# CLEAR Calculus

## Calculus 2 Labs

Calculus 1 review

This lab reviews the basic terminology and notation required to develop derivatives and definite integrals in terms of approximations. Students approximate the derivative using the difference quotient and a definite integral using left and right sums. The focus is on clearly identifying the exact value, approximations, errors, and error bounds numerically, algebraically, and graphically.

Integral approximations

These labs develop the left, right, midpoint, trapezoid, and simpsons methods for approximating definite integrals and the corresponding error bound formulas.

This lab develops the geometry of approximating integrals using the left, right, midpoint, trapezoid, and Simpson's methods. Students connect numeric and graphical representations of the integral, approximations, and errors then make geometic arguments for whether the approximations are underestimates or overestimates.

Students use error bound formulas to approximate a definite integral (of a function without an elementary antiderivative) to within a desired degree of accuracy using various methds.

Arclength

Students use secant lines and the distance formula to approximate the arclength of the graph of a function. They generalize the formula for breaking the curve into an increasing number of secant lines for use with a calculator or CAS.

Students identify why their approximations from Lab 4 were not Riemann sums, then apply the mean value theorem to convert it into one and eventually a definite integral.

Modeling with definite integrals

Students develop definite integrals to model physcal quantities. Each integral is more complex than the simplest product template, so students must carefully track the meaning of each factor in their integral.

Students work in jigsaw groups to extend several of the integral models developed in Lab 6 to answer more general questions.

Series

Mystery constants are represented as an infinite sum, which students must approximate to within a desired degree of accuracy. They represent the exact value, approximations, errors, and error bounds in multiple representations.

Students work in jigsaw groups to explore similaritites and differences in the various types of sequences from Lab 8. They determine the number of terms required to i) get within a very small tolerance for the convergent series, or ii) grow beyond any bound for a divergent series.

Taylor series

Students explore the nature of pointwise convergence of Taylor series by plotting convergent sequences above several x-values in the interval of convergence and focusing on the size of the error for the partial sums at each point.

Students explore the derivative and antiderivative of Taylor series.

Students use the Lagrange error bound to answer three basic types of questions for Taylor series, fixing two of thw quantities xn, and ε, then solving for the third.

The formal definition of sequence convergence

Lab 13: Constructing a formal definition of sequence convergence    [pdf]

Students create a as many qualitatively different sequences that converge to 5 and sequences that do not converge to 5. They then engage in multiple cycles of i) writing a definition that includes all of their examples while excluding all of their non-examples, ii) testing their definition against the examples and non-examples, iii) identifying problems with their definition, and iv) revising their definition. Through the process the students either develop their own definition of sequence convergence and come to appreciate the reasons for each component of the standard formal definition.

Dynamic Graphs | Graphs with ε-lines | Graphs with

Lab 14: Applying the formal definition of sequence convergence     [pdf]

Students use the formal definition of sequence convergence to prove some basic statements and adapt the definition to the limit of a sequence derived from a limit of Riemann sums for an integral and the limit of a sequence of functions for a Taylor series. They identify the meaning of each quantity from the formal definition in each of these settings.