Question

a. Approximate the given quantities using Taylor polynomials with \(n=3\) b. Compute the absolute error in the approximation assuming the exact value is given by a calculator. $$\tan (-0.1)$$

Short answer

Answer: The approximation for the tangent function using a Taylor polynomial of degree 3 at \(-0.1\) is \(-0.1 - \frac{2}{300}\). The absolute error of this approximation when compared to the exact value given by a calculator is approximately \(0.00017\).

Step-by-step solution

Find the Taylor polynomial for the tangent function

To approximate the tangent function using Taylor polynomials, we need to first compute its Taylor expansion around \(0\). Recall that the Taylor polynomial of degree \(n\) of a function \(f(x)\) centered at \(a\) is given by: $$P_n(x) = \sum_{k=0}^n\frac{f^{(k)}(a)}{k!}(x-a)^k$$ Here, we want to find the Taylor polynomial for the tangent function, i.e., \(f(x) = \tan (x)\), with respect to \(x = -0.1\) (centered at \(a = 0\)), up to \(n = 3\) terms. To do this, we first need to find the derivative of tangent function: - \(f(x) = \tan (x)\) - \(f'(x) = \sec^2(x)\) - \(f''(x) = 2\sec(x)\sec(x)\tan(x)\) - \(f'''(x) = 2\sec^2(x)\tan^2(x) + 2\sec^4(x)\) Now, we will substitute \(a = 0\) in these derivatives: - \(f(0) = \tan (0) = 0\) - \(f'(0) = \sec^2(0) = 1\) - \(f''(0) = 2\sec(0)\sec(0)\tan(0) = 0\) - \(f'''(0) = 2\sec^2(0)\tan^2(0) + 2\sec^4(0) = 2\) Now, we can find the Taylor polynomial: $$P_3(x) = f(0) + f'(0)(x - 0) + \frac{f''(0)}{2!}(x - 0)^2 + \frac{f'''(0)}{3!}(x - 0)^3$$

Plug in the value of x

Now, we will substitute the value of \(x = -0.1\) into the Taylor polynomial to get our approximation: $$P_3(-0.1) = 0 + 1(-0.1) + \frac{0}{2!}(-0.1)^2 + \frac{2}{3!}(-0.1)^3 = -0.1 - \frac{2}{300}\, .$$

Compute the exact value and absolute error

To compute the absolute error, we first need to calculate the exact value of the tangent function using a calculator: Exact value: \(\tan (-0.1) \approx -0.09983\) Now, let's compute the absolute error between the approximation and the exact value: Absolute error \(= \lVert P_3(-0.1) - \tan (-0.1) \rVert = \lVert -0.1 - \frac{2}{300} + 0.09983 \rVert \approx \lVert -0.00017 \rVert = 0.00017\) Therefore, the absolute error in the approximation is approximately \(0.00017\).

Key concepts

Tangent Function

The tangent function, often denoted as \( \tan(x) \), is a fundamental trigonometric function. It represents the ratio of the sine of an angle to the cosine of the same angle. In mathematical terms, \( \tan(x) = \frac{\sin(x)}{\cos(x)} \).
The tangent function has a periodicity of \( \pi \) radians, meaning it repeats its values every \( \pi \) units. It is undefined where the cosine equals zero, leading to vertical asymptotes in its graph.
This function is essential in various fields such as geometry, physics, and engineering, where angles and rotations are involved. Particularly in calculus, the derivatives of the tangent function are crucial for solving problems related to motion and rates of change.

Absolute Error

Absolute error is a measure used to quantify the difference between an approximate value and the exact value.
It is defined as \( \lVert \, \text{exact value} - \text{approximate value} \, \rVert \).
To compute absolute error, you follow these steps:

• Calculate the approximate value using a method such as a Taylor polynomial.
• Determine the exact value using precise calculation methods, typically a calculator for trigonometric functions.
• Subtract the approximate value from the exact one and take the absolute of the result, ensuring a positive error value.

Understanding absolute error helps in evaluating the reliability of an approximation method, especially important in engineering and scientific calculations where precision is critical.

Derivatives

Derivatives are at the core of calculus, representing the rate of change of a function with respect to its variables.
A derivative provides a mathematical way to understand how a function behaves at any given point.
When working with functions like \( \tan(x) \), knowing the derivatives is crucial:

• The first derivative of the tangent function is \( \sec^2(x) \), which describes how \( \tan(x) \) changes as \( x \) changes.
• The second derivative adds another layer of complexity, explaining the curvature of \( \tan(x) \).
• Higher-order derivatives, like the third derivative, give insights into finer details of the function's behavior.

In Taylor polynomials, we use derivatives to construct polynomial approximations, capturing the function's behavior near a specific point.

Approximation Methods

Approximation methods like Taylor polynomials are vital tools in mathematics for simplifying complex functions.
They provide a polynomial which behaves similarly to the function near a chosen point.
The Taylor polynomial is expressed as:\[P_n(x) = \sum_{k=0}^n\frac{f^{(k)}(a)}{k!}(x-a)^k\]This formula uses derivatives of a function evaluated at a specific point \(a\).
By using Taylor polynomials, we approximate the value of functions which are not easily computed directly, such as \( \tan(-0.1) \).
This method is incredibly beneficial in various applications:

• Simplifies calculations in physics and engineering where exact equations are too complex.
• Enhances numerical methods used in computer algorithms.
• Allows for error estimation in numerical approximations, supporting precise scientific work.

Approximations are approximate, not exact, so understanding their limitations through factors like absolute error is essential.