# Calculus I Repository

## Purpose of Website

As part of the Active Learning Initiative, the mathematics department decided in 2017 to review its biggest Calculus I course MATH1110. The goal was to transform this large multi-section course so that the standard teaching method would focus on the concepts (rather than the procedures) and include a lot of active learning. In order to achieve this goal, learning objectives and worksheets for pre- and in-class activities were created. They have been used since spring 2018.While they are not tied to a particular textbook (except for a few references to theorems), they overall follow the structure of Thomas’ Calculus, early transcendental, 14 edition.

One of the goals when creating these worksheets was to make their use as easy as possible for the instructors. To do so, the files are organized by chapters (the downloadable zip files). In each of them, you will find a list of sections/topics that contain:

• an info sheet (“info”) that contains the learning objectives of the chapter as well as the goals of each activity (pre-class and in-class),
• the pre-class activities (“pc”),
• the worksheet or in-class activities (“ws”),
• some supplemental activities (“sup”).
There are some instructor files with comments for the instructor and student files that can be printed out right away. Moreover, all tex files are provided.

## How To Use The Repository

We have divided the repository into Chapter which roughly follow the chapters of Thomas. Within each Chapter archive is a sequence of Topics comprised of pre-class worksheets, in-class worksheets, supplemental activities, and active reviews. Prefacing each Topic is an info-sheet detailing each each worksheet and activity with descriptions and suggestions for the instructor. Each of these worksheets and activities are represented in the archive as both compilable TeX files and printable PDFs.

## Repository Contents

### Chapter 1 - Functions

Topic Learning Objectives
Topic 1: Functions
• list the pieces of information that define a function
• determine when two functions are equal or not by comparing their algebraic definitions
• draw the graph of important functions such as $$\sin x, \cos x, \tan x$$, quadratic functions, absolute value of $$x$$
• explain what the inverse function of a given function is
• correctly use the conventions of dots and empty circles when drawing the graph of a function
• shift and compress a function vertically or horizontally
• compute the inverse function of simple'' polynomials
Topic 2: Function Review
• students can identify and solve algebraic problems involving polynomials, trigonometric functions, exponentials, logarithms, and their function properties
• students can clearly explain in their own words and diagrams to their peers their solution to a mathematical problem

### Chapter 2 - Limits

Topic Learning Objectives
Topic 1: Limits
• explain in their own words the definition of a limit and one-sided limit
• compute limits and one-sided limits using limit laws for polynomial and rational functions as well as common methods studied in class
• give examples that illustrate the different cases where a limit or a one-sided limit fails to exist
• know important limits such as $$\displaystyle \lim_{x \to 0} \frac{\sin x}{x}$$ and $$\displaystyle \lim_{x \to 0} \sin (1/x)$$
• appropriately use the squeeze theorem to compute limits. This includes being able to:
• give the statement of the theorem
• recognize situations in which the theorem applies and can be useful
• follow a procedure to use the theorem in order to compute the limit of a given function
• explain the relationship between the existence of a limit and one-sided limits
Topic 2: Continuity
• explain with their own words the definition of the continuity of a function
• list the different types of discontinuities that a function may have
• list examples of continuous and discontinuous functions
• prove the continuity of a function at a given point using the definition and/or the theorems
• use the Intermediate Value Theorem (IVT) to show the existence of solutions to given equations. This includes being able to:
• state the theorem
• recognize when we can apply the theorem
• follow a procedure to show the existence of a root using the theorem
Topic 3: Asymptotes
• compute limits at infinity using limit laws
• explain in words what an asymptote is
• compute the equation of a horizontal or vertical asymptote
Topic 4: Continuity Review
• differentiate between the concept of a limit and of continuity
• identify and use the definition of continuity to verify continuity of functions
• provide examples of discontinuous functions and explain in words why these functions have discontinuities

### Chapter 3 - Derivatives

Topic Learning Objectives
Topic 1: Derivatives
• explain in words what the definition of the derivative means
• use the definition of the derivative to compute the derivative of a function
• use the definition of the derivative to compute the slope and equation of the tangent line at a given point
• given the graph of a function, qualitatively draw the graph of its derivative, and conversely, given two graphs recognize the graph of a function and of its derivative
• using the definition, determine on which intervals a function is differentiable and on which it is not. This implies being able to compute one-sided derivatives and be able to determine when it does not exist
• list the cases where a function is not differentiable and draw the corresponding graphs
• recognize on a graph where a function fails to be differentiable
Topic 2: Differentiation Rules
• correctly use the differentiation rules presented in the section (derivative of a constant, power rule, constant multiple rule, sum rule, natural exponential rule, product rule, quotient rule)
• compute the equation of the tangent line at a given point using these rules
Topic 3: Trigonometric Derivatives
• compute the derivative of trigonometric functions $$sin x, cos x, tan x$$
Topic 4: Chain Rule
• recognize when the chain rule is needed
• appropriately apply the chain rule to compute derivatives of functions
Topic 5: Implicit Differentiation
• explain the interest of using implicitly defined functions
• explain how to implicitly differentiate functions and when it applies
• recognize when implicit differentiation applies and use it correctly to differentiate implicitly defined functions
• use this process to compute the equations of tangent or normal lines to a given curve
Topic 6: Derivatives of Inverse Functions
• compute the derivative of the inverse of a function
• compute the derivatives of $$\ln (x), a^x, \log_a (x)$$
• explain in mathematical terms why the derivative of the inverse function is the reciprocal of the derivative
Topic 7: Related Rates
• build an appropriate mathematical model for word problems. This includes:
• assign variables to appropriate quantities
• identify which numerical information is relevant and/or needed
• relate the variables using appropriate equations taking into account the numerical information provided
• solve word problems using the differentiation techniques seen earlier in the term
• for a given problem, clearly explain with words, mathematical symbols and equations their reasoning, in particular, what is known, what we are looking for and the steps of the procedure to solve the question
Topic 8: Linearization
• explain in words what the process of linearization consist of and why it is interesting
• use the linear approximation of a function at a given point to compute an approximate value of the function
• using the graph of a function explain if a linear approximation gives an underestimate or overestimate of the true value of the function
• explain in general terms what the conditions are for the process to give a reasonable'' approximation

### Chapter 4 - Applications

Topic Learning Objectives
Topic 1: Indeterminate Forms and L'Hospital's Rule
• explain in words what an indeterminate form is
• explain what L'Hospital's rule is, when we can use it, and what kinds of limits we can compute with it
• correctly use L'Hospital's rule to compute limits
Topic 2: Extreme Value Theorem
• define the notions of local/absolute min and max, and critical point
• explain the extreme value theorem (in particular its hypotheses) and exhibit counter-examples'', i.e. functions that don't have an absolute min or max
• find the absolute min and max of a continuous function on a closed interval $$[a, b]$$
Topic 3: Mean Value Theorem
• explain in words the Mean Value Theorem as well as its corollaries
• explain the importance of the Mean Value Theorem
• use the Mean Value Theorem to prove properties of a function based on information about its derivative
Topic 4: The First Derivative Test
• use the first derivative test to determine the nature of an extremum
Topic 5: Concavity and Curve Sketching
• explain the difference between concave up and concave down
• use the second derivative of a function to determine:
• on what interval(s) a curve is concave up, respectively concave down
• where the inflection points are
• the nature of a local extremum
• qualitatively sketch the graph of a function using the information provided by the first and second derivatives
• given the algebraic expression of a function as well as its graph (e.g. using a graphing software), qualitatively verify that the curve corresponds to the given function
Topic 6: Optimization
• build an appropriate mathematical model for word problems. This includes:
• assign variables to appropriate quantities
• identify which numerical information is relevant and/or needed
• relate the variables using appropriate equations taking into account the numerical information provided
• solve word problems using the differentiation techniques seen earlier in the term and determine the optimal solution
• for a given problem, clearly explain with words, mathematical symbols and equations their reasoning, in particular, what is known, what we are looking for and the steps of the procedure to solve the question
Topic 7: Antiderivatives
• compute the antiderivative of simple'' functions with or without initial value
• verify that a function is the antiderivative of another function

### Chapter 5 - Integrals

Topic Learning Objectives
Topic 1: Area and Estimating with Finite Sums
• explain how the process of estimating an area with finite sums work and use it
• use this process to estimate lengths, speeds and the average value of a function
Topic 2: Sigma Notation and Limits of Finite Sums
• use the sigma notation for sums and compute sums
• explain in words what the Riemann sums of a given function are
Topic 3: The Definite Integral
• explain in words what the definite integral is, both in terms of Riemann sums and of area under a curve
• give examples of functions that are integrable and functions that are not
• compute definite integrals using known areas and rules satisfied by definite integrals
• justify the rules satisfied by definite integrals using area arguments
Topic 4: The Fundamental Theorem of Calculus
• explain in words the two parts of the fundamental theorem of calculus and why it is important
• mathematically state the two parts of the fundamental theorem of calculus
• use the fundamental theorem of calculus to compute definite integrals

## Contact Information

 Steve Bennoun Active Learning Lecturer Department of Mathematics Cornell University s 'dot' bennoun 'at' cornell 'dot' edu Collaborators include: Tara Holm and Matt Hin