Numerical Analysis Questions and Answers – Gauss’s Forward Interpolation Formula

This set of Numerical Analysis Multiple Choice Questions & Answers (MCQs) focuses on “Gauss’s Forward Interpolation Formula”.

1. What will be the solution for the following table using Gauss’s forward interpolation formula, where x = 3?

 x f(x) 1 1 2 1 3 1

a) 1
b) 0.328
c) 0.327
d) 0.322

Explanation:
Gauss's forward method to find solution
h=2-1=1
The central difference table is

 x p=(x-2)/1 y Δy 1 -1 1 0 2 0 1 0 3 1 1

Solution of Gauss's forward interpolation is y(3)=1.

2. What will be the solution for the following table using Gauss’s forward interpolation formula, where x = 2?

 x f(x) 1 2 2 3 3 4

a) 0.2452
b) 0.2864
c) 0.2862
d) 3

Explanation:
Gauss's forward method to find solution
h=2-1=1
Now the central difference table is

 x p=(x-2)/1 y Δy Δ2y 1 -1 2 1 2 0 3 0 1 3 1 4
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Solution of Gauss's forward interpolation is y(2)=3.

3. What will be the solution for the following table using Gauss’s forward interpolation formula, where x = 1?

 x f(x) 1 2 2 3 3 4 4 8

a) 11.152
b) 2
c) 11.122
d) 11.128

Explanation: Given,
Gauss's forward method to find solution
h=2-1=1
Now the central difference table is

 x p=(x-2)/1 y Δy Δ2y Δ3y 1 -1 2 1 2 0 3 0 1 3 3 1 4 3 4 4 2 8

Solution of Gauss's forward interpolation is y(1)=2.

4. What will be the solution for the following table using Gauss’s forward interpolation formula, where x = 2.5?

 x f(x) 1 2 2 6 3 4 4 8 5 10 6 12

a) 1
b) 4.1774
c) 4.1797
d) 2

Explanation:
Gauss's forward method to find solution
h=2-1=1
Now the central difference table is

 x p=(x-3)/1 y Δy Δ2y Δ3y Δ4y Δ5y 1 -2 2 4 2 -1 6 -6 -2 12 3 0 4 6 -20 4 -8 30 4 1 8 -2 10 2 2 5 2 10 0 2 6 3 12

Solution of Gauss's forward interpolation is y(2.5)=4.1797.

5. What will be the solution for the following table using Gauss’s forward interpolation formula, where x = 4?

 x f(x) 1 2 2 6 3 4 4 8 5 10 6 12 7 24 8 26 9 28

a) 8
b) 9
c) 7
d) 6

Explanation:
Gauss's forward method to find solution
h=2-1=1
Now the central difference table is

 x p=(x-5)/1 y Δy Δ2y Δ3y Δ4y Δ5y Δ6y Δ7y Δ8y 1 -4 2 4 2 -3 6 -6 -2 12 3 -2 4 6 -20 4 -8 30 4 -1 8 -2 10 -32 2 2 -2 -4 5 0 10 0 8 -36 138 2 10 -38 134 6 1 12 10 -30 98 12 -20 60 7 2 24 -10 30 2 10 8 3 26 0 2 9 4 28

Solution of Gauss's forward interpolation is y(4)=8.

6. What will be the solution of the equation 4x+1 using Gauss’s forward interpolation formula where x1
= 2 and x2 = 4 and x = 2.1, step value (h) = 0.25?
a) 8.4
b) 9.4
c) 8
d) 5

Explanation:
Gauss's forward method to find solution
h=2.25-2=0.25
Now the central difference table is

 x p=(x-3)/0.25 y Δy Δ2y 2 -4 9 1 2.25 -3 10 0 1 2.5 -2 11 0 1 2.75 -1 12 0 1 3 0 13 0 1 3.25 1 14 0 1 3.5 2 15 0 1 3.75 3 16 0 1 4 4 17

Solution of Gauss's forward interpolation is y(2.1)=9.4.

7. What will be the solution of the equation 2x 2 using Gauss’s forward interpolation formula, where x1 = 1 and x2 = 2 & x = 2 & step value (h) = 0.25?
a) 2
b) 7
c) 5
d) 8

Explanation:
Gauss's forward method to find solution
h=1.25-1=0.25

Now the central difference table is

 x p=(x-1.5)/0.25 y Δy Δ2y Δ3y 1 -2 2 1.125 1.25 -1 3.125 0.25 1.375 0 1.5 0 4.5 0.25 1.625 0 1.75 1 6.125 0.25 1.875 2 2 8

Solution of Gauss's forward interpolation is y(2)=8.

8. What will be the solution of the equation 3x using Gauss’s forward interpolation formula, where x1 =
2 and x2 = 4 & x = 2 & step value (h) = 0.25?
a) 8
b) 6
c) 5
d) 1

Explanation:
Gauss's forward method to find solution
h=2.25-2=0.25
Now the central difference table is

 X p=(x-3)/0.25 y Δy Δ2y 2 -4 6 0.75 2.25 -3 6.75 0 0.75 2.5 -2 7.5 0 0.75 2.75 -1 8.25 0 0.75 3 0 9 0 0.75 3.25 1 9.75 0 0.75 3.5 2 10.5 0 0.75 3.75 3 11.25 0 0.75 4 4 12

Solution of Gauss's forward interpolation is y(2)=6.

9. What will be the solution of the equation x + 1 using Gauss’s forward interpolation formula, where x1 = 1 and x2 = 2 & x = 1 & step value (h) = 0.25?
a) 1
b) 3
c) 2
d) 4

Explanation:
Gauss's forward method to find solution
h=1.25-1=0.25
Now the central difference table is

 X p=(x-1.5)/0.25 y Δy Δ2y 1 -2 2 0.25 1.25 -1 2.25 0 0.25 1.5 0 2.5 0 0.25 1.75 1 2.75 0 0.25 2 2 3

Solution of Gauss's forward interpolation is y(1)=2.

10. What will be the solution of the equation x using Gauss’s forward interpolation formula, where x1 = 1 and x2 = 2 & x = 1 & step value (h) = 0.25?
a) 4
b) 2
c) 3
d) 1

Explanation:
Gauss's forward method to find solution
h=1.25-1=0.25
Now the central difference table is

 X p=(x-1.5)/0.25 y Δy Δ2y 1 -2 1 0.25 1.25 -1 1.25 0 0.25 1.5 0 1.5 0 0.25 1.75 1 1.75 0 0.25 2 2 2

Solution of Gauss's forward interpolation is y(1)=1.

11. What will be the solution of the equation sin(x) using Gauss’s forward interpolation formula, where x1 = 1 and x2 = 2 & x = 1 & step value (h) = 0.25?
a) 4
b) 2
c) 3
d) 0.0167

Explanation:
Gauss's forward method to find solution
h=1.25-1=0.25
Now the central difference table is

 X p=(x-1.5)/0.25 y Δy Δ2y Δ3y Δ4y 1 -2 0.8415 0.1075 1.25 -1 0.949 -0.059 0.0485 -0.003 1.5 0 0.9975 -0.062 0.0039 -0.0135 0.0008 1.75 1 0.984 -0.0612 -0.0747 2 2 0.9093

Solution of Gauss's forward interpolation is y()=0.0167.

12. What will be the solution of the equation cos(x) using Gauss’s forward interpolation formula, where x1 = 1 and x2 = 2 & x = 1 & step value (h) = 0.25?
a) 0.9496
b) 2
c) 3
d) 0.0167

Explanation:
Gauss's forward method to find solution

h=1.25-1=0.25
Now the central difference table is

 X p=(x-1.5)/0.25 y Δy Δ2y Δ3y Δ4y 1 -2 0.5403 -0.225 1.25 -1 0.3153 -0.0196 -0.2446 0.0152 1.5 0 0.0707 -0.0044 0.0003 -0.249 0.0155 1.75 1 -0.1782 0.0111 -0.2379 2 2 -0.4161

Solution of Gauss's forward interpolation is y()=0.9496.

13. What will be the solution of the equation tan(x) using Gauss’s forward interpolation formula, where x1 = 1 and x2 = 2 & x = 1 & step value (h) = 0.25?
a) 0.9496
b) 4190.0569
c) 3
d) 0.0167

Explanation:
Gauss's forward method to find solution
h=1.25-1=0.25

Now the central difference table is

 X p=(x-1.5)/0.25 y Δy Δ2y Δ3y Δ4y 1 -2 1.5574 1.4522 1.25 -1 3.0096 9.6397 11.0919 -40.3533 1.5 0 14.1014 -30.7137 94.0241 -19.6218 53.6708 1.75 1 -5.5204 22.9571 3.3353 2 2 -2.185

Solution of Gauss's forward interpolation is y()=4190.0569.

14. What will be the solution of the equation cot(x) using Gauss’s forward interpolation formula, where x1 = 1 and x2 = 2 & x = 1 & step value (h) = 0.25?
a) 0.9496
b) 4190.0569
c) 3.3389
d) 0.0167

Explanation:
Gauss's forward method to find solution
h=1.25-1=0.25
Now the central difference table is

 X p=(x-1.5)/0.25 y Δy Δ2y Δ3y Δ4y 1 -2 0.6421 -0.3098 1.25 -1 0.3323 0.0485 -0.2614 -0.0392 1.5 0 0.0709 0.0093 0.0054 -0.2521 -0.0337 1.75 1 -0.1811 -0.0244 -0.2765 2 2 -0.4577

Solution of Gauss's forward interpolation is y()=3.3389.

15. What will be the solution of the equation sec(x) using Gauss’s forward interpolation formula, where x1 = 1 and x2 = 2 & x = 1 & step value (h) = 0.25?
a) 4191.0168
b) 4190.0569
c) 3.3389
d) 0.0167

Explanation:
Gauss's forward method to find solution
h=1.25-1=0.25
Now the central difference table is

 X p=(x-1.5)/0.25 y Δy Δ2y Δ3y Δ4y 1 -2 1.8508 1.3205 1.25 -1 3.1714 9.6449 10.9655 -40.3575 1.5 0 14.1368 -30.7125 94.0243 -19.7471 53.6668 1.75 1 -5.6102 22.9543 3.2072 2 2 -2.403

Solution of Gauss's forward interpolation is y()=4191.0168.

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