Problem 6 - Calculus for Biology and Medicine

Question

Use the formula $$f(x) \approx f(a)+f^{\prime}(a)(x-a)$$ to approximate the value of the given function. Then compare your result with the value you get from a calculator. \(\tan (0.01)\)

Step-by-Step Solution

Answer

Using linear approximation, \( \tan(0.01) \approx 0.01 \), closely matching the calculator result of \( 0.010000333 \).

Step 1: Identify Values and Function

We want to approximate \( \tan(0.01) \) using the formula \( f(x) \approx f(a) + f^{\prime}(a)(x-a) \). Let's define \( f(x) = \tan(x) \), \( x = 0.01 \), and choose \( a = 0 \) as it's a simple calculation point.

Step 2: Calculate \( f(a) \)

Evaluate \( f(a) \) by finding \( \tan(a) \) at \( a = 0 \):\[ f(0) = \tan(0) = 0. \]

Step 3: Find the Derivative, \( f^{\prime}(x) \)

The derivative of \( f(x) = \tan(x) \) is \( f^{\prime}(x) = \sec^2(x) \).

Step 4: Calculate \( f^{\prime}(a) \)

Evaluate the derivative at \( a = 0 \):\[ f^{\prime}(0) = \sec^2(0) = 1^2 = 1. \]

Step 5: Apply the Linear Approximation Formula

Substitute \( f(a) \), \( f^{\prime}(a) \), \( x \), and \( a \) into the linear approximation formula:\[ \tan(0.01) \approx 0 + 1 \times (0.01 - 0) = 0.01. \]

Step 6: Compute the Exact Value Using a Calculator

Now use a calculator to find the exact value of \( \tan(0.01) \). A calculator gives:\[ \tan(0.01) \approx 0.010000333. \]

Step 7: Compare the Results

The approximation \( 0.01 \) is very close to the calculator value \( 0.010000333 \), showing that the linear approximation method provides a good estimate for small angles close to zero.

Key Concepts

Tangent Function

The tangent function, often denoted as \( \tan(x) \), is one of the fundamental trigonometric functions. It is defined in terms of sine and cosine, specifically as the ratio:

• \( \tan(x) = \frac{\sin(x)}{\cos(x)} \)

This function is periodic, with a period of \( \pi \), meaning it repeats its pattern every \( \pi \) units along the x-axis. The tangent function is particularly useful in problems involving right triangles and modeling periodic phenomena.
One important aspect of the tangent function is its behavior near zero. As angles approach zero, the value of the tangent function also approaches zero, which simplifies calculations such as linear approximations.

• This makes the tangent function particularly suitable for approximations of small values, which is exactly what happens in our discussion of \( \tan(0.01) \). In this range, \( \tan(x) \approx x \).

Understanding the nature of the tangent function helps us appreciate how linear approximations serve as straightforward methods for predicting function behavior around simple points like zero.

Derivative of Trigonometric Functions

Derivatives provide insight into how a function changes at any given point and are central to techniques like linear approximation. For trigonometric functions like the tangent function, understanding the derivative is crucial.
The derivative of \( \tan(x) \) is \( \sec^2(x) \) because:

• The derivative of \( \tan(x) = \frac{\sin(x)}{\cos(x)} \) can be found using the quotient rule of derivatives: \[ f'(x) = \frac{d}{dx}\left(\frac{\sin(x)}{\cos(x)}\right) = \sec^2(x) \]

Calculating this derivative at specific points helps in forming linear approximations. For example, calculating at \( a = 0 \) gives us \( \sec^2(0) = 1 \), providing a rate of change of 1 at zero.
This derivative points to how quickly the tangent function grows near zero, which is key in establishing that for very small \( x \), the function behaves almost linearly.

Differences between Approximation and Exact Values

Understanding the differences between approximation and exact values is crucial for evaluating the effectiveness of mathematical techniques like linear approximation. By approximating \( \tan(0.01) \), the linear approximation yields a value of 0.01, whereas the exact value from a calculator is approximately 0.010000333.
These differences occur because:

• Approximations use simple linear expressions to estimate the value of a function around a certain point.
• Exact calculations consider the entire curve of the function, resulting in more precise values.
• For small angles close to zero, the differences between the linear approximation and the actual value are negligible, making approximations highly useful and efficient.

However, it's essential to recognize that as the input values move further from the point of approximation (in this case, zero), the error may increase. Therefore, linear approximation is most reliable where the function behaves almost linearly, which underscores the importance of choosing appropriate points and understanding the limits of approximation strategies.