Numerical differentiation and integration – MCQs – EE

1. Which method approximates the derivative using the function value at a current and next point?
(A) Forward difference
(B) Backward difference
(C) Central difference
(D) Simpson’s rule

2. Which method uses values from both sides of the point to approximate the derivative?
(A) Forward difference
(B) Backward difference
(C) Central difference
(D) Trapezoidal rule

3. Which method uses the current and previous point to approximate the derivative?
(A) Forward difference
(B) Backward difference
(C) Midpoint rule
(D) Simpson’s rule

4. What type of errors affect numerical differentiation?
(A) Truncation error
(B) Rounding error
(C) Both truncation and rounding errors
(D) None

5. How many points does a three-point differentiation formula use?
(A) One
(B) Two
(C) Three
(D) Four

6. Which numerical integration method uses straight-line approximation of the curve?
(A) Trapezoidal rule
(B) Simpson’s rule
(C) Midpoint rule
(D) Euler method

7. The trapezoidal rule is exact for which type of function?
(A) Linear
(B) Quadratic
(C) Cubic
(D) Exponential

8. Simpson’s 1/3 rule requires what type of interval?
(A) Odd number of intervals
(B) Even number of intervals
(C) Any number of intervals
(D) Prime number of intervals

9. Romberg integration is based on which combination?
(A) Trapezoidal rule and extrapolation
(B) Simpson’s rule only
(C) Euler method only
(D) Midpoint rule only

10. The midpoint rule uses which value to approximate the integral?
(A) Left endpoint
(B) Right endpoint
(C) Midpoint of interval
(D) Average of endpoints

11. Numerical differentiation becomes inaccurate due to what?
(A) Too small step size
(B) Too large step size
(C) Both small and large step size
(D) None

12. Higher-order differentiation formulas help to:
(A) Reduce truncation error
(B) Increase rounding error
(C) Both reduce truncation and increase rounding error
(D) None

13. Trapezoidal rule approximates the area using:
(A) Rectangles
(B) Trapezoids
(C) Parabolas
(D) Cubic curves

14. Simpson’s 3/8 rule uses which type of approximation?
(A) Linear
(B) Quadratic
(C) Cubic
(D) Quartic

15. Forward difference is less accurate than central difference because:
(A) It uses more points
(B) It has higher truncation error
(C) It uses midpoint
(D) It ignores endpoints

16. Decreasing the step size in differentiation may:
(A) Always improve accuracy
(B) Increase rounding error
(C) Have no effect
(D) Reduce computational time

17. Which methods can be used for equally spaced points?
(A) Forward difference
(B) Backward difference
(C) Central difference
(D) All of the above

18. Simpson’s 1/3 rule uses how many points per segment?
(A) Two
(B) Three
(C) Four
(D) Five

19. Composite rules in integration are used to:
(A) Increase accuracy
(B) Reduce computation
(C) Reduce step size
(D) Ignore error

20. Trapezoidal rule error depends on which derivative?
(A) First derivative
(B) Second derivative
(C) Third derivative
(D) Fourth derivative

21. Simpson’s 1/3 rule error depends on which derivative?
(A) First derivative
(B) Second derivative
(C) Third derivative
(D) Fourth derivative

22. Numerical differentiation is mainly used when:
(A) Analytical function is available
(B) Only discrete data points are available
(C) Function is simple
(D) Function is linear

23. Which method is preferred for smooth functions requiring high accuracy?
(A) Trapezoidal rule
(B) Simpson’s rule
(C) Midpoint rule
(D) Euler method

24. Forward difference for second derivative uses how many consecutive points?
(A) Two
(B) Three
(C) Four
(D) Five

25. Central difference for second derivative is:
(A) Less accurate than forward difference
(B) More accurate than forward difference
(C) Equal in accuracy
(D) Not used

26. Simpson’s 3/8 rule requires how many intervals per segment?
(A) Two
(B) Three
(C) Four
(D) Five

27. Numerical differentiation formulas are derived using:
(A) Taylor series expansion
(B) Fourier series
(C) Laplace transform
(D) Z-transform

28. Romberg integration improves trapezoidal rule using:
(A) Extrapolation
(B) Euler method
(C) Simpson’s rule
(D) Midpoint rule

29. Midpoint rule approximates the integral by:
(A) Left endpoint value
(B) Midpoint value
(C) Right endpoint value
(D) Average of endpoints

30. For small step size and smooth function, the most accurate method is:
(A) Trapezoidal rule
(B) Simpson’s 1/3 rule
(C) Forward difference
(D) Midpoint rule