# ModelCourses/Differential Calculus

## Contents

## General Description

- Freshman level differential calculus course
- Pre-requisite: Pre-Calculus
- This homework set is based on a course taught at Saint Louis University

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

Students will:

- Review properties of elementary functions
- Understand the definition of continuity
- Determine if a function is continuous or not
- Explore limits: both concept and computation
- Symbolically compute derivatives
- Find derivatives graphically and numerically
- Solve optimization problems
- Solve related rates problems
- Understand local linearity
- Understand the geometric interpretation of the integral
- Be able to compute simple Riemann Sums
- Integrate basic functions
- Use the fundamental theorem of calculus

## Problem sets

### Use of Problem Sets

The problem sets were assembled to allow for personalization by individual faculty. The topics covered are fairly standard in a first semester reform calculus course, but faculty can rearrange the topics and delete any sections they do not wish to cover, or wish to assess by other means. The names of the problem sets are meant to be descriptive and the learning objectives will help you evaluate if the set should be included or not.

### Download the problem sets

A copy of the course can be found at Differential Calculus at the MAA website

The course can be downloaded here <insert link>.

To use the files remove the .txt from the end. The .tgz can be added. This file can now be directly uploaded into your own course:

- go to Filemanager
- Upload the file
- etc <provide enough detail to allow for easy installation by anyone>

### Description of 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