Calculus I Spring 2016 Lab 4: Different Types of Limits Lab Preparation: Answer the following questions individually and bring your write-up to class. For easy comparisons between the two contexts, create two columns for your write-up where the answers to A. are in the left column and the answers to similar questions for B. are on the right. A. In Lab 3, you approximated the height of a hole in a graph above the x-axis. Suppose the hole occurred at x = 1, and the function, f , is decreasing. (NOTE: The function f is a generic function, different from whatever function you had in Lab 3.) B. In class we investigated the motion of a bolt fired from a crossbow straight up into the air with an initial velocity of 49 m/s. Accounting for wind resistance proportional to the speed of the bolt, its height above the ground is given by the equation h(t) = 7350245t 7350et/25 meters (with t measured in seconds). i. You were approximating the height of the hole above the xaxis. Why couldn't you compute that height directly from the formula for f ? ii. How did you find an underestimate? an overestimate? iii. How did you know that your underestimate was an underestimate, and your overestimate was an overestimate? iv. How did you determine an error bound for these approximations? i. We were approximating the speed of the bolt when t = 2 seconds. Why couldn't we determine the speed directly using h/t? ii. How did we find an underestimate? an overestimate? iii. How did we know that your underestimate was an underestimate, and your overestimate was an overestimate? iv. How did we determine an error bound for these approximations? CLEAR Calculus 2010 Calculus I Spring 2016 Lab 4 (read carefully): Throughout this course, we will be dealing with many different types of limits. The basic structure will be the same: a quantity being approximated, approximations, errors, and error bounds. But the mathematical objects used (and the quantities they represent) in that basic structure will be different. The key to understanding the entire course is i). Developing a strong understanding of the basic limit/approximation structure ii). Developing an ability to identify the relevant quantities and interpret their meanings This lab will help lay out this basic structure and distinguish its application in different types of limits. For the following, work with your group on all problems. We encourage you to collaborate both in and out of class, but you must write up your responses to ALL problems individually. Answer the questions in numerical order. For easy comparisons between the two contexts, create two columns for your write-up, where the height-of-the-hole answers are in the left column and the speed answers are in the right column. Limits/Approximations for the height of a hole in a graph Limits/Approximations for instantaneous speed In Lab 3, we approximated the height of a hole in a graph. Now we In class we investigated the motion of a bolt fired from a crossbow are going to be intentional about answering this question in each of straight up into the air with an initial velocity of 49 m/s. Accountfour different representations (numerical, graphical, algebraic, or ing for wind resistance, its height above the ground is given by the descriptive). Suppose the hole occurred at x = 1, and the function, function h(t) = 7350245t 7350et/25 meters (with t measured f , is decreasing. in seconds). We approximated the speed of the bolt when t = 2 (NOTE: The function f is a generic function, different from what- seconds. ever function you had in Lab 3.) 2. (Verbal representations) 1. (Verbal representations) Answer the following briefly using only words (no numbers Answer the following briefly using only words (no numbers except the t-value 2, no algebraic expressions, etc.). except the x-value 1, no algebraic expressions, etc.). a. What was being approximated? a. What was being approximated? b. How did you find an underestimate? How did you know it was an underestimate? b. How did you find an underestimate? How did you know it was an underestimate? c. What is the error for your underestimate? c. What is the error for your underestimate? d. How did you determine an error bound for these approximations? d. How did you determine an error bound for these approximations? CLEAR Calculus 2010 Calculus I 3. (Numerical representations) Answer the following using the table of values for f . 0.8 0.95 1 1.03 x f (x) 4.216 4.18 dne 4.07 Spring 2016 1.1 3.9 4. (Numerical representations) Answer the following using the table of values for h. 1 1.5 2 2.4 2.6 t h(t) 43.198 60.531 75.095 84.789 88.994 a. Can you write down an exact number for the height of the hole? Why or why not? a. Can you write down an exact number for the speed of the bolt at t = 2 seconds? Why or why not? b. What is an underestimate? an overestimate? b. What is an underestimate? an overestimate? c. Can you write down numerical values of the errors for your two approximations? Why or why not? c. Can you write down numerical values of the errors for your two approximations? Why or why not? d. What is an error bound for your approximations? d. What is an error bound for your approximations? 5. (Algebraic representations) 6. (Algebraic representations) Answer the following using only algebraic expressions. The Answer the following using only algebraic expressions. The only only number you can use is 1. You can use the variables a and number you can use is 2. You can use the variables t1 and t2 where b where a < 1 < b. t1 < 2 < t2 . a. What are you approximating? (Hint: it's not f (1), because that value doesn't exist.) Use limit notation to answer the question. Choose a single letter for a variable name to represent this value for subsequent questions. b. What is an underestimate? an overestimate? b. What is an underestimate? an overestimate? c. What is the error for your underestimate? c. What is the error for your underestimate? d. What is the error bound for your approximations? d. What is the error bound for your approximations? CLEAR Calculus 2010 a. What are you approximating? Use limit notation to answer the question. Choose a single letter for a variable name to represent this value for subsequent questions. Calculus I Spring 2016 7. (Graphical representations) 8. (Graphical representations) Draw a large graph on half of a page that could represent y = Draw a large graph on the other half the page of h vs. t. Be sure to f (x). Be sure to include the hole, label your axes, and mark label your axes, and mark t = 2. Answer the following questions x = 1. Answer the following questions by identifying a line by identifying a line in the graph whose slope is the requested segment in the graph whose length is the requested value. Draw value OR by identifying two lines where the difference between and label your answers very clearly! the slopes is the requested value. Draw and label your answers very clearly! a. What are you approximating? b. What is an underestimate? an overestimate? a. What are you approximating? c. What is the error for your underestimate? b. What is an underestimate? an overestimate? d. What is the error bound for your approximations? c. What is the error for your underestimate? d. What is the error bound for your approximations? 9. Suppose you had a formula for f and needed an approximation 10. Suppose you needed an approximation for the speed of the bolt at for the height of the hole above the x-axis accurate to within t = 2 accurate to within 1 m/s. Explain what you would do. 0.001. Explain what you would do. (Hint for 9 and 10: Think back to questions 6 and 7 on Lab 3.) CLEAR Calculus 2010