ModelCourses/Differential Calculus
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Contents
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 economy)
- Find maxima and minima in the numerical setting (from economy)
- 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
