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dx2 dy [₁²√(²)² + (1) ² 2) The Arc Length formula from section 8.1 becomes x = 3t², y=2t³, 0≤t≤2 dt. Use this formula to find the exact length of curve

dx2 dy [₁²√(²)² + (1) ² 2) The Arc Length formula from section 8.1 becomes x = 3t², y=2t³, 0≤t≤2 dt. Use this formula to find the exact length of curve

Calculus For The Life Sciences
Calculus For The Life Sciences
2nd Edition
ISBN:9780321964038
Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Publisher:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Chapter6: Applications Of The Derivative
Section6.3: Implicit Differentiation
Problem 43E
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In this sections, we would like to explore parametric equations with calculus: dy dt to find the derivative of y with respect to x. However, for the second derivative dx dt 1) We use the formula d²y dx² x = t - lnt, y = t + ln t. need to use this formula a) Calculate dy dx b) Calculate || d² y d² y by dx² dx² by d² y ď² y dx² dx² d (v) dt dx dt dy² dt² d² x dt² d dt dx dt part i with respect to t; iii) Calculate d² y dx² " (This formula is 2 where y' where y' dt dx dt - 2) The Arc Length formula from section 8.1 becomes X = 3t², y= 2t³, 0≤t≤2 , = dy dt dx dt correct but just want to demonstrate the difference). dy dt dx dt with the following steps: i) calculate Now let's demonstrate that the two formulas are different: Consider the parametric equation where y dy dt dx dt B dx dy 2 · [₁² √ ( ² ) ² + ( ¹² ) ³² dt dt dy dx || dy dt d² y dx² dx dt dy² dt² d² x dt² -; ii) calculate d is not correct!!!!! Instead, we dt (v) (y) by taking derivative of dt. Use this formula to find the exact length of curve
Transcribed Image Text:In this sections, we would like to explore parametric equations with calculus: dy dt to find the derivative of y with respect to x. However, for the second derivative dx dt 1) We use the formula d²y dx² x = t - lnt, y = t + ln t. need to use this formula a) Calculate dy dx b) Calculate || d² y d² y by dx² dx² by d² y ď² y dx² dx² d (v) dt dx dt dy² dt² d² x dt² d dt dx dt part i with respect to t; iii) Calculate d² y dx² " (This formula is 2 where y' where y' dt dx dt - 2) The Arc Length formula from section 8.1 becomes X = 3t², y= 2t³, 0≤t≤2 , = dy dt dx dt correct but just want to demonstrate the difference). dy dt dx dt with the following steps: i) calculate Now let's demonstrate that the two formulas are different: Consider the parametric equation where y dy dt dx dt B dx dy 2 · [₁² √ ( ² ) ² + ( ¹² ) ³² dt dt dy dx || dy dt d² y dx² dx dt dy² dt² d² x dt² -; ii) calculate d is not correct!!!!! Instead, we dt (v) (y) by taking derivative of dt. Use this formula to find the exact length of curve
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