Calculus I

Here is a collection of activities and worksheets for Calculus I. The activities and section numbers follow the numbering in Stewart, Calculus, 8th edition, but the material is new, and the activities do not rely on specific content from Stewart. Links take you to Word versions of the files so that you can make changes for your own local conditions. If links are broken, please let me know.

I used these activities in Math 1310 at Bowling Green State University in Summer 2020. After using each activity, I revised the activity to be more clear to students and to ask questions that guided students more reliably toward the right thought processes. I will be happy to revise the activities so that they work better, please write to me at zirbel@bgsu.edu with suggestions.

One request: do not post or distribute solutions to students. That way, more people can use these activities, either as activities, for ungraded assessments, or for graded assignments.

Here is how I used these activities in Summer 2020.

Some activities are designed so that students can work on them without any lecture before, these are generally marked as "ARL" for "Activity which Replaces a Lecture". Others are simply worksheets and these are generally marked as "WS". Some are hard to classify and so don't have an annotation like that.
After giving whatever introduction was needed, I organized students into groups and asked them to work together on each activity. In an online meeting, I asked students to take turns sharing their screen and writing in real time, so that students in each group could all be working on the same thing at the same time.
Early in the semester, it's good to assign the groups something to talk about, for instance, to introduce themselves, tell what they are studying, what classes they are taking, etc. The more you can get students talking with each other, the more comfortable they will be asking questions in class, and the more likely they will be to come to class.
While students work in groups, I went from group to group to see how they were doing, make sure they were on track, answering any questions, and asking questions of the person who is sharing their screen and the people who are not sharing their screens. This does pretty well to make sure that all students are participating and staying on track.All students were required to turn in their own solutions to the activity, using their own thoughts that went beyond the group work.
A danger here is that one student does all the work and the others don't help but just write down the solutions, right or wrong. It's important to train the groups to work well together. It helps to have a leader, who is in charge of making sure that people take turns sharing their screen. Choose a different leader day to day or week to week, for example, by designating the person who is taking the most/least credit hours, the person who is taking the highest/lowest level class, whatever. This gets the students talking with each other, and that's always good.
I think it's a good idea to reassign the groups every two weeks. One week is too short. Two weeks lets students get to know other students in the same class. I think it's helpful to assign them randomly, so weaker students don't latch on to a stronger student and just write down what they write.

Here are links to the activities and brief descriptions.

2.0_Algebra_review_ARL reviews algebra skills that are important for calculating, rewriting, and simplifying difference quotients. This "just in time" review should be given at the start of the semester, with a quick turnaround so you can see how weak or strong their algebra skills are. What they write may shock you! 2 pages.
2.0_Equation_of_line_in_calculus_form_ARL introduces a somewhat new way to write the equation of a line, as y=y_0+m(x-x_0). Why? The number one reason is that when students know how to calculate derivatives, they can write the equation of the tangent line as y=f(a)+f'(a)(x-a) and won't have to do arithmetic, which they often get wrong. Second reason is that this is the start of the Taylor series for f near x=a; they will be more ready to write y=f(a)+f'(a)(x-a)+(1/2)f''(a)(x-a)^2 and add more terms. The idea of an "anchor point" is also useful when graphing quadratics like y=3+(x-5)^2 and absolute values like y=3+|x-5|. 2 pages.
2.1_Calculating_slopes_of_secant_lines_ARL guides students to calculate slopes of secant lines numerically, using points that are closer and closer together, and to start thinking about using a limiting operation to find the slope of the tangent line. Graphs show how a "curve" looks like a straight line when you zoom in. The last page outlines how to do this algebraically. This is a good first-day activity. The activity will take more than one day to complete. 4 pages.
2.1_Average_rate_of_change_CO2_ ARL has students calculate average rates of change of the level of carbon dioxide in the atmosphere, from 1959 to 2019. 10-year average rates of change have increased since 1959, and a graph at the end helps them understand what that looks like. 2 pages.
2.1_Slopes_of_2_to_the_x_WS has students approximate the slope of the tangent lines for y=2^x at several values of x to many decimal places. The function y=2^x and ln(x) features in several activities, because it is interesting to investigate numerically. Maybe by the end of the semester they will recognize the number 0.69314718... The activity makes clear that the slope of the tangent line at x depends on the value of x. 2 pages.
2.2_Investigate_special_limits_numerically_WS has students use calculators or Wolfram Alpha to investigate two limits numerically. 1 page.
2.2_Limits_and_rate_of_convergence_ARL has students investigate limits numerically, with special emphasis on how quickly the decimal digits get "locked in". This helps students learn that different functions (x^2, x^3, square root of x, etc.) converge to their limits at different rates, which is an important lesson from calculus class. A later activity covers the same idea for limits as x goes to infinity. 2 pages.
2.2_Identifying_limits_from_a_graph_WS is a worksheet for students to read a graph with discontinuities and asymptotes and identify one sided and two-sided limits. 1 page.
2.3_Using_limit_laws_WS guides students through the evaluation of limits using algebra and limit laws. It is very helpful for students to be able to identify, step by step, what limit law they are using. In this activity, they need to use one limit law per step, but in later activities they can combine steps. The last three limits use algebra to resolve 0/0 forms. 2 pages.
2.4_Algebra_review_ARL covers a number of algebra topics that students need when taking limits as x goes to infinity, when calculating difference quotients with powers, and when solving inequalities, for example, in elementary delta-epsilon proofs. Students really, really struggle with 5d, 5e, 5f, 9b, 9c, 10a, 10b. That's unfortunate, because these are important in a number of later problems. 2 pages.
2.4_Using_the_definition_to_check_limits_ARL is a modest attempt to get students to work with the inequalities you would need in delta-epsilon proofs. Since we do not cover these deeply, there are a few purely numerical examples (equivalent to "What is delta when epsilon = 0.0001?") and a few delta-epsilon examples. If you find a good way to get students to be able to solve these, without spending days on it in class, let me know! 2 pages.
2.5_Checking_continuity_piecewise_function_WS has students use algebra to check limits of a function defined in 3 parts. They struggle to use the right formula for limits from the right and limits from the left. It also asks them to show that the function is continuous at all points in certain intervals, and they struggle to write an argument for a generic value x=a. Good challenges here for all students. 2 pages.
2.6_Limits_at_infinity_numerical_and_with_inequalities_ARL has students investigate limits as x goes to infinity numerically for 7 different functions, which all go to 0 but at different rates, and then work with inequalities to answer questions equivalent to, "To get f(x) < epsilon, how large does x need to be?". 1 page.
2.6_Limits_at_infinity_using_algebra_WS has students use algebra to change infinity over infinity indeterminate forms to limits that can be evaluated using limit laws. Some of the algebra is difficult for students; the activity "2.4 Algebra Review" is helpful for that. 2 pages.
2.7_Calculating_derivatives_with_limits_ARL leads students through calculating limits of difference quotients, all of which are 0/0 indeterminate forms, and using algebra to reduce them so they can use limit laws. 2 pages.
2.7_Interpreting_derivatives_ARL asks students to read a graph of cumulative rainfall over time and approximate instantaneous rates of change in a variety of ways, including by annotating the graph. You can tell pretty quickly from their annotations how well they understand the problem. 2 pages.
2.8_Graph_the_derivative_from_a_graph_ARL has students study a graph to determine where the derivative is zero, positive, negative, +1, -1, large positive, large negative, and then graph the derivative below the original function. 2 pages.