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First Derivative f(x,) – f(x,-1) 2h Error f'(x) = -f(x,) + 8ƒ(x,.) – 8ƒ(x}-1) + ƒ(x_-2) 12h f(x) = O(h") Second Derivative f(x)- 2f(x) +f(x}-1) f"(x) = O(4) -f(x2) + 16f(x) - 30f(x) + 16f (x-1) – f(x_-2) 12A f"(x) = O(h") Third Derivative f(x,) – 2f(x,.) + 2ƒ(x_}) – f(x,-2) f" (x) = O(h) 2h -f(x,,) + 8ƒ(x,»2) – 13f(x,) + 13ƒ(x,-1) – 8ƒ(x__2) + ƒ(x,--) f"(x,) = 8h Fourth Derivative f(X2) – 4f(x,»1) + 6ƒ(x) – 4f(x,-1) + f(x\.2) f"(x) = ht O(h*) fm)=f(x;,3) +12f(x,,2)– 39f (x,.1)+56ƒ (x,)– 39 f (x-1)+12f(x_2)-ƒ (x-3) 6h* In the above table, the formulations to calculate different order derivatives of a function are given by using the central difference method. Eor the function f (x) = In (x), obtain the first, second, third and fourth. order derivatives of this function by using the above methods for the neighborhood step h = 0.01 at the point x = 4.0. Soru çözüm formatı oluşturması adına birinci türevin elde edilme yöntemi aşağıda verilmiştir f(x) = In (x) ƒ (4.0) = ? f"(4.0)=? f"(4.0)=? f "(4.0) = ? h = 0.01 için x, = 4.00 x, = 4.01 x1 = 3.99 x = 4.02 x = 3.98 İki nokta için birinci türev f(4.01)- f (3.99) 1.3888 –1.3838 =0.25 S'(4.0) = 2(0.01) 0.02 Dört nokta için birinci türev L'(4.0) -/(4.02)+8ƒ(4.01) – 8ƒ (3.99)+ S(3.98) _ (-1.3913)+8(1.3888)– 8(1.3838)+1.3813 = 0.25 12 (0.01) 12(0.01) Analitik çözüm f(x) = In (x) → f(x) =1/x → f'(4.0)= 0.25 Using the solution format given above, obtain the second, third and fourth order derivatives of the function f (x) = In (x). Compare the results you get with the numerical solution with the derivatives you get with the analytical solution for the relevant function.

First Derivative f(x,) – f(x,-1) 2h Error f'(x) = -f(x,) + 8ƒ(x,.) – 8ƒ(x}-1) + ƒ(x_-2) 12h f(x) = O(h") Second Derivative f(x)- 2f(x) +f(x}-1) f"(x) = O(4) -f(x2) + 16f(x) - 30f(x) + 16f (x-1) – f(x_-2) 12A f"(x) = O(h") Third Derivative f(x,) – 2f(x,.) + 2ƒ(x_}) – f(x,-2) f" (x) = O(h) 2h -f(x,,) + 8ƒ(x,»2) – 13f(x,) + 13ƒ(x,-1) – 8ƒ(x__2) + ƒ(x,--) f"(x,) = 8h Fourth Derivative f(X2) – 4f(x,»1) + 6ƒ(x) – 4f(x,-1) + f(x\.2) f"(x) = ht O(h*) fm)=f(x;,3) +12f(x,,2)– 39f (x,.1)+56ƒ (x,)– 39 f (x-1)+12f(x_2)-ƒ (x-3) 6h* In the above table, the formulations to calculate different order derivatives of a function are given by using the central difference method. Eor the function f (x) = In (x), obtain the first, second, third and fourth. order derivatives of this function by using the above methods for the neighborhood step h = 0.01 at the point x = 4.0. Soru çözüm formatı oluşturması adına birinci türevin elde edilme yöntemi aşağıda verilmiştir f(x) = In (x) ƒ (4.0) = ? f"(4.0)=? f"(4.0)=? f "(4.0) = ? h = 0.01 için x, = 4.00 x, = 4.01 x1 = 3.99 x = 4.02 x = 3.98 İki nokta için birinci türev f(4.01)- f (3.99) 1.3888 –1.3838 =0.25 S'(4.0) = 2(0.01) 0.02 Dört nokta için birinci türev L'(4.0) -/(4.02)+8ƒ(4.01) – 8ƒ (3.99)+ S(3.98) _ (-1.3913)+8(1.3888)– 8(1.3838)+1.3813 = 0.25 12 (0.01) 12(0.01) Analitik çözüm f(x) = In (x) → f(x) =1/x → f'(4.0)= 0.25 Using the solution format given above, obtain the second, third and fourth order derivatives of the function f (x) = In (x). Compare the results you get with the numerical solution with the derivatives you get with the analytical solution for the relevant function.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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First Derivative Error f(x)-f(x-1) f'(x) = O(h?) 2h -f(x,2) + 8f (x,-1) – 8ƒ(x,-1) + f(x,-2) 12h f'(x) = O(h") Second Derivative f(x)- 2f(x) +f(x-) f"(x) = O(h) h? -f (x2) + 16f(x)-30f(x) + 16f (x)-1) – f(x)-2) 12½² f"(x) = Third Derivative f(x,2) – 2f(x,) + 2f(x,1) – f (x,-2) 2h f" (x) = O(h) -f(x,s) + 8ƒ(x;+2) – 13ƒ(x,+1) + 13ƒ(x,-1) – 8ƒ (x,--2) + f (x)--) f"(x,) = O(h") 8h Fourth Derivative f(x2)-4f(x) +6f(x) – 4f(x;-1) + f(x}-2) f"' (x) = O(h") (mr)=f(xs)+12f (x;u2)– 39f (*1)+ 56ƒ (x, )– 39 f(x-1) +12f (x,-_2)– f (*-3) 6h* In the above table, the formulations to calculate different order derivatives of a function are given by using the central difference method. For the function f (x) = In (x), obtain the first, second, third and fourth order derivatives of this function by using the above methods for the neighborhood step h = 0.01 at the point x = 4.0. Soru çözüm formatı oluşturması adına birinci türevin elde edilme yöntemi aşağıda verilmiştir f(x) = In (x) ƒ(4.0) = ? ƒ "(4.0)=? f"(4.0) = ? f "(4.0) = ? h = 0.01 için x, = 4.00 x = 4.01 x,-1 = 3.99 x,2 = 4.02 x-2 = 3.98 İki nokta için birinci türev f (4.01)– ƒ (3.99) 1.3888–1.3838 2(0.01) S'(4.0) = = 0,25 0.02 Dört nokta için birinci türev -(4.02)+ 8f (4.01) – 8f (3.99)+ f(3.98) _ (-1.3913)+8(1.3888)– 8(1.3838)+1.3813 S'(4.0) = = 0.25 12 (0.01) 12(0.01) Analitik çözüm f(x) = In (x) f (x) = 1 / x → f '(4.0) = 0.25 Using the solution format given above, obtain the second, third and fourth order derivatives of the function f (x) = In (x). Compare the results you get with the numerical solution with the derivatives you get with the analytical solution for the relevant function.
Transcribed Image Text:First Derivative Error f(x)-f(x-1) f'(x) = O(h?) 2h -f(x,2) + 8f (x,-1) – 8ƒ(x,-1) + f(x,-2) 12h f'(x) = O(h") Second Derivative f(x)- 2f(x) +f(x-) f"(x) = O(h) h? -f (x2) + 16f(x)-30f(x) + 16f (x)-1) – f(x)-2) 12½² f"(x) = Third Derivative f(x,2) – 2f(x,) + 2f(x,1) – f (x,-2) 2h f" (x) = O(h) -f(x,s) + 8ƒ(x;+2) – 13ƒ(x,+1) + 13ƒ(x,-1) – 8ƒ (x,--2) + f (x)--) f"(x,) = O(h") 8h Fourth Derivative f(x2)-4f(x) +6f(x) – 4f(x;-1) + f(x}-2) f"' (x) = O(h") (mr)=f(xs)+12f (x;u2)– 39f (*1)+ 56ƒ (x, )– 39 f(x-1) +12f (x,-_2)– f (*-3) 6h* In the above table, the formulations to calculate different order derivatives of a function are given by using the central difference method. For the function f (x) = In (x), obtain the first, second, third and fourth order derivatives of this function by using the above methods for the neighborhood step h = 0.01 at the point x = 4.0. Soru çözüm formatı oluşturması adına birinci türevin elde edilme yöntemi aşağıda verilmiştir f(x) = In (x) ƒ(4.0) = ? ƒ "(4.0)=? f"(4.0) = ? f "(4.0) = ? h = 0.01 için x, = 4.00 x = 4.01 x,-1 = 3.99 x,2 = 4.02 x-2 = 3.98 İki nokta için birinci türev f (4.01)– ƒ (3.99) 1.3888–1.3838 2(0.01) S'(4.0) = = 0,25 0.02 Dört nokta için birinci türev -(4.02)+ 8f (4.01) – 8f (3.99)+ f(3.98) _ (-1.3913)+8(1.3888)– 8(1.3838)+1.3813 S'(4.0) = = 0.25 12 (0.01) 12(0.01) Analitik çözüm f(x) = In (x) f (x) = 1 / x → f '(4.0) = 0.25 Using the solution format given above, obtain the second, third and fourth order derivatives of the function f (x) = In (x). Compare the results you get with the numerical solution with the derivatives you get with the analytical solution for the relevant function.
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