Interpolation and Approximation: Master Numerical Methods for Pharmacy

Question 1

Which of the following is essential for successful Lagrange interpolation?

Accepted Answer

Distinct interpolation points

Answer

Approximating polynomial degree less than interpolation point count

Answer

Continuous function being interpolated

Answer

Differentiable function being interpolated

Question 2

What is the primary advantage of least squares approximation compared to interpolation?

Accepted Answer

Improved fit with noisy data

Answer

Simplified computation

Answer

Guaranteed exact fit to data points

Answer

Always superior accuracy to interpolation

Question 3

The polynomial representation of the least squares approximation of f(x) up to degree n is:

Accepted Answer

f(x) ≈ a_0 + a_1 * x + ... + a_n * x^n

Answer

f(x) ≈ a_0 + a_1 * x^2 + ... + a_n * x^(n+1)

Answer

f(x) ≈ a_0 + a_1 * x + ... + a_n * sin(x)

Answer

f(x) ≈ a_0 + a_1 * cos(x) + ... + a_n * cos(nx)

Question 4

Which interpolation method constructs a unique polynomial of degree n-1 that passes through n distinct data points?

Accepted Answer

Lagrange interpolation

Answer

Spline interpolation

Answer

Least squares approximation

Question 5

In least squares approximation, the objective is to:

Accepted Answer

Minimize the sum of the squared differences between the polynomial and the given data points

Answer

Minimize the sum of the absolute differences between the polynomial and the given data points

Answer

Minimize the sum of the relative differences between the polynomial and the given data points

Question 6

In cubic spline interpolation, the number of unknown coefficients is:

Accepted Answer

2n-1

Answer

n

Answer

n-1

Question 7

The interpolation error can be defined as:

Accepted Answer

The difference between the interpolated value and the true value of the function

Answer

The difference between the interpolated value and the data point used for interpolation

Answer

The difference between the interpolated value and the polynomial used for interpolation

Question 8

The Runge phenomenon is characterized by:

Accepted Answer

Excessive oscillations between data points when using high-degree polynomials

Answer

Underfitting of the data

Answer

Overfitting of the data

Question 9

Which interpolation method uses polynomial segments to approximate a function?

Accepted Answer

Spline Interpolation

Answer

Lagrange Interpolation

Answer

Newton's Divided Difference Interpolation

Answer

Least Squares Approximation

Question 10

What is the main advantage of using least squares approximation over polynomial interpolation?

Accepted Answer

It minimizes the sum of squared errors

Answer

It is more efficient

Answer

It is more accurate for large datasets

Answer

It is easier to implement

Question 11

Which of the following is the formula for the least squares approximation of a function f(x) by a polynomial of degree n on the interval [-1, 1]?

Accepted Answer

Pn(x) = sum(i=0 to n) (ai * x^i), where ai = integral from -1 to +1 of f(t) * Pi(t) dt

Answer

Pn(x) = sum(i=0 to n) (ai * x^i), where ai = integral from 0 to 1 of f(t) * Pi(t) dt

Answer

Pn(x) = sum(i=0 to n) (ai * x^i), where ai = integral from -inf to inf of f(t) * Pi(t) dt

Question 12

What is the purpose of using orthogonal polynomials in least squares approximation?

Accepted Answer

To simplify the computation of the coefficients

Answer

To reduce the number of terms in the approximation

Answer

To improve the accuracy of the approximation

Question 13

What is the significance of using numerical methods for interpolation and approximation in pharmacy?

Accepted Answer

It provides efficient and accurate solutions to complex problems

Answer

It is more precise than analytical methods

Answer

It reduces the need for experimental data

Question 14

Which of these is *not* a property of cubic splines?

Accepted Answer

They are not continuously differentiable twice.

Answer

They interpolate the given data points.

Answer

They have continuous second derivatives.

Answer

They are piecewise polynomials of degree 3.

Question 15

The least squares approximation of a function f(x) by a polynomial of degree *m* minimizes the sum of which of the following?

Accepted Answer

Squared errors

Answer

Absolute errors

Answer

Weighted errors

Answer

Relative errors

Question 16

What is one advantage of using a piecewise polynomial approximation?

Accepted Answer

Flexibility in modeling different regions of the function.

Answer

Reduced computational cost

Answer

Improved accuracy over the entire domain

Question 17

How are the coefficients of a least squares polynomial approximation found?

Accepted Answer

By solving a system of linear equations

Answer

By solving a system of nonlinear equations

Answer

By solving a system of differential equations

Answer

By solving a system of integral equations

Question 18

What is a potential limitation of interpolation and approximation methods?

Accepted Answer

They can lead to instability if the data is not well-behaved.

Answer

They are computationally expensive.

Answer

They always provide exact solutions.

Question 19

Which of the following is a common application of polynomial approximations in pharmacy?

Accepted Answer

Modeling drug release kinetics

Answer

Solving differential equations

Answer

Estimating population growth

Answer

Predicting weather patterns