Exercise 4.4 [Approximation of derivatives using interpolation polynomials]

EXERCISE 4.4 [Approximation of derivatives using interpolation polynomials]

1. Find the first and second derivative of √x at x = 15 from the table below.

 [Ans. 0.1289, -0.004]

2. Obtain the second derivative of y at x = 0.96 from the data.

 [Ans. -1.91666]

3. Find the first three derivatives of the function at x = 1.5 from the table below. [A.U N/D 2013]

[Ans. 4.75, 9.0, 6.0]

4. The following data give the corresponding values for pressure and specific volume of a superheated steam.

 [Ans. -52.4]

5. Find the minimum value of f (x) which has values.

 [Ans. 2.25]

6. Find the extremum values of y given the table below :

 [Ans. x = 4.42 min y = 2]

7. In a machine, a slider moves along a fixed straight rod. Its distance x cms along the rod is given below for various values of time t secs. Find the velocity and acceleration of the slider when t = 0.3

 [Ans. 5.375 cm/sec, -46.083 cm/sec]

8. Evaluate, y' and y'' at x = 2 given

 [Ans. -16, 4]

9. Determine f' (4) from the following table.

 (Note: Intervals are unequal)

[Ans. 2.83]

10. Find the maximum value of f (x) from the given table.

 [Ans. x = 1.58]

11. From the table given below, find f' (30), f' (31) and f' (35).

 [Ans. f' (30) 1.403, f' (31) = 0.806, f' (35) = 0.894]

12. Find f' (8) given ƒ (6) = 1.556, ƒ (7) 1.690, ƒ (9) 1.908, = f (12) = 2.158.

[Ans. 0.109]

13. A curve passes through the points (0, 18), (1, 10), (3, -18) and (6, 90).

Find the slope of the curve at x = 2.0.SI 2581.18

[Ans. -16]

14. Find the first and second derivatives of the function y = f (x) tabulated below at the point x = 1.1

 [Ans. 0.630; 6.6]

15. From the following data find dy/dx at x = 0.00.

 [Ans. 1.99999]

16. Give the following tabulated values of function, prepare a difference table and evaluate y' at x = 1.2

 [Ans. 3.3205]

17. Determine the first derivative of f (x) at x = 0.04 from the following table :

 [Ans. 0.2458]

18. Calculate dy/dx and d²y/dx² at x = 1.35 from the following data:

 [Ans. 29.34, 71.33]