See attached image. Draw a large graph of y = x2 from x = -1 to x = 2. This exercise focuses on finding the arc lengthof this graph segment. (Include your graph.)%3D1. First, draw line segments connecting (consecutively) the points on the graph where x is -1, 0, 1, and then estimate the length of the graph by calculating the lengths of thesesegments using the distance formula. There will be 3 lengths. Does this overestimate orunderestimate the arc length? Explain.2. In a similar manner, find an estimate of the length using segments connecting the pointson the graph where x is -1, -0.5, 0, 0.5, 1, 1.5, and 2. There will be 6 lengths this time.Explain why this estimate is better than the one in problem 1.3. In Problems 1 and 2 the arc length, L, is approximated by the sum L =EVAx? + Ay².In problem 1, Ax = 1 and in problem 2, Ax = 0.5. What would you have to do to find the%3Dexact value of L?4. If y = f (x), and f is a differentiable function, then any term in the sum in problem 3 canbe written exactly as AL = /1+ [f'(c)]²Ax where c is some number in the respectivesubinterval. Consult the text and identify the theorem that is the basis for this fact.5. Explain why, in the limit, AL can be written as dL = dx2 + dy?.6. Find dL for y = x?. Then write L exactly as a definite integral.7. Find a decimal approximation for the exact arc length by evaluating the integral inproblem 6 numerically. (Hint: Check the text for information about how to obtain anexact answer by completing the integration by trigonometric substitution.)8. What did you learn from completing this exercise?

Note: It is a multipart question. No specific part has been asked.So, First six parts are solved as…

Draw a large graph of y = x from x = -1 to x = 2. This exercise focuses on finding the arc length of this graph segment. (Include your graph.) 1. First, draw line segments connecting (consecutively) the points on the graph where x is - 1,0, 1, and then estimate the length of the graph by calculating the lengths of these segments using the distance formula. There will be 3 lengths. Does this overestimate or underestimate the arc length? Explain. 2. In a similar manner, find an estimate of the length using segments connecting the points on the graph where x is -1, -0.5, 0, 0.5, 1, 1.5, and 2. There will be 6 lengths this time. Explain why this estimate is better than the one in problem 1. 3. In Problems 1 and 2 the arc length, L, is approximated by the sum L EAx2 + Ay?. In problem 1, Ax = 1 and in problem 2, Ax = 0.5. What would you have to do to find the exact value of L? 4. If y = f (x), and f is a differentiable function, then any term in the sum in problem 3 can be written exactly as AL = /1 + [f'(c)]²Ax where c is some number in the respective subinterval. Consult the text and identify the theorem that is the basis for this fact. %3D 5. Explain why, in the limit, AL can be written as dL = dx² + dy2. 6. Find dL for y = x2. Then write L exactly as a definite integral. 7. Find a decimal approximation for the exact arc length by evaluating the integral in problem 6 numerically. (Hint: Check the text for information about how to obtain an exact answer by completing the integration by trigonometric substitution.) 8. What did you learn from completing this exercise?

7th Edition

ISBN:9781337614085

Author:Alexander, Daniel C.; Koeberlein, Geralyn M.

Publisher:Alexander, Daniel C.; Koeberlein, Geralyn M.

ChapterP: Preliminary Concepts

SectionP.CT: Test

Problem 1CT