Untitled Flashcards Set

Why is approximate error often used instead of true error?
a) True error can only be calculated when the true value is known. ✅

Which of the following statements about numerical errors is true?
c) Floating-point precision errors occur because not all decimal numbers can be exactly represented in binary. ✅

What is the main advantage of using Simpson’s 1/3 rule over the trapezoidal rule?
b) It requires fewer function evaluations for the same level of accuracy. ✅

Which of the following best explains why increasing the number of terms in a Taylor series improves accuracy?
c) It reduces the truncation error by including higher-order terms. ✅

Which of the following best describes how machine epsilon (ϵ\epsilonϵ) affects numerical computations?
b) It limits how accurately a computer can represent numbers. ✅

What is the primary reason that round-off error occurs in numerical methods?
b) The limitations in how numbers are stored in a computer. ✅

How can truncation error in numerical differentiation be minimized?
c) Using higher-order finite difference approximations. ✅

Which of the following statements about error propagation is true?
b) Errors can accumulate and significantly affect final results. ✅

Why is it important for a matrix to be non-singular when finding its inverse?
a) A singular matrix has no inverse because its determinant is zero. ✅

What is the primary limitation of using numerical integration methods like Simpson’s rule and the trapezoidal rule?
c) They introduce error because they approximate continuous functions with discrete values. ✅

What is the main difference between accuracy and precision?
b) Accuracy refers to how close a computed/measured value is to the true value, while precision refers to how close individual computed/measured values are to one another. ✅

Which of the following contributes to round-off error in numerical calculations?
c) The limitation of how numbers are stored and represented in a computer. ✅

Which of the following is an example of a floating-point precision issue?
a) When a small number is added to a large number, the small number is lost due to limited precision. ✅

Which of the following statements is true about truncation error?
a) It occurs when a mathematical process is approximated by a finite number rather than by an exact representation. ✅

A matrix must be __________ in order for its inverse to exist.
b) Square and non-singular. ✅

What is the correct way to access the element in the second row and first column of matrix B in Python?
a) B[1,0]. ✅ (Python uses zero-based indexing.)

How can you minimize truncation error in numerical methods?
a) Increasing the number of terms in a Taylor series expansion. ✅

Which of the following correctly describes the purpose of numerical integration?
a) To approximate an integral by summing discrete values rather than computing an exact integral. ✅

What is the best way to decrease numerical integration error using the trapezoidal rule?
b) Increase the number of segments. ✅

What happens when you use Simpson's 1/3 rule with an odd number of subintervals?
b) The rule does not apply because Simpson's 1/3 rule requires an even number of segments. ✅

When implementing while loops, Python requires:
c) a logical expression that is evaluated at the beginning of each loop. ✅

When creating a function in Python, which of the following is not required?
d) that all variables used in the function be declared at the beginning. ✅

True error is defined as
b) True Value – Approximate Value. ✅

When should you use approximate error instead of true error?
a) The actual value is unknown. ✅

Truncation error is caused by approximating
d) exact mathematical procedures. ✅

For the matrix A = np.array([[1,2,3],[4,5,6],[7,8,9]]), A[:,1] =
c) [2,5,8]. ✅

Machine epsilon gives
b) the spacing between the two smallest numbers in the system. ✅

Which of the following is not true of the first-order Taylor series approximation of f(x) at a?
c) its truncation error always approaches zero as x approaches a. ✅

The Taylor series represents a function as an infinite sum of terms calculated from
c) values of the function's derivatives at a single point. ✅

Which of the following will not tend to decrease truncation error in Taylor series approximations?
b) increasing the number of significant digits. ✅

The Taylor series error propagation formula for a function of two variables assumes that variation due to one variable
d) is independent of the variation due to another variable. ✅

Numerical differentiation allows us to compute rates of change without knowing
c) the underlying function. ✅

Numerical integration involves evaluating a definite integral from
c) a set of values. ✅

Simpson's 1/3 rule requires
d) all of the above (an even number of segments, an odd number of data points, evenly spaced segments).