Question

Derivative of a Function. The derivative of a continuous function \(f(x)\) is defined by the equation $$ \frac{d}{d x} f(x)=\lim _{\Delta x=0} \frac{f(x+\Delta x)-f(x)}{\Delta x} $$ In a sampled function, this definition becomes $$ f^{\prime}\left(x_{i}\right)=\frac{f\left(x_{i+1}\right)-f\left(x_{i}\right)}{\Delta x} $$ where \(\Delta x=x_{i+1}-x_{i}\) Assume that a vector vect contains nsamp samples of a function taken at a spacing of \(d x\) per sample. Write a function that will calculate the derivative of this vector from Equation (5.12). The function should check to make sure that \(\mathrm{dx}\) is greater than zero to prevent divide-by-zero errors in the function. To check your function, you should generate a data set whose derivative is known, and compare the result of the function with the known correct answer. A good choice for a test function is \(\sin x\). From elemen\(\operatorname{tary}\) calculus, we know that \(\frac{d}{d x}(\sin x)-\cos x\). Generate an input vector containing 100 values of the function \(\sin x\) starting at \(x=0\) and using a step size \(\Delta x\) of \(0.05\). Take the derivative of the vector with your function, and then compare the resulting answers to the known correct answer. How close did your function come to calculating the correct value for the derivative?

Short answer

The short answer to calculate the derivative of a sampled function vector and test its accuracy with a sine function is as follows: 1. Define a function to calculate the derivative of a sampled function vector, checking that dx > 0. 2. Generate a sin(x) vector with 100 samples, starting from x=0 and a step size of 0.05. 3. Calculate the derivative vector using the function defined in (1) with the sin(x) vector and dx=0.05. 4. Generate a reference cosine vector corresponding to the input sin(x) vector values. 5. Compare the calculated derivative vector with the reference cosine vector, computing an error metric (e.g., mean squared error) to assess the accuracy of the derivative calculation function.

Step-by-step solution

Define the function for derivative calculation

Define a function to calculate the derivative of a sampled function vector. The function should take the input vector and the step size dx as arguments, and return the derivative vector. It should also check if dx > 0 to avoid errors.

Generate sin(x) vector

Create an input vector containing 100 values of the sine function starting at x=0 and using a step size ∆x of 0.05. This vector will be used to check the accuracy of our derivative calculation function.

Calculate derivative vector

Call the function defined in Step 1 with the sin(x) vector as input and dx=0.05, getting the derivative vector as output.

Generate comparison cosine vector

Generate a reference vector containing the cosine function values corresponding to the input vector values. Since the derivative of sin(x) is cos(x), this vector will be used to compare with the calculated derivative.

Compare calculated derivative with the cosine vector

Compare the calculated derivative vector with the reference cosine vector element-wise to see how close our derivative calculation function came to finding the correct values. Calculate the difference between the two vectors, and then compute an error metric (e.g., mean squared error) to quantify the difference between the calculated and known correct values.

Key concepts

MATLAB Function Programming

Understanding MATLAB function programming is crucial when carrying out mathematical computations like derivatives. In MATLAB, a function is a group of statements that together perform a task. To write a function, you typically define inputs, a set of instructions, and outputs. When calculating the derivative of a sampled function, you need to program a MATLAB function that takes in a vector of sampled points and a spacing value and then employs a formula to calculate and return the derivative of that sampled function.

• Start with the keyword function followed by output and input variables.
• Check for input validity (such as dx > 0).
• Employ a loop or vectorized operations to compute the derivative.
• Return the output from the function.

Numerical Differentiation

Numerical differentiation is a technique to approximate the derivative of a function using values of the function at certain points. Unlike analytical differentiation, where an explicit formula for the derivative is found, numerical methods provide an approximation that is particularly useful when dealing with sampled data or complex functions. For a set of data samples, the derivative is approximated by calculating the difference between successive points divided by the spacing between those points, known as a finite difference.

Finite Difference Approximation

• The forward difference uses the subsequent point
• The backward difference uses the previous point
• The central difference uses both surrounding points for a more balanced estimate

Calculus in MATLAB

In MATLAB, calculus operations like differentiation can be performed both symbolically and numerically. When working with sampled data or discretized functions, numerical methods become particularly important. MATLAB's versatile environment allows for implementation of calculus concepts directly into code, such as by writing functions that calculate derivatives, integrals, and solve differential equations.

Implementing Derivatives

For derivative approximation, one might use functions like diff for discrete differences or gradient for more sophisticated estimates accounting for variable spacing between data points. Additionally, MATLAB supports symbolic differentiation using the syms functionality, which can define and differentiate symbolic expressions analytically.

Sampled Function Derivative

For a function sampled at discrete intervals, its derivative can be approximated numerically. This is particularly relevant when we have empirical data or the function cannot be differentiated analytically. The derivative of a sampled function is calculated by finding the rate of change between each successive sample point, using the concept of finite differences. In MATLAB, this involves iterating over the data points in the vector and applying the formula for numerical differentiation. This requires careful handling of array indices to avoid out-of-bounds errors.

Data Alignment

• Ensure indices are properly aligned to avoid mismatched operations.
• Consider edge cases, such as the first or last point in the dataset
• Apply boundary conditions or assumptions as necessary for the problem

Vector-Based Calculations

Vector-based calculations in MATLAB utilize arrays to perform operations on a series of values simultaneously, which is highly efficient. When calculating derivatives, instead of using loops to process each data point individually, you can apply vectorized operations to compute the differences across all points in one go. This not only simplifies the code but usually leads to significant performance improvements, especially for large datasets. In the context of numerical differentiation, MATLAB's vectorization allows for compact and fast computation of the discrete derivative by operating on the entire vector at once.

Advantages of Vectorization

• Reduces the need for explicit loops
• Increases computational efficiency
• Makes code more readable and concise