First Derivative Error f(x,.,) – f(x,-1) 2h f'(x) = O(h) -f(x,2) + 8ƒ (x,1) – 8f(x,-1) + f(x,2) 12h f'(x) = O(h") Second Derivative f(x)- 2f(x) +f(x}-1) f"(x) = O(h) -f(x,,2) + 16ƒ(x,) – 30ƒ(x) + 16ƒ(x,-1) – f(x,-2) 12A2 f"(x) = O(h*) Third Derivative f(x,,,) – 2f (x,1) + 2f(x,_;) – f(x_2) 2h f"(x) = O(h) -f(x,,) + 8ƒ(x,,2) – 13f(x,) + 13ƒ(x,-1) – 8ƒ(x,-2) + ƒ(x,-) f"(x) = O(h") 8h Fourth Derivative f(x)- 4f(x1) + 6f(x) – 4f(x,-1) + f(x}-2) f™(x) = O(h") -{(x;,s) +12f(x,»2)– 39f (x,1)+56f (x, )–39 f(x-1)+12f (x-2)-S(x-3) f"(x,) = 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) = ? f "(4.0) =? f "(4.0) = ? f "(4.0) = ? h = 0.01 için x, = 4.00 x- = 4.01 x,- = 3.99 x,-2 = 4.02 x,-2 = 3.98 İki nokta için birinci türev S'(4.0) =- f (4.01)– ƒ (3.99) 1.3888 –1.3838 = 0.25 2(0.01) 0.02 Dört nokta için birinci türev -S (4.02)+8ƒ (4.01) –8ƒ (3.99)+ ƒ(3.98) _ (-1.3913)+8(1.3888)– 8(1.3838)+1.3813 =0.25 f'(4.0) =- 12 (0.01) 12(0.01) Analitik çözüm f(x) = In (x) → S(x) = 1/x → S'(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 Error f(x,.,) – f(x,-1) 2h f'(x) = O(h) -f(x,2) + 8ƒ (x,1) – 8f(x,-1) + f(x,2) 12h f'(x) = O(h") Second Derivative f(x)- 2f(x) +f(x}-1) f"(x) = O(h) -f(x,,2) + 16ƒ(x,) – 30ƒ(x) + 16ƒ(x,-1) – f(x,-2) 12A2 f"(x) = O(h*) Third Derivative f(x,,,) – 2f (x,1) + 2f(x,_;) – f(x_2) 2h f"(x) = O(h) -f(x,,) + 8ƒ(x,,2) – 13f(x,) + 13ƒ(x,-1) – 8ƒ(x,-2) + ƒ(x,-) f"(x) = O(h") 8h Fourth Derivative f(x)- 4f(x1) + 6f(x) – 4f(x,-1) + f(x}-2) f™(x) = O(h") -{(x;,s) +12f(x,»2)– 39f (x,1)+56f (x, )–39 f(x-1)+12f (x-2)-S(x-3) f"(x,) = 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) = ? f "(4.0) =? f "(4.0) = ? f "(4.0) = ? h = 0.01 için x, = 4.00 x- = 4.01 x,- = 3.99 x,-2 = 4.02 x,-2 = 3.98 İki nokta için birinci türev S'(4.0) =- f (4.01)– ƒ (3.99) 1.3888 –1.3838 = 0.25 2(0.01) 0.02 Dört nokta için birinci türev -S (4.02)+8ƒ (4.01) –8ƒ (3.99)+ ƒ(3.98) _ (-1.3913)+8(1.3888)– 8(1.3838)+1.3813 =0.25 f'(4.0) =- 12 (0.01) 12(0.01) Analitik çözüm f(x) = In (x) → S(x) = 1/x → S'(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.
Functions and Change: A Modeling Approach to College Algebra (MindTap Course List)
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Chapter1: Functions
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Can you help me? it's about numerical numbers question. I want to fourth derivative like this first derivative solution. And with analytical solution.
Transcribed Image Text:ll %25 12:43 PM
In the above table - Salt okunur
Bu hesapla cihazınızda düzenlemeye izin veril..
First Derivative
Error
f (x+1) – f(x;-1)
2h
f'(x) =
O(h)
-f(x+2) + 8f (x) – 8ƒ(x,-1) + f(x_-2)
f'(x) =
O(h")
12h
Second Derivative
f(x;-1) – 2ƒ(x) +f(xj-1)
f"(x) =
O(h)
h?
-f(x+2) + 16f(x+) - 30f(x) + 16f (x,-1) – f(x,-2)
12½²
f"(x) =
O(h*)
Third Derivative
f(x,,2) – 2f (x,.,) + 2f(xj_1) – f (x,_2)
f"(x) =
O(h?)
2h
cr(e--f(x,) + 8ƒ(x;,2) – 13f(x;+1) + 13f(x,-1) – 8f(x_-2) + f(x_2)
8h
O(h*)
Fourth Derivative
f(X+2) – 4f(x;=1) +6f(x) – 4f(x-1)+ f(xj-2)
f" (x) = -
O(h")
-f(x;,3) +12f(x;,2)– 39f (x;,-1)+ 56f (x; )– 39 f (x;-1)+12f (x;-2)-5 (x-3)
f "(x,) =.
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-1 = 4.01 x-1=3.99 x,42 = 4.02 x-2 = 3.98
İki nokta için birinci türev
f (4.01)– ƒ (3.99) 1.3888–1.3838
= 0.25
S'(4.0) =
2(0.01)
0.02
Dört nokta için birinci türev
-f (4.02)+8ƒ(4.01) – 8ƒ (3.99)+ ƒ (3.98) (-1.3913)+8(1.3888)–8(1.3838)+1.3813
= 0.25
f'(4.0) =
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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