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Unit 5: Interpolation, Differentiation and Integration - Practice Quiz

ECE183 — Mathematics For Robotics 60 Questions
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1 Newton's divided-difference interpolation is mainly used to estimate the value of a function between known data points. What type of data is required?

Newton's divided-difference interpolating polynomials Easy
A. Known values of and
B. Only the values of
C. Only the first derivative values
D. Only the values of

2 What is the first divided difference of two data points and ?

Newton's divided-difference interpolating polynomials Easy
A.
B.
C.
D.

3 Newton's divided-difference polynomial is especially useful when the -values are:

Newton's divided-difference interpolating polynomials Easy
A. Unequally spaced
B. Restricted to integers only
C. All equal
D. Always zero

4 In Lagrange interpolation, the polynomial is written as a sum of:

Lagrange interpolating polynomials Easy
A. Only constant terms
B. Basis polynomials multiplied by function values
C. Derivatives multiplied by step sizes
D. Integrals divided by interval lengths

5 What is the value of the Lagrange basis polynomial at its own interpolation point ?

Lagrange interpolating polynomials Easy
A.
B.
C.
D.

6 A Lagrange interpolating polynomial passing through three distinct data points has degree at most:

Lagrange interpolating polynomials Easy
A. 2
B. 4
C. 3
D. 1

7 Newton's forward interpolation formula is generally used near the:

Newton's forward and backward interpolating polynomials Easy
A. Largest derivative value
B. Beginning of the table
C. End of the table
D. Middle of an unrelated table

8 Newton's backward interpolation formula is generally used near the:

Newton's forward and backward interpolating polynomials Easy
A. End of the table
B. Smallest function value
C. Origin of the coordinate system
D. Beginning of the table

9 Newton's forward and backward interpolation formulas usually require the data points to be:

Newton's forward and backward interpolating polynomials Easy
A. All at the origin
B. Equally spaced
C. Measured only in radians
D. Randomly ordered

10 In finite-difference notation, the forward difference of is written as:

Newton's forward and backward interpolating polynomials Easy
A.
B.
C.
D.

11 Numerical integration is used to approximate:

Numerical integration Easy
A. A polynomial degree
B. Only a maximum value
C. A matrix inverse
D. A definite integral

12 Geometrically, the definite integral represents the signed area between the curve and the:

Numerical integration Easy
A. Line only
B. -axis only
C. -axis
D. Line only

13 Why are numerical integration methods useful in robotics?

Numerical integration Easy
A. They approximate integrals from sampled data
B. They eliminate all measurement errors
C. They replace every sensor with a motor
D. They guarantee an exact result for every function

14 Newton-Cotes formulae approximate a function over an interval using:

Newton-Cotes formulae Easy
A. A lookup table with no function values
B. Only the interval midpoint
C. Random derivative values
D. Function values at selected points

15 The closed Newton-Cotes formulas include function values at:

Newton-Cotes formulae Easy
A. Neither endpoint
B. Only points outside the interval
C. Only the left endpoint
D. Both endpoints

16 The trapezoidal rule is a Newton-Cotes formula based on a polynomial of degree:

Newton-Cotes formulae Easy
A. 0
B. 2
C. 3
D. 1

17 What is the single-interval trapezoidal rule for approximating ?

Trapezoidal rule Easy
A.
B.
C.
D.

18 The geometric shape used by the trapezoidal rule to approximate the area under a curve is a:

Trapezoidal rule Easy
A. Rectangle
B. Circle
C. Trapezoid
D. Triangle only

19 Simpson's rule approximates the integrand using a polynomial of degree:

Simpson's 1/3 rule Easy
A. 4
B. 2
C. 3
D. 1

20 For the single application of Simpson's rule, the number of equal subintervals must be:

Simpson's 1/3 rule Easy
A. 3
B. 4
C. 2
D. 1

21 Given the data points , , and , what is the interpolated value at using a second-degree Newton divided-difference polynomial?

Newton's divided-difference interpolating polynomials Medium
A.
B.
C.
D.

22 For the points , , and , what is the coefficient of the quadratic term in the Newton divided-difference polynomial?

Newton's divided-difference interpolating polynomials Medium
A.
B.
C.
D.

23 Using the data , , and , which Newton interpolating polynomial is obtained?

Newton's divided-difference interpolating polynomials Medium
A.
B.
C.
D.

24 Using Lagrange interpolation for , , and , estimate the value at .

Lagrange interpolating polynomials Medium
A.
B.
C.
D.

25 The data points , , and are interpolated by a quadratic Lagrange polynomial. What is its value at ?

Lagrange interpolating polynomials Medium
A.
B.
C.
D.

26 For Lagrange interpolation using three data points, what is the degree of the interpolating polynomial unless the data produce a lower-degree result?

Lagrange interpolating polynomials Medium
A. At most degree
B. At least degree
C. Exactly degree
D. At most degree

27 For equally spaced data with , , and values , , , and , estimate using Newton's forward formula.

Newton's forward and backward interpolating polynomials Medium
A.
B.
C.
D.

28 For equally spaced data at with values , estimate the value at using Newton's backward formula.

Newton's forward and backward interpolating polynomials Medium
A.
B.
C.
D.

29 Which interpolation formula is generally most convenient when the required value is close to the beginning of an equally spaced data table?

Newton's forward and backward interpolating polynomials Medium
A. A cubic spline formula
B. Newton's backward formula
C. A central difference formula
D. Newton's forward formula

30 In Newton's backward interpolation formula, the parameter used for interpolation is typically defined as:

Newton's forward and backward interpolating polynomials Medium
A.
B.
C.
D.

31 What is the main purpose of numerical integration in a robotics application?

Numerical integration Medium
A. To estimate an accumulated quantity from sampled data
B. To determine the roots of a polynomial exactly
C. To convert a nonlinear model into a linear one
D. To replace all sensor measurements with constants

32 Using one trapezoidal panel, approximate .

Numerical integration Medium
A.
B.
C.
D.

33 If the step size is reduced while applying a convergent numerical integration method to a smooth function, what generally happens to the approximation?

Numerical integration Medium
A. The error necessarily increases
B. The error always becomes exactly zero
C. The error usually decreases
D. The approximation becomes independent of function values

34 Newton-Cotes formulae approximate an integral by:

Newton-Cotes formulae Medium
A. Differentiating the integrand symbolically
B. Interpolating the integrand with a polynomial
C. Replacing the interval with a single endpoint
D. Finding the exact antiderivative

35 Which Newton-Cotes rule uses equally spaced points and a quadratic interpolating polynomial over two subintervals?

Newton-Cotes formulae Medium
A. The midpoint rule
B. Boole's rule
C. The trapezoidal rule
D. Simpson's rule

36 Using the composite trapezoidal rule with , approximate .

Trapezoidal rule Medium
A.
B.
C.
D.

37 For a strictly convex function such as on an interval, the single-panel trapezoidal rule generally:

Trapezoidal rule Medium
A. Underestimates the integral
B. Produces zero error for every interval
C. Overestimates the integral
D. Matches the midpoint rule

38 Using Simpson's rule with one application, evaluate numerically.

Simpson's 1/3 rule Medium
A.
B.
C.
D.

39 Apply Simpson's rule to approximate using the points .

Simpson's 1/3 rule Medium
A.
B.
C.
D.

40 A composite Simpson's rule is applied over equal subintervals. Which condition must hold?

Simpson's 1/3 rule Medium
A. must be odd
B. must equal
C. must be even
D. must be prime

41 Using the data , , and , construct the Newton divided-difference interpolating polynomial and determine its value at .

Newton's divided-difference interpolating polynomials Hard
A.
B.
C.
D.

42 For the points , , and , evaluate the Lagrange interpolating polynomial at .

Lagrange interpolating polynomials Hard
A.
B.
C.
D.

43 Let be interpolated by a quadratic Lagrange polynomial at the nodes . Using the standard interpolation remainder bound on , which is a valid upper bound for the error at ?

Lagrange interpolating polynomials Hard
A.
B.
C.
D.

44 The equally spaced data for are . Using Newton's forward interpolation formula, find the interpolated value at .

Newton's forward and backward interpolating polynomials Hard
A.
B.
C.
D.

45 For , the corresponding values are . Using Newton's backward formula about , determine the value at .

Newton's forward and backward interpolating polynomials Hard
A.
B.
C.
D.

46 Given , , , and , what derivative at is obtained by differentiating the cubic Newton forward interpolating polynomial?

Newton's forward and backward interpolating polynomials Hard
A.
B.
C.
D.

47 For the data , , , and , determine the third divided difference .

Newton's divided-difference interpolating polynomials Hard
A.
B.
C.
D.

48 A composite trapezoidal calculation gives and, with doubled step size, . Assuming the leading error is proportional to , estimate the integral using Richardson extrapolation.

Numerical integration Hard
A.
B.
C.
D.

49 Which closed Newton-Cotes rule on four equally spaced nodes has the correct weighted form for approximating ?

Newton-Cotes formulae Hard
A.
B.
C.
D.

50 For five equally spaced nodes with spacing , which expression represents Boole's closed Newton-Cotes rule?

Newton-Cotes formulae Hard
A.
B.
C.
D.

51 The three-point open Newton-Cotes rule over uses interior points , , and , where . Which formula is correct?

Newton-Cotes formulae Hard
A.
B.
C.
D.

52 Apply the composite trapezoidal rule with to approximate . What is the resulting absolute error?

Trapezoidal rule Hard
A.
B.
C.
D.

53 Suppose on . For the composite trapezoidal rule with , which is a valid upper bound for the absolute integration error?

Trapezoidal rule Hard
A.
B.
C.
D.

54 For a twice continuously differentiable function satisfying , what is the leading Euler-Maclaurin error term for the composite trapezoidal rule on ?

Trapezoidal rule Hard
A.
B.
C.
D.

55 Apply Simpson's rule with two subintervals to . What approximation is obtained?

Simpson's 1/3 rule Hard
A.
B.
C.
D.

56 Using Simpson's rule with on , which approximation is obtained?

Simpson's 1/3 rule Hard
A.
B.
C.
D.

57 Which statement correctly describes the degree of exactness of the composite Simpson's rule when applied over an even number of equal subintervals?

Simpson's 1/3 rule Hard
A. It is exact for all polynomials of degree at most two.
B. It is exact for all polynomials of degree at most three.
C. It is exact for all polynomials of degree at most five.
D. It is exact for all polynomials of degree at most four.

58 Why can the standard composite Simpson's rule not be applied directly with equal subintervals over an interval?

Simpson's 1/3 rule Hard
A. Its endpoint weights become negative.
B. Its interpolation polynomial has degree five.
C. Its step size cannot be defined for odd .
D. Its paired-panel construction requires an even number of subintervals.

59 For a sufficiently smooth function, the composite Simpson's error is . If throughout , how does the Simpson approximation compare with the exact integral?

Simpson's 1/3 rule Hard
A. It equals the exact integral for every mesh size.
B. It is strictly greater than the exact integral.
C. It is strictly less than the exact integral.
D. Its direction cannot be determined from the sign of .

60 A quadrature method has error expansion . Which linear combination of approximations at and cancels the term?

Numerical integration Hard
A.
B.
C.
D.
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