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MATH 11012: INTUITIVE CALCULUS SOME SUGGESTIONS BY NANCY SMITH (FORMER COORDINATOR) It is important that you cover the entire departmental syllabus. You’ll need to make a day-by-day schedule for yourself in order to keep up with this departmental syllabus. If you’re teaching a 2-day/week class you may want to remind students (and yourself?) that each class meeting is 11/2 regular classes (meaning 3 hours homework for the next class is not unreasonable). COMMENTS: Your students will have poor algebra skills. Do not take class time to review algebra skills, but you might: give review sheets, or assign problems from Chapter 1. Do explain algebra as you work calculus. Stress ‘why’ not ‘proof’. This is “Intuitive” or experience-based. Remember to teach by example. Notation: It is important throughout the course to stress notation (standard notation not student inventions!). Help them to understand the meaning and value of notation and how to use it correctly. I try to stress writing the solution not just finding the answer. Chapter 2. You might want to briefly review (x + 1)3 = . . . , fractions, point-slope line form, supply, demand, and split functions as they come up. Try to help students separate notions of limit, continuity, and differentiability. Try to help students make the connection between geometric, algebraic and numeric explanations. Draw lots of pictures and graphs. Work lots of problems. The students have a hard time with the notion of function. I use: “X is ‘where’ and Y is ‘how tall.’” This text does not do limits as x approaches until chapter 6. You might consider introducing this idea here, enabling you to relate horizontal asymptotes and limits in chapters 3 and 4. Chapter 2. Section 1. There is a nice (for both traditional and calculator sections) calculator example showing a hole in a function on page 85, Graphing Calculator Exploration. Because the hole in the example occurs at an integer one can use ZINTEGER or ZDECIMAL on the TI-82 or 83 (TI-81: Zoom: INTEGER). Chapter 3. Sections 1-2 (graphing). Teach by example. I attempt to quickly cover the complete graph. Make sure students can graph polynomials, then take on corners or rational functions. To help students get organized, make a list of steps and always follow this list. Note: Using the graphing calculator does not seem to make this topic significantly easier for students. The object is not to create the perfect graph, but to relate the derivative and the graph. I tend to take off extra points if a student has the correct graph, but incorrect intervals of increasing/decreasing for example. Probably the most useful feature of the graphing calculator is the table (TI-82&83) for functional values. Chapter 3, Sections 3-5 (optimization). Our students have a very hard time with story problems. Many tell me they can’t do them or they never have done them. I tell them now they will learn! Try to help them to break down the story phrase by phrase. Sometimes I have them price each piece: cost of front fence, cost of side fence, etc. To help them get organized give the steps and do every problem following those steps. I have students write their plan: To Max Area: A' = 0 Be sure to do economic problems, remembering that most of our students are business majors. If you’ve extra time Chapter 3 can use it. Chapter 4 (logs & exponents). These students have studied logarithms and exponents, they just forgot! Don’t get bogged down here. Review quickly. They need to know the derivative of the natural logarithm and exponential functions. I encourage my students to use their calculator and to give decimal solutions so they have some feel for size of their answer. (I require a calculator in my traditional classes.) I go over the use of the log and exponential features of the various calculators. The notion of present value is important. Notation: The book uses some nonstandard notation here. In Section 1, the usual A=Pert is given as (Value after n years)=Pern. Compound interest is given as (Value after n periods)=P(1+r)n. Also in Section 4.3 E(p) denotes consumer expenditure while in Section 4.4 E(p) denotes elasticity of demand. I suggest CE(p) for the former. Chapter 4, Section 4 (elasticity): (See preceding comment on notation.) Be sure to cover relative rates first. The book does a nice job of explaining elasticity using relative rates. If your class has students who have had economics, you might ask for their input. Students have offered insulin as very elastic and heroin as ‘perfectly’ elastic starting a nice conversation. The Application Preview, blue section, looks interesting. Chapter 5 (integration).. Consider introducing section 6 (substitution) earlier in the chapter. Chapter 5, Section 4 (areas): Asking for “set-up” only is a good ploy for tests. (Some instructors of calculator sections allow students to use the numeric integration feature for 2 3 some of the area problems. 0 x dx is approximated by math 9: fnInt (x3,x,0,2) on the TI- 82 and 83. Be aware some functions take a very long time or give inaccurate results.) You may wish to supplement this exercise set. Goldstein has good exercises. Chapter 5. Section 6 (substitution): For students of weak algebra skills this section is very hard and may take several days or returning to the subject several times. You might wish to supplement the exercise set with more difficult substitution problems. (Goldstein has some.) Students must be given problems of sufficient difficulty to force them to write out all the steps, otherwise they think they understand when they do not. Be aware that the exercise set in our text includes a number of examples which cannot be done with substitution. Chapter 5. Section 5. Be sure to do Consumers' and Producers' Surplus. (I have other texts with explanations, if you’d like to read up on the subject.) It is important to our students that they understand that this is a sum and an area. You might ask members of your class who have had economics to explain equilibrium point. Students tell me that they calculate Consumers' and Producers' Surplus in their economics class. They use linear supply and demand functions and find areas of triangles. The Gini Index of Income Distribution is both accessible and interesting to our students. It does not take long to explain. Note: Recall that the Business School has specifically asked us to cover many applications in this course. By Nancy Smith, December, 2002 Revised August 2005 by Darci Kracht 2