Use your slope of tangent line 0 1 2 3 b You should notice a pattern in the above table. Therefore, the limit does not exist, even as an infinite limit. Limits are treated from descriptive, graphical, numerical, and algebraic points of view. Illustrate that many functions such as x 2 and x 2 sin x look locally linear, and discuss the relationship of this property to the concept of the tangent line. May not be scanned, copied, or duplicated, or posted to a publicly accessible website, in whole or in part. Each group should also be given a large copy of one of the graphs, perhaps on acetate.
Have some respect for people who take time to answer your question and follow the posting rules. If a group finishes early, have them discuss where f is increasing and where it is decreasing. Have the students check if lim x x lim 5 x x 5 x 5 x 5 exists, and then compute left- and right-hand limits. As a increases, the slope of the curve at x 0 is increasing, as can be seen below. A9 Appendix B Coordinate Geometry and Lines Exercises p. May not be scanned, copied, or duplicated, or posted to a publicly accessible website, in whole or in part.
If so, what is its value? Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Find c and m, or explain why they do not exist. Give a physical interpretation of your answer. Allow the students plenty of time to do the first three questions, which should help them to internalize and understand the formal definition of a limit. When the students are finishing up, it is crucial to pass out Problem 2. May not be scanned, copied, or duplicated, or posted to a publicly accessible website, in whole or in part.
Solutions Manuals are available for thousands of the most popular college and high school textbooks in subjects such as Math, Science , , , Engineering , , , and more. See Problems 12—14 on page 272, Problem 13 on page 363, Problems 16—17 on page 426, and Problem 8 on page 986. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. Reading 001 002 003 004. May not be scanned, copied, or duplicated, or posted to a publicly accessible website, in whole or in part. Stewart was shivering in Alaska; Shasta was in Hawaii, warm and happy. It appears to stop at t 2.
A polynomial is continuous everywhere, so lim exists and is equal to. Its units are dollars percent per year. Back up your estimate by looking at the function, and discussing 1 21 x why your estimate is probably correct. On any interval where positive or negative. Based on Problems 4 and 5, guess the slope of the tangent line at any point x 53 a, for a 1. Many will not believe that it is. Its units are thousands of surgeries per year.
Notice that one limit exists, and one does not. Describe the inverse function f 1 in words. Calculus is an exciting subject, justly considered to be one of the greatest achievements of the human intellect. If y you wish to foreshadow f , add its graph below. Look for any corners on the graph.
May not be scanned, copied, or duplicated, or posted to a publicly accessible website, in whole or in part. Examples include the following: By definition, the slope of the tangent line is the limit of the slopes of secant lines. Give reasons for your answer. Notice the difference in the π 2 quality of the tangent line approximations. May not be copied, scanned, or duplicated, in whole or in part. YouBook is available within Enhanced WebAssign. The system also includes Active Examples, in which students are guided in step-bystep tutorials through text examples, with links to the textbook and to video solutions.
Similarly, there is a root in 1 2. When the water from the hot water tank starts coming out, is large and positive as increases to the temperature of the water in the tank. ¤ 117 The discrepancy can be explained by the choice of the viewing window. May not be scanned, copied, or duplicated, or posted to a publicly accessible website, in whole or in part. Note that the answers for parts b and c must be the same for these graphs because as soon as the velocity for runner B overtakes the velocity for runner A, the distance between the runners starts to decrease.