# CLEAR Calculus

## Calculus 1 Labs

Rate of Change

The purpose of these labs is to help students talk and write in meaningful ways about mathematics. Specifically to describe quantities and changes in quantities clearly in terms of context, to make rigorous arguments about how such quantities are related, and to make connections between these features in the contexts and on graphs.

This outlines a discussion that surfaces the need to talk precisely about quantities and models doing so with a distance-time relationship.

Each group of students work on two of the numbered problems in the lab focusing on different aspect of the relationship between height and volume of water in bottles of various shapes.

Limit of a function

The goals of this lab are to i) develop a graphical, numerical and algebraic sense of the meaning of the limit of a function at a point, and ii) develop the terminology and structure of the approximation framework that will serve as the overarching structure for everything defined in terms of a limit in the course.

Each group of students works with one function that has a removable singularity. The algebra makes it easy to see the x-value where the singularity occurs, and the students can see the hole when graphing the function on their calculator. The overall task is to evaluate the function at nearby points to approximate the height of the hole (limit of the function).

Different types of limits

This lab is intended to help students make one of the most difficult shifts in the course, from the limit of a function at a point to the limit of a difference quotient.

Students compare the limit of a function at a point (as developed in Lab 3) side-by-side with the limit of a difference quotient for the definition of the derivative (in the distance-time-velocity context of a crossbow bolt fired into the air).

Lab 4: In-class solution

This is an extensive write-up of the crossbow bolt example as responses to the questions in Labs 5 and 6. It is helpful to cover most of this example in class prior to doing Lab 4. This document can be distributed to students as a summary of the in-class example, as a reference when working on Lab 4, and as an example of how to write up solutions to Labs 5 and 6.

The Derivative at a point

Students begin to generalize their ideas of derivative previously developed in a familiar context of distance-time-velocity to other rate of change contexts.

Students duplicate the mathematics of the limit of a difference quotient that was covered in the crossbow bolt example from Lab 4, but in new contexts

Students work in groups composed of students that worked on different contexts for Lab 5, teach each other their contexts and approximation methods, then must find approximations within a small specified tolerance for each of the contexts.

These labs engage students in finding and using linear (tangent line) approximations and quadratic (second-order Taylor series) approximations to a function.

Students are given a velocity-time function and a few position-time points. They must find and use the equations of tangent lines to a position-time function to estimate distances traveled.

Labs 7-8: Hints on linear and quadratic approximation

This is a handout for students that works through a linear and quadratic approximation for the function. This can serve as a helpful example as they are writing up their own lab.

Optimization

These labs provide practice with modeling skills through challenging optimization problems.

Students identify relevant constant and variable quantities in their optimization problems to prepare for the modeling required in the lab.

These are challenging optimization problems. Each group works on one numbered problem, which contains two separate tasks.

Students work “jigsawed” with students who worked on different problems in Lab 9. The problems are slightly generalized by introducing parameters for one or more of the constants. In many cases the result that the optimal solutions don’t depend on some of the parameters is surprising.

Related rates

This lab provides practice with modeling skills through related rates problems.

Definite integrals

These labs develop the idea of the definite integral and engage students in modeling various phenomena with Riemann sums and definite integrals to generalize the concepts.

Students generalize the Riemann sum structure for a context other than distance-time-velocity.

An extensive write-up of Riemann sums applied to a distance-time-velocity context developing all of the ideas students will need for Lab 12. This also serves as a good model for the students’ work on the new contexts.

Students continue to work on the context they started in Lab 12, but now must find a Riemann sum approximation within a desired accuracy, requiring a very large number of terms. They also generate a formula for determining how many terms they need to achieve any desired accuracy.

Dynamic graphs: Example

An extensive write-up of Riemann sums applied to a distance-time-velocity context developing all of the ideas students will need for Lab 13.

Students work “jigsawed” on all of the Lab 12-13 contexts. Additional emphasis is placed on writing the definite integral and interpreting the meaning of each factor in terms of the context.

An extensive write-up of Riemann sums applied to a distance-time-velocity context developing all of the ideas students will need for Lab 14.

The fundamental theorem of calculus

These labs have students develop proofs of the fundamental theorem of calculus using the approximation ideas developed throughout the course and categorize the various ways in which the theorem can be used.

Develops a proof of the fundamental theorem of calculus, part 1.

Develops a proof of the fundamental theorem of calculus, part 2.

Students categorize and practice different uses of the fundamental theorem of calculus.