A vertical line can intersect the graph of a function of x at most once. Putnam Exam Challenges Putnam Exam questions appear in selected sections. Average Speed You drive to the beach at a rate of 120 kilometers per hour. Then connect the points with a smooth curve, as shown in Figure R2. On the return trip, you drive at a rate of 60 kilometers per hour. Matching In Exercises 1-4, match the equation with its Testing for Symmetry In Exercises 27-38, test for symmetry graph. Polynomial functions and rational functions are examples of algebraic functions.
The symmetry tests in Section P. From the scatter plot, you can see that the data do not appear to be linear. For a review of absolute value, see Appendix C. Cmtrrj for MrJictl fr A Smiltej md Gnum flvnfwj in I Use a graphing uiiliiy to 1ml linear model', fix ihe health care expenditures hi rt and Ihe pcfiubrlkin rl. Length A right triangle is formed in the first quadrant by the v- and y-axes and a line through the point 3, 2 see figure.
Printed in the United States of America 1 2 3 4 5 6 7 16 15 14 13 12 Copyright 2012 Cengage Learning. The Area Problem In the tangent line problem, you saw how the limit process can be applied to the slope of a line to find the slope of a general curve. The graph of a function of v cannot have symmetry with respect to the v-axis. Except for cases involving a vertical tangent line, the problem of finding the tangent line at a point P is equivalent to finding the slope of the tangent line at P. The models in Example 6 were developed using a procedure called least squares regression see Section 13. M Fit a quadratic model to a real-life data set.
Least squares regression line The graph of the model and the data are shown in Figure P. Determine whether the data can be modeled by a linear function, a quadratic function, or a trigonometric function, or that there appears to be no relationship between x and y. Census Bureau X 20 30 40 50 60 70 y 10. Using the negative reciprocal of the slope of the given line, you can determine that the slope of a line perpendicular to the given line is — §. To print an enlarged copy of the graph, go to MathGraphs.
Bureau of Labor Statistics 941, 111 , 1001, 754 , 1043, 770 , 1111, 791 , 1151, 812 , 1198, 844 , 1248, 883 , 1275, 923 , 1303, 937 a Plot the data. That is, you want the model to be simple enough to be workable, yet accurate enough to produce meaningful results. Lemniscate Let d 1 and d 2 be the distances from the point x, y to the points —1,0 and 1,0 , respectively, as shown in the figure. This title is available in a variety of formats — digital and print. The height of the basketball is recorded 23 times at intervals of about 0. An average rate of change is always calculated over an interval. R1 Graphs and Models 3 One disadvantage of point plotting is that to get a good idea about the shape of a graph, you may need to plot many points.
To check the accuracy of this model, a weather almanac was used to find the numbers of minutes of daylight on different days of the year at the location of 20°N latitude. Further permissions questions can be emailed to permissionrequest cengage. Slope is not defined for vertical lines. To print an enlarged copy of the graph, go to MathGraphs. This means that the graph is unchanged by a rotation of 1 80° about the origin. If it is false, explain why or give an example that shows it is false.
M The rate of change found in Example 3 is an average rate of change. Sketch the region R and find its area. Calculus is the mathematics of change. How well does the model fit the data? The online questions are identical to the textbook questions except for minor wording changes necessary for Web use. By the end, readers will learn how to interpret research and to address and resolve controversies. To view this article, go to MathArticles.
Solution Begin by sketching a scatter plot of the data, as shown in Figure P. Describe the changes in the positions of the plotted points and the change in the equation of the line. To discover the value of g experimentally, you could record the heights of a falling object at several increments, as shown in Example 2. Technology Throughout the book, technology boxes show you how to use technology to solve problems and explore concepts of calculus. Try to keep track of where you are relative to the three stages involved in the study of calculus.