ModelCourses/Differential Calculus
Revision as of 11:57, 10 March 2013 by Anneke Bart (talk | contribs)
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- Estimate the slope at a point given a graph
- Estimate the limit of the difference quotient
- Find the average rate of change
- Set 10 The Derivative at a Point
Students will be able to- Determine from a graph where the derivative is greatest, least or zero
- Find the derivative using the limit definition
- Analyze the difference between a secant and a tangent line approximation
- Set 11 The Derivative Function
Students will be able to- Estimate the derivative given a graph
- Estimate the derivative given a table
- Determine the graph of the derivative function
- Set 12 Interpretations of the Derivative
Students will be able to- Interpret the derivative in specific real world settings.
- Set 13 The Second Derivative
Students will be able to- Estimate the second derivative at a point
- Determine the sign of the second derivative form a graph
- Connect the second derivative to concavity
- Set 14 Differentiability
Students will be able to- Determine where a graph is differentiable given a graph
- Connect the concepts of continuity and differentiability
- 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