Full Instructors Manual

# Calculus

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Notes on the Text: Classroom Experience Chapter 1 makes a start on three topics: functions in general, the sine and cosine in particular, and computing. Here are brief comments so you will know what is needed later. 1. finctions and graphs (essential): Section 1.1 is a solid starting point for calculus. You may give it more than a day, especially if you introduce f (t + 2) and f (t) + 2 and f (2t) and 2 f (t). These notes offer ideas about other graphing activities. One purpose is to maintain the interest of those who have already taken calculus, without giving them an enormous advantage. My favorite is the forward-back function graphed on page 4. Section 1.2 goes on to other piecewise linear models like income tax - and says explicitly that "I hope you like them but you don't have to learn them." Any of these examples, especially the delta function mentioned briefly, can be passed over. It is Section 1.3 that compares average to instantaneous for y = z2. 2. Sines and cosines (these are optional in Chapter 1): My intention is to see trigonometry in use - for points on a circle as well as sides of a triangle. Many classes will not spend substantial time on the review, but it must be available. The figure on page 31 leads neatly to cos(s - t). Section 2.4 computes derivatives of sin x and cos z in the normal way from 2. The limits of and 9 are fully developed there. But students may understand these functions better (and also the motion described by x = cost, y = sin t) by following a point on a circle. That is the outstanding example of a parameter. 3. Computing in calculus (optional): The computing section is placed in a way that allows you to discuss it or not. This topic is especially dependent on the local situation. (M.I.T. does not do much computing in the first year, and does nothing with graphing calculators.) But calculators are so convenient that we will see them more and more. They have the advantage of requiring less faculty time, as well as being personal and portable and not too expensive. The valuable thing is to see graphs (better than numbers). The example of 3= versus zs is quite good - those graphs are surprisingly close for 2.2 < z < 3.2. It is a challenge to find their intersection. It is a real challenge to find the only value of b for which bz never goes below zb for positive z. The main point is to see how this happens - the graphs of ez and ze are tangent at z = e. In Section 6.2 we know the derivatives and verify eZ 2 ze. Computing needs to be separate from the stream of ideas that launch calculus in Chapter 2. '1000 Points of Lightn is purely for entertainment. See the College Mathematics Journal of November 1990, and a forthcoming American Mathematical Monthly paper by Richert. Graphing Activities Ithaca College has developed a course that begins with a study of graphs and includes larger projects throughout the course. They find that ''pqsing questions about graphs results in abundant class participation and student response." I can confirm this from the Boston Workshop for Mathematics Faculty.

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