Students create a as many qualitatively different sequences that converge to 5 and sequences that do not converge to 5. Lab 13: Constructing a formal definition of sequence convergence The formal definition of sequence convergence Students use the Lagrange error bound to answer three basic types of questions for Taylor series, fixing two of thw quantities x, n, and ε, then solving for the third. Students explore the derivative and antiderivative of Taylor series.
Arc length calculus 2 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. 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. Students work in jigsaw groups to explore similaritites and differences in the various types of sequences from Lab 8. They represent the exact value, approximations, errors, and error bounds in multiple representations. Mystery constants are represented as an infinite sum, which students must approximate to within a desired degree of accuracy. Students work in jigsaw groups to extend several of the integral models developed in Lab 6 to answer more general questions. Lab 7: Modeling with definite integrals - part 2 Each integral is more complex than the simplest product template, so students must carefully track the meaning of each factor in their integral. Students develop definite integrals to model physcal quantities. Lab 6: Modeling with definite integrals - part 1 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.
They generalize the formula for breaking the curve into an increasing number of secant lines for use with a calculator or CAS.
Students use secant lines and the distance formula to approximate the arclength of the graph of a function.
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. Lab 3: Approximating definite integrals - part 2 Students connect numeric and graphical representations of the integral, approximations, and errors then make geometic arguments for whether the approximations are underestimates or overestimates. This lab develops the geometry of approximating integrals using the left, right, midpoint, trapezoid, and Simpson's methods. Lab 2: Approximating definite integrals - part 1 These labs develop the left, right, midpoint, trapezoid, and simpsons methods for approximating definite integrals and the corresponding error bound formulas. The focus is on clearly identifying the exact value, approximations, errors, and error bounds numerically, algebraically, and graphically. Students approximate the derivative using the difference quotient and a definite integral using left and right sums. This lab reviews the basic terminology and notation required to develop derivatives and definite integrals in terms of approximations.
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