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Magazine Chapter 1: Problem 8
Use an anonymous function to evaluate the Taylor polynomial \(T_{4}(x)=1-\frac{1}{2} x^{2}+\frac{1}{24} x^{4}\) at the given value of \(x .[8]\) (a) 0 (b) \(\frac{1}{2}\) (c) 1 (d) \(\pi\)
Short Answer
Expert verified
(a) 1; (b) 0.8776; (c) 0.5417; (d) 0.1239.
Step by step solution
01
Define the Anonymous Function
In Python, an anonymous function is created using `lambda`. Let's define the function \( T_4(x) = 1 - \frac{1}{2}x^2 + \frac{1}{24}x^4 \) using `lambda`: ```pythonT4 = lambda x: 1 - (1/2)*x**2 + (1/24)*x**4```This line of code effectively creates a function that can be used to compute values for various inputs.
02
Evaluate at x=0
Substitute \( x = 0 \) into the function:```pythonT4_0 = T4(0)```Calculate it using the defined `lambda` function:\[ T_4(0) = 1 - (1/2)(0)^2 + (1/24)(0)^4 = 1 \]
03
Evaluate at x=1/2
Substitute \( x = \frac{1}{2} \) into the function:```pythonT4_half = T4(0.5)```Perform the calculation:\[ T_4\left(\frac{1}{2}\right) = 1 - \frac{1}{2}(0.5)^2 + \frac{1}{24}(0.5)^4 = 1 - \frac{1}{2}\times 0.25 + \frac{1}{24}\times 0.0625 \]Calculating further, we have:\[ 1 - 0.125 + 0.0026041667 \approx 0.8776041667 \]
04
Evaluate at x=1
Substitute \( x = 1 \) into the function:```pythonT4_1 = T4(1)```Calculate it:\[ T_4(1) = 1 - \frac{1}{2}(1)^2 + \frac{1}{24}(1)^4 = 1 - 0.5 + 0.04166667 \]The result is:\[ 0.54166667 \]
05
Evaluate at x=蟺
Substitute \( x = \pi \) into the function:```pythonimport mathT4_pi = T4(math.pi)```Calculate it using \( \pi \approx 3.14159265 \):\[ T_4(\pi) = 1 - \frac{1}{2}(3.14159265)^2 + \frac{1}{24}(3.14159265)^4 \]Perform the calculation:\[ 1 - \frac{1}{2}\times 9.8696 + \frac{1}{24}\times 97.4091 \]Calculating further, we get:\[ 1 - 4.9348 + 4.0587 = 0.1239 \]
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Anonymous Functions
Anonymous functions are a convenient tool in programming, particularly in Python. They allow you to define functions quickly without the need for a formal function declaration. This can be particularly handy in situations where you require a simple function for short-term use.
- Anonymous functions are often used for short, throwaway tasks where defining a full function might seem cumbersome.
- They are created using the `lambda` keyword in Python, which is why they are often referred to as lambda functions.
- The structure of a lambda function is straightforward: `lambda parameters: expression`. It returns the result of the expression calculated from its parameters.
- The expression within a lambda function is usually limited to a single statement.
Lambda Expressions in Python
Lambda expressions in Python form the core of anonymous functions. They are small and anonymous and help perform operations in a compact syntax. Let's see how they work through the syntax and an example from the exercise.
However, if the logic becomes complicated, named functions (`def`) should be considered due to better readability and debugging ease.
- The syntax is simple: `lambda
: `. - In the example, for the Taylor polynomial, the lambda expression is defined as `T4 = lambda x: 1 - (1/2)*x**2 + (1/24)*x**4`.
- This expression quickly sets up a function that computes the value of the Taylor polynomial for a given `x` value without defining a named function.
However, if the logic becomes complicated, named functions (`def`) should be considered due to better readability and debugging ease.
Numerical Evaluation
Numerical evaluation is a process of calculating the value of an expression with particular input numbers. In the context of the original exercise, it's applied to evaluate the Taylor polynomial at specific points: 0, 1/2, 1, and \( \pi \).
- Numerical evaluation often involves substituting numbers into equations or expressions, as done in the symptoms of this exercise.
- It's crucial for checking the behavior of mathematical models or functions at various points to understand how they perform or predict results.
- The calculation steps outlined in the exercise showed how the Taylor polynomial equation was evaluated step by step for each point, picking individual numerical values and performing operations.
- Python, being a numerical computing powerhouse due to its libraries like `math`, simplifies the process of numerical evaluation.
Polynomial Approximation
Polynomial approximation is a cornerstone concept in mathematical and numerical analysis. It revolves around approximating complex expressions or functions using polynomials, which are easier to compute over certain intervals.
- Taylor polynomials are specific polynomial approximations that simplify more complex functions by focusing on their behavior near a specific point (usually around zero, known as the "Maclaurin series").
- In the exercise, the function of \( T_4(x) \), which is the Taylor polynomial of degree 4, is used to approximate a function by calculating it at several points, showcasing how close the approximation is to the actual function values.
- Polynomial approximation is often used in numerical methods to simplify and solve larger problems in engineering, physics, and computer science.
