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Lesson 3 Find the Point of Inflection
Michael Cain
seniors/calculus    45 min Lesson created on 8/6/1999 12:33:49 PM EST. Abstract  (help) Students will use the data gathered on the previous lesson and find the point of inflection first using the CMBDERIV program on the TI-83 and then using the best-fit curve method used in lesson one. Once again this will complete the cross-curricular connection with chemistry which was introduced in lesson two.

National Standards  (help) Mathematics instructional programs should include attention to data analysis, statistics, and probability so that all students Pose questions and collect, organize, and represent data to answer those questions Interpret data using methods of exploratory data analysis Develop and evaluate inferences, predictions, and arguments that are based on data Mathematics instructional programs should emphasize connections to foster understanding of mathematics so that all students Recognize and use connections among different mathematical ideas Understand how mathematical ideas build on one another to produce a coherent whole Recognize, use and learn about mathematics in contests outside of mathematics. Mathematics instructional programs should emphasize mathematical representations to foster understanding of mathematics so that all students Create and use representations to organize, record, and communicate mathematical ideas Develop a repertoire of mathematical representations that can be used purposefully, flexibly, and appropriately Use representations to model and interpret physical, social, and mathematical phenomena.

Pre-requisite Skills  (help) Use of TI-83 Calculator How to find first and second derivatives on the TI-83

Teacher Information  (help) CMBDERIV is a program which is part of the CHEM/BIO appication. The entire procedure is on page 4-18 in the TI CHEM/BIO 1999 Participant Packet.

Assessment  (help) Students will be able to use the data stored in their calculators to find the point of reflection by both methods. At the end of the unit, students will be tested by taking a set of data, entering it on the calculator and find the point of inflection by both methods learned.

Student Activity  (help) Part 1: Students will use the CBMDERIV program to determine from the sample titration data the equivalence point and concentration of a weak acid. Students must have data from lesson 2 in lists 1 and 2 in the TI-83 calculator. Students enter the CBMDERIV program and press enter. They will be given the option to view the graph, first derivative or second derivative. By selecting 1 students can trace along the titration graph and view the y values to estimate the point when the delta y changes from increasing to decreasing. (point of inflection) By selecting 2 students can trace the derivative. The point of infleciton is where the derivative is at its maximum point. By selecting 3 students can trace the second derivative. The point of inflection is where the derivative is a zero. Part 2: Students will find the best-fit curve and equation using the logistic regression function as in lesson one. (note: this function wants the first y value to be near 0, therefore one must subtract the initial y value for when x=0 from all data, apply the regression and then ad the amount subtracted to the resulting equation. Students will then discuss the point of inflection as in lesson one.

Technology Requirements/Integration  (help) Use of CBL Use of TI-83 calculator (CHEM/BIO application, CMBDERIV program, nDeriv function and regression functions)

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