ModelCourses/Differential Calculus

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
• Set 23 Linear Approximation and the Derivative
• Set 24 Theorems about Differentiable Functions
• Set 25 Using First and Second Derivatives
• Set 26 Optimization
• Set 27 Families of Functions
• Set 28 Optimization Geometry and Modeling
• Set 29 Applications to Marginality (optional)
• Set 30 Rates and Related Rates
• Set 31 L’Hopital’s Rule, Growth, and Dominance
• Set 32 Parametric Equations
• Set 33 Introduction to the definite integral
• Set 34 The Definite Integral
• Set 35 The Fundamental Theorem and Interpretations
• Set 36 Theorems about Definite Integrals
• Set 37 Antiderivatives Graphically and Numerically
• Set 38 Constructing Antiderivatives Analytically
• Set 39 Differential Equations
• Set 40 Second Fundamental Theorem of Calculus
• Set 41 The Equations of Motion