Homework Help is Here – Start Your Trial Now!
Math
Advanced Math
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
See similar textbooks
icon
Related questions
Question
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.
Expert Solution
steps

Step by step

Solved in 2 steps with 9 images

SEE SOLUTIONCheck out a sample Q&A here
Blurred answer
Similar questions
  • SEE MORE QUESTIONS
Recommended textbooks for you
Advanced Engineering Mathematics
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Mathematics For Machine Technology
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
Basic Technical Mathematics
Basic Technical Mathematics
Advanced Math
ISBN:
9780134437705
Author:
Washington
Publisher:
PEARSON
Topology
Topology
Advanced Math
ISBN:
9780134689517
Author:
Munkres, James R.
Publisher:
Pearson,
SEE MORE TEXTBOOKS