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Mathematics Calculus Wolfram Language

Differentiate equation TC with respect to T

VANI MADHAVI TUKKAPURAM

VANI MADHAVI TUKKAPURAM, Hyderabad

Posted 3 years ago

TC == 1/T {ChD1((\[Lambda](c(s-1)-td)(T-td)^2)/2+ (a(T+c(1-s))(T-td)^b+1)/(b+1)-(2a(T-td)^b+2)/((b+1)(b+2))+ T^2/2-((T^3)\[Lambda])/3 + T^2\[Lambda]td-(3T\[Lambda]td^2)/2+(\[Lambda]td^3)/6)+K} /.{D1 -> 4000, c -> 10, Ch -> 2, s -> 0.1, K -> 200,a -> 0.2, b -> 1.5,td -> 20/365, \[Lambda] -> 0.2} D[TC, T]==0 Please evaluate.

    TC == 1/T {ChD1((\[Lambda](c(s-1)-td)(T-td)^2)/2+
    (a(T+c(1-s))(T-td)^b+1)/(b+1)-(2a(T-td)^b+2)/((b+1)(b+2))+
    T^2/2-((T^3)\[Lambda])/3 + 
    T^2\[Lambda]td-(3T\[Lambda]td^2)/2+(\[Lambda]td^3)/6)+K}
    /.{D1 -> 4000, c -> 10, Ch -> 2, s -> 0.1, K -> 200,a -> 0.2, b -> 1.5,td -> 20/365, 
\[Lambda] -> 0.2}
    D[TC, T]==0

Please evaluate.

POSTED BY: VANI MADHAVI TUKKAPURAM

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Mariusz Iwaniuk

Mariusz Iwaniuk, engineer, math hobbyist.

Posted 3 years ago

ClearAll["`"] TC = Rationalize[ 1/T {ChD1 ((\[Lambda](c(s - 1) - td)(T - td)^2)/ 2 + (a(T + c(1 - s))(T - td)^b + 1)/(b + 1) - (2a(T - td)^b + 2)/((b + 1)(b + 2)) + T^2/2 - ((T^3) \[Lambda])/3 + T^2\[Lambda]td - (3 T\[Lambda]td^2)/ 2 + (\[Lambda]td^3)/6) + k} /. {D1 -> 4000, c -> 10, Ch -> 2, s -> 0.1, k -> 200, a -> 0.2, b -> 1.5, td -> 20/365, \[Lambda] -> 0.2}, 0][[1]] // FullSimplify ((200 + 8000 (2334134/5835255 - 4/35 (2 + 2/5 (-(4/73) + T)^(3/2)) - ( 11203068469727 (-(4/73) + T)^2)/12372526449169 - (24 T)/26645 + ( 373 T^2)/730 - T^3/15 + 2/25 (-(4/73) + T)^(3/2) (9 + T)))/T) func = D[TC, T] // FullSimplify (-(2850630093888538400/903194430789337) + 960 Sqrt[-(4/73) + T] + (-(156639186421396108130600/ 101075585555203914333) + 151040/511 Sqrt[-(4/73) + T])/T^2 + ( 18880 Sqrt[-(4/73) + T])/(7 T) - (3200 T)/3) Solve[func == 0, T] // N ({{T -> -3.03984 - 0.158213 I}, {T -> -3.03984 + 0.158213 I}, {T -> 0.582782 - 0.712647 I}, {T -> 0.582782 + 0.712647 I}})

ClearAll["`*"]
TC = Rationalize[
    1/T {Ch*D1 ((\[Lambda]*(c*(s - 1) - td)*(T - td)^2)/
            2 + (a*(T + c*(1 - s))*(T - td)^b + 1)/(b + 
              1) - (2*a*(T - td)^b + 2)/((b + 1)*(b + 2)) + 
           T^2/2 - ((T^3) \[Lambda])/3 + 
           T^2*\[Lambda]*td - (3 T*\[Lambda]*td^2)/
            2 + (\[Lambda]*td^3)/6) + k} /. {D1 -> 4000, c -> 10, 
      Ch -> 2, s -> 0.1, k -> 200, a -> 0.2, b -> 1.5, 
      td -> 20/365, \[Lambda] -> 0.2}, 0][[1]] // FullSimplify
 (*(200 + 8000 (2334134/5835255 - 4/35 (2 + 2/5 (-(4/73) + T)^(3/2)) - (
     11203068469727 (-(4/73) + T)^2)/12372526449169 - (24 T)/26645 + (
     373 T^2)/730 - T^3/15 + 2/25 (-(4/73) + T)^(3/2) (9 + T)))/T*)

 func = D[TC, T] // FullSimplify
 (*-(2850630093888538400/903194430789337) + 
  960 Sqrt[-(4/73) + T] + (-(156639186421396108130600/
    101075585555203914333) + 151040/511 Sqrt[-(4/73) + T])/T^2 + (
  18880 Sqrt[-(4/73) + T])/(7 T) - (3200 T)/3*)

  Solve[func == 0, T] // N
  (*{{T -> -3.03984 - 0.158213 I}, {T -> -3.03984 + 0.158213 I}, {T -> 
     0.582782 - 0.712647 I}, {T -> 0.582782 + 0.712647 I}}*)

POSTED BY: Mariusz Iwaniuk

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