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:First Derivative
Error
f'(x)% =
O(h)
2h
-f(x)+ 8f(x)-8f(x) + f(x,-2)
f (x) =
12h
Second Derivative
f(x)-2f(x) +f(x-)
f"(x) =
-f(K) + 16f(x,)- 30ƒ(x) + 16ƒ(x,-1) – f(x,-)
12
f"(x) =-
O(k")
Third Derivative
f(x,,2) – 2f(x,) + 2f(x_;) – f(x__2)
f" (x,) =
O(h)
2h
-f(x,) + 8ƒ(x,,2) - 13f(x,,) + 13f(x,-1) – 8ƒ(x, 2) + f(x,)
8h
f"(x) =
Fourth Derivative
f(x)-4f(x,)+6f(x) – 4f(x,-1) + f(x}-2)
f"(x) =
-f) + 12f(x,,) + 39ƒ(x,,) + 56ƒ(x) – 39f(x_.,) + 12fK,--) + f(x_)
f"(x) =
In the above table, the formulations to calculate different order derivatives of a function are given by using the
çentral 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)=? ƒ ""(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
f (4.01)–f (3.99) 1.3888 –1.3838
= 0.25
f'(4.0) =
2(0.01)
0.02
Dört nokta için birinci türev
-f (4.02)+8ƒ(4.01) – 8ƒ (3.99)+ f (3.98) (-1.3913)+8(1.3888) – 8(1.3838)+1.3813
f'(4.0) =
= 0.25
12 (0.01)
12(0.01)
Analitik çözüm
f (x) = In (x) → ƒ(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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