New PDF release: Elementary Calculus: An Infinitesimal Approach, 2nd Edition

By H. Jerome Keisler

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Additional info for Elementary Calculus: An Infinitesimal Approach, 2nd Edition

Example text

Prove the identities. ¨ (a) tan ␪ sin ␪ ϩ cos ␪ ෇ sec ␪ b FIGURE FOR PROBLEM 5 (b) 2 tan x ෇ sin 2x 1 ϩ tan 2x 8. Find all values of x such that sin 2x ෇ sin x and 0 ഛ x ഛ 2␲. 9. Sketch the graph of the function y ෇ 1 ϩ sin 2x without using a calculator. Answers to Diagnostic Test D: Functions 1. (a) 5␲͞3 (b) Ϫ␲͞10 6. 2. 6Њ 8. 0, ␲͞3, ␲, 5␲͞3, 2␲ 1 15 (4 ϩ 6 s2 ) 9. 3. 2␲ cm 4. (a) s3 (b) Ϫ 12 5. (a) 24 sin ␪ (b) 24 cos ␪ y 2 (c) 2 _π 0 π x If you have had difficulty with these problems, you should look at Appendix C of this book.

For instance, Figure 2 shows a graph of the linear function f ͑x͒ ෇ 3x Ϫ 2 and a table of sample values. 3. So f ͑x͒ increases three times as fast as x. Thus the slope of the graph y ෇ 3x Ϫ 2, namely 3, can be interpreted as the rate of change of y with respect to x. 5 FIGURE 2 v EXAMPLE 1 Interpreting the slope of a linear model (a) As dry air moves upward, it expands and cools. If the ground temperature is 20ЊC and the temperature at a height of 1 km is 10ЊC, express the temperature T (in °C) as a function of the height h (in kilometers), assuming that a linear model is appropriate.

One possibility is the line that passes through the first and last data points. 07 1 Equation 1 gives one possible linear model for the carbon dioxide level; it is graphed in Figure 5. C 380 370 360 350 FIGURE 5 Linear model through first and last data points 340 1980 1985 1990 1995 2000 2005 t Notice that our model gives values higher than most of the actual CO2 levels. A better linear model is obtained by a procedure from statistics called linear regression. If we use 28 CHAPTER 1 FUNCTIONS AND MODELS A computer or graphing calculator finds the regression line by the method of least squares, which is to minimize the sum of the squares of the vertical distances between the data points and the line.

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Elementary Calculus: An Infinitesimal Approach, 2nd Edition by H. Jerome Keisler


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