How come about I remarked on the question? 154%d-r√²+h²√(d-x)² + hShow that at this stationary value f(x) has a minimum.sin ₁ =Exercise 1.21Show that the functionals- S drS₁ [y] =sin 8₂ =-f. dr.dr (1+ry') y' and S₂]y] =dx xy¹²,Exitwhere b> a>0, y(b) = B and y(a) = A, are both stationary on the same curve,namelyy(x) = A + (B - A)In(x/a)In(b/a)Explain why the same function makes both functionals stationary.132%38Solution 1.21Observe thatSi[y] = S₂[y] +howhow me ??+ fºdrzdx y(x) = S₂[y] + B - A,8 = 2€that is, the values of the two functionals differ by a constant, independent of thepath. Hence the stationary paths of the two functionals are the same.Consider the difference & S₂y + eg] - S2[y], where g(a) = g(b) = 0:26 ["* dx xy' (x)g'(x) +0(2²)Chapter Iso that 8 = O(²) if xy'(x) = c, where c is a constant. Integrating this equationgives y(x) = d+cln(x/a), where d is another constant. The boundary conditionsnow give

Answered: Image /qna-images/answer/aa2dbc98-8c58-4789-a2df-711eb4af6a42.jpg

sin 8₂ = ਸ 1²+1 Show that at this stationary value f(r) has a minimum. sin 8₁ = Exercise 1.21 Show that the functionals La S₁[y] = dr (1+ry') y' and S₂[y] = d-r √(d-x)² + h where b> a>0, y(b) = B and y(a) = A, are both stationary on the same curve, namely In(x/a) In(b/a) y(x) = A + (B-A): Explain why the da r 38 Solution 1.21 Observe that -Sa Si[y] = S₂[y] + how ?? Come dry (r) = S₂[y]+B-A, 8 = 2€ = 26 [* dx xy' (a)g'(x) + 0(e²) Chap that is, the values of the two functionals differ by a constant, independent of the path. Hence the stationary paths of the two functionals are the same. Consider the difference & Saly+eg] - S2[y], where g(a) = g(b) = 0: so that 8 = O(²) if ry'(x) = e, where e is a constant. Integrating this equation gives y(x) = d+ cln(x/a), where d is another constant. The boundary condition

sin 8₂ = ਸ 1²+1 Show that at this stationary value f(r) has a minimum. sin 8₁ = Exercise 1.21 Show that the functionals La S₁[y] = dr (1+ry') y' and S₂[y] = d-r √(d-x)² + h where b> a>0, y(b) = B and y(a) = A, are both stationary on the same curve, namely In(x/a) In(b/a) y(x) = A + (B-A): Explain why the da r 38 Solution 1.21 Observe that -Sa Si[y] = S₂[y] + how ?? Come dry (r) = S₂[y]+B-A, 8 = 2€ = 26 [* dx xy' (a)g'(x) + 0(e²) Chap that is, the values of the two functionals differ by a constant, independent of the path. Hence the stationary paths of the two functionals are the same. Consider the difference & Saly+eg] - S2[y], where g(a) = g(b) = 0: so that 8 = O(²) if ry'(x) = e, where e is a constant. Integrating this equation gives y(x) = d+ cln(x/a), where d is another constant. The boundary condition

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 45E

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