Mt 217 Accelerated Calculus - Fall 2003

Assignment 12: Prelude to Final Things

November 25, 2003

Purpose

I'm not quite sure exactly the order of the topics we will cover over the next couple of weeks. I list here important themes of this course with some suggestion of where you can find these topics in your text.

Class Activities and Homework Assignments

Problem Solving:

• Problem solving with Sequences and Series

  Project #4: Drug Dosage is due on Wednesday, December 3. There are two series in this problem -- one gives the levels of the drug in the blood just before an injection, and the other gives the levels just after the injection. The goal of this project is to determine a dosage level and schedule that will make the drug both effective and safe.

• Problem Solving with Definite Integrals and Antiderivatives

  Possible topics include

  • areas of regions bounded by curves -- including areas bounded by parametric curves (Section 6.1),
  • average values of a function (Section 6.4), and
  • applications to economics (Section 6.6 in your text).

• Problem Solving with Derivatives

  There are some interesting problems in the Focus on Problem Solving at the end of Chapter 4. (See pages 340 - 343.)

Calculations of Calculus:

• Calculating antiderivatives, definite integrals, derivatives, and limits

  We will do some review problems in class each class period. You can find a lot of good review problems in the Review sections at the ends of Chapters 2, 3, and 5. There are some study problems suggested on the Benchmark Study Guide.

• Prepare for Benchmark #3.

  Make an appointment with me to take this Benchmark outside of class between December 1 - 8. To have the opportunity to retake this benchmark, plan to take it early in this window of opportunity.

Concept of Function:

• Functions are a primary tool supporting a mathematician's way of thinking. We will review the ideas in the definitions of the derivative and the integral -- and the ways that these processes are themselves functions. See especially Section 2.8: The Derivative as a Function, Section 5.3 (the discussion of Indefinite Integrals), and Section 5.4: The Fundamental Theorem of Calculus.

• You should be able to use ideas about derivatives and antiderivatives to identify in a graph showing two or three functions which graph represents the function, which represents its derivative, and/or which represents its antiderivative.

Special Theorems of Calculus:

• Intermediate Value Theorem

  See Section 2.5 -- especially page 126, and the examples that follow. There are a number of problems in the exercises on pages 128 - 130 that use the Intermediate Value Theorem. See also problem #8, page 187.

• Mean Value Theorem

  See Section 4.3 -- especially page 281 for both the statement of this theorem and an example which makes use of it. See exercises #45 - 48, page 290.

• Fundamental Theorem of Calculus

  See Section 5.4.