ModelCourses/Differential Calculus

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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

Review of Functions and Their Properties

  • 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

Continuity and Limits

  • 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

Conceptual Introduction to the Derivative

  • 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 function is differentiable given a graph
    • Connect the concepts of continuity and differentiability

Computing Derivatives

  • Set 15 Derivatives of Powers and Polynomials
    Students will be able to
    • Find the derivative symbolically of a power function
    • Find the derivative symbolically of a polynomial
    • Find the derivative after some minor algebraic manipulation
  • Set 16 Derivative of the Exponential Function
    Students will be able to
    • Find the derivative of exponential functions
    • Solve some population problems using the derivative of exponentials
  • Set 17 The Product and Quotient Rules
    Students will be able to
    • Find derivatives of products of functions
    • Find derivatives of quotients of functions
  • Set 18 The Chain Rule
    Students will be able to
    • Find derivatives of compositions of functions
  • Set 19 Derivatives of Trigonometric Functions
    Students will be able to
    • Find derivatives of the basic trig functions (sine, cosine, tangent)
    • Use trigonometric derivatives to solve word problems
  • Set 20 The Chain Rule and Inverse Functions
    Students will be able to
    • Find the derivative of ln(x)
    • Find the derivative of arctan(x)
    • Find the derivative of arcsin(x)
    • Use the derivatives of these functions combined with other differentiation rules
  • Set 21 Implicit Functions
    Students will be able to
    • Find the derivative using implicit differentiation
  • Set 22 Hyperbolic Functions
    Students will be able to
    • Find derivatives of (composite) functions involving sinh(x) and cosh(x)
  • Set 23 Linear Approximation and the Derivative
    Students will be able to
    • Find tangent line approximations
    • Find a formula for the error E(x) in the tangent line approximation
    • Use substitution techniques to find tangent line approximations
    • Use differentials to estimate the (maximum) possible error
  • Set 24 Theorems about Differentiable Functions
    Students will be able to
    • Check the hypotheses of the Mean Value Theorem
    • Use the Racetrack Principle to show that one function is greater than another

Applications of Derivatives

  • Set 25 Using First and Second Derivatives
    Students will be able to
    • Estimate the x-values of any critical points based on graphical data
    • Estimate the x-values of any inflection points based on graphical data
    • Find and classify the critical points
    • Find the inflection points
    • Find all intervals where the function is increasing
    • Find all intervals where the function is decreasing
  • Set 26 Optimization
    Students will be able to
    • Find the exact global maximum and minimum values of a function
  • Set 27 Families of Functions
    Students will be able to
    • Find local extrema of functions written with general constants
    • Find formulas of functions given a general form of the function and some specific data (location of max, min, and/or roots)
  • Set 28 Optimization Geometry and Modeling
    Students will be able to
    • Find maxima and minima in an applied setting
  • Set 29 Applications to Marginality
    Students will be able to
    • Find maxima and minima using functions (from economics)
    • Find maxima and minima in the numerical setting (from economics)
    • Find cost functions
    • Find revenue functions
    • Find marginal cost
    • Find marginal revenue
  • Set 30 Rates and Related Rates
    Students will be able to
    • Find the rate of change in related rates problems
    • Solve applied related rates problems
  • Set 31 L’Hopital’s Rule, Growth, and Dominance
    Students will be able to
    • Determine the limit of a ratio of two functions, given graphical data
    • Find a limit using l'Hopital's rule
  • Set 32 Parametric Equations
    Students will be able to
    • Find the slope of a parametrically defined function
    • Find the speed of a particle
    • Write a parameterization for a curve
    • Find an equation of the tangent line to a parametrically defined function

Introduction to Integration

  • Set 33 Introduction to the definite integral
    Students will be able to
    • Estimate distance traveled based on a table of velocities
    • Find left- and right-hand sums using small values of n
    • Determine if the estimated distance traveled is an underestimate or an overestimate
  • Set 34 The Definite Integral
    Students will be able to
    • Estimate the value of the definite integral given numerical data
    • Use a calculator or computer to find the value of the definite integral
    • Estimate the area between a curve and the x-axis between 2 given x-values
    • Find definite integrals based on graphical data
  • Set 35 The Fundamental Theorem and Interpretations
    Students will be able to
    • Find the units of a specific definite integral
    • Find the average value of a function
    • Estimate the definite integral based on a table of data for f(x)
  • Set 36 Theorems about Definite Integrals
    Students will be able to
    • Given the graph of the derivative determine the relative size of the values of f at given x-values
    • Estimate the area between 2 functions
  • Set 37 Antiderivatives Graphically and Numerically
    Students will be able to
    • Estimate f(x) for given values, given a table for the derivative f′(x)
    • Estimate f(x) for given values, given the graph of the derivative f′(x)
    • Sketch graphs of f and f′ given the graph of f″
  • Set 38 Constructing Antiderivatives Analytically
    Students will be able to
    • Find antiderivatives symbolically for elementary functions and thier sums and differences
    • Find antiderivatives symbolically using simple guess and check procedure
    • Find the exact value between two curves
  • Set 39 Differential Equations
    Students will be able to
    • Find the general solution of a differential equation
    • Find the solution of an initial value problem
    • Solve problems involving position, velocity and acceleration
  • Set 40 Second Fundamental Theorem of Calculus
    Students will be able to
    • Evaluate functions defined in terms of a definite integral (with one variable limit of integration)
    • Construct a function based on the derivative function and an initial value.
    • Find the derivative of an integral
  • Set 41 The Equations of Motion
    Students will be able to
    • Solve problems involving position, velocity and acceleration