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
  • Find antiderivatives symbolically for sums and differences of elementary functions
  • 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