The Trigonometric Functions''' <br> Students will be able to |
** |
** |
||
− | * '''Set 20 The Chain Rule and Inverse Functions''' Students will be able to |
+ | * '''Set 20 The Chain Rule and Inverse Functions''' <br> Students will be able to |
** |
** |
||
− | * '''Set 21 Implicit Functions''' Students will be able to |
+ | * '''Set 21 Implicit Functions''' <br> Students will be able to |
** |
** |
||
− | * Set 22 Hyperbolic Functions |
+ | * '''Set 22 Hyperbolic Functions''' <br> Students will be able to |
− | * |
+ | ** |
− | + | ||
− | * Set |
+ | * '''Set 23 Linear Approximation and the Derivative''' <br> Students will be able to |
− | * |
+ | ** |
− | + | ||
− | * Set |
+ | * '''Set 24 Theorems about Differentiable Functions ''' <br> Students will be able to |
− | * |
+ | ** |
− | + | ||
− | * Set |
+ | * '''Set 25 Using First and Second Derivatives''' <br> Students will be able to |
− | * |
+ | ** |
− | + | ||
− | * Set |
+ | * '''Set 26 Optimization''' <br> Students will be able to |
− | * |
+ | ** |
− | + | ||
− | * Set |
+ | * '''Set 27 Families of Functions''' <br> Students will be able to |
− | * |
+ | ** |
− | + | ||
− | * Set |
+ | * '''Set 28 Optimization Geometry and Modeling''' <br> Students will be able to |
− | * |
+ | ** |
+ | |||
+ | * '''Set 29 Applications to Marginality''' <br> Students will be able to |
||
+ | ** |
||
+ | |||
+ | * '''Set 30 Rates and Related Rates''' <br> Students will be able to |
||
+ | ** |
||
+ | |||
+ | * '''Set 31 L’Hopital’s Rule, Growth, and Dominance''' <br> Students will be able to |
||
+ | ** |
||
+ | |||
+ | * '''Set 32 Parametric Equations ''' <br> Students will be able to |
||
+ | ** |
||
+ | |||
+ | * '''Set 33 Introduction to the definite integral''' <br> Students will be able to |
||
+ | ** |
||
+ | |||
+ | * '''Set 34 The Definite Integral''' <br> Students will be able to |
||
+ | ** |
||
+ | |||
+ | * '''Set 35 The Fundamental Theorem and Interpretations''' <br> Students will be able to |
||
+ | ** |
||
+ | |||
+ | * '''Set 36 Theorems about Definite Integrals ''' <br> Students will be able to |
||
+ | ** |
||
+ | |||
+ | * '''Set 37 Antiderivatives Graphically and Numerically''' <br> Students will be able to |
||
+ | ** |
||
+ | |||
+ | * '''Set 38 Constructing Antiderivatives Analytically''' <br> Students will be able to |
||
+ | ** |
||
+ | |||
+ | * '''Set 39 Differential Equations''' <br> Students will be able to |
||
+ | ** |
||
+ | |||
+ | * '''Set 40 Second Fundamental Theorem of Calculus''' <br> Students will be able to |
||
+ | ** |
||
+ | |||
+ | * '''Set 41 The Equations of Motion''' <br> Students will be able to |
||
+ | ** |
Revision as of 16:50, 9 March 2013
General Description
- Freshman level differential calculus course
- Pre-requisite: Pre-Calculus
Possible textbooks include, but are not limited to:
- Deborah Hughes-Hallett, Andrew Gleason, William McCallum et al. Calculus, Fifth Edition. New York, NY: John Wiley & Sons, Inc., 2009. (or a later edition)
Course Objectives
- Properties of Elementary Functions
- Introduction to continuity
- Introduction to limits
- Explore differentiation from graphical, numerical and analytical viewpoints
- Optimization and modeling
- The definite integral
- Explore anti-derivatives from graphical, numerical and analytical viewpoints.
- Fundamental Theorem of Calculus
Problem sets
- Set 01 Functions and Change
Students will be able to- Find equations of lines
- Find equations of perpendicular lines
- Find equations of parallel lines
- Find the domain and range of functions
- Set 02 Exponential Functions
Students will be able to- Construct exponential functions based on given numerical data
- Construct exponential functions based on given graphical data
- Find the concavity of a function based on graphical data
- Set 03 New Functions from Old
Students will be able to- Evaluate compositions of functions
- Determine if a function is invertible or not
- Interpret the value of an inverse function
- Evaluate an inverse function
- Set 04 Logarithmic Functions
Students will be able to- Solve exponential equation using logarithms
- Find doubling times
- Identify the growth rate of an exponential function
- Set 05 Trigonometric Functions
Students will be able to- Find the period and amplitude of trigonometric functions
- Find the equation of a function based on the graph
- Apply concepts to problems in an applied setting
- Set 06 Powers, Polynomials, and Rational Functions
Students will be able to- Find horizontal asymptotes
- Find vertical asymptotes
- Find the equation of a polynomial given a graph
- Apply concepts to problems in an applied setting
- Set 07 Introduction to Continuity
Students will be able to- Apply the Intermediate Value Theorem
- Determine how to find parameters so that a piece-wise defined function is continuous
- Determine where a function is continuous
- Set 08 Limits
Students will be able to- Use a graph to estimate limits
- Use a table to estimate limits
- Determine the limit of sums, differences, products and quotients of two functions if the limits of the individual functions are known
- Set 09 Introduction to the derivative
Students will be able to
- Set 10 The Derivative at a Point
Students will be able to
- Set 11 The Derivative Function
Students will be able to
- Set 12 Interpretations of the Derivative
Students will be able to
- Set 13 The Second Derivative
Students will be able to
- Set 14 Differentiability
Students will be able to
- Set 15 Derivatives of Powers and Polynomials
Students will be able to
- Set 16 Derivative of the Exponential Function
Students will be able to
- Set 17 The Product and Quotient Rules
Students will be able to
- Set 18 The Chain Rule
Students will be able to
- Set 19 The Trigonometric Functions
Students will be able to
- Set 20 The Chain Rule and Inverse Functions
Students will be able to
- Set 21 Implicit Functions
Students will be able to
- Set 22 Hyperbolic Functions
Students will be able to
- Set 23 Linear Approximation and the Derivative
Students will be able to
- Set 24 Theorems about Differentiable Functions
Students will be able to
- Set 25 Using First and Second Derivatives
Students will be able to
- Set 26 Optimization
Students will be able to
- Set 27 Families of Functions
Students will be able to
- Set 28 Optimization Geometry and Modeling
Students will be able to
- Set 29 Applications to Marginality
Students will be able to
- Set 30 Rates and Related Rates
Students will be able to
- Set 31 L’Hopital’s Rule, Growth, and Dominance
Students will be able to
- Set 32 Parametric Equations
Students will be able to
- Set 33 Introduction to the definite integral
Students will be able to
- Set 34 The Definite Integral
Students will be able to
- Set 35 The Fundamental Theorem and Interpretations
Students will be able to
- Set 36 Theorems about Definite Integrals
Students will be able to
- Set 37 Antiderivatives Graphically and Numerically
Students will be able to
- Set 38 Constructing Antiderivatives Analytically
Students will be able to
- Set 39 Differential Equations
Students will be able to
- Set 40 Second Fundamental Theorem of Calculus
Students will be able to
- Set 41 The Equations of Motion
Students will be able to