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 multisection 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 inclass 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 (preclass and inclass),
 the preclass activities (“pc”),
 the worksheet or inclass 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.
We would love to hear feedback on these worksheets and activities, so feel free to contact us (email below)! We are also happy to answer any question about this project.
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 preclass worksheets, inclass worksheets, supplemental activities, and active reviews. Prefacing each Topic is an infosheet 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
Download Chapter 1 Archive (.ZIP)
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
Download Chapter 2 Archive (.ZIP)
Topic 
Learning Objectives 
Topic 1: Limits 
 explain in their own words the definition of a limit and onesided limit
 compute limits and onesided 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 onesided 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 onesided 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
Download Chapter 3 Archive (.ZIP)
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 onesided 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
Download Chapter 4 Archive (.ZIP)
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 ``counterexamples'', 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
Download Chapter 5 Archive (.ZIP)
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
