Problem 28 - Calculus Single Variable

Question

Choose an appropriate function \(f\) and point \(c\), and use the differential approximation of \(f\) in order to estimate the given number. Compute the absolute error. $$ \sin (0.48 \pi) $$

Step-by-Step Solution

Answer

The approximate value is 1, with an absolute error of 0.04894.

Step 1: Choose an Appropriate Function and Point

We want to approximate the value of \( \sin(0.48\pi) \). Let's choose the function \( f(x) = \sin(x) \) and select a point \( c \) near \( 0.48\pi \) where we know the exact value of the sine function. An appropriate point is \( c = 0.5\pi \), since \( \sin(0.5\pi) = 1 \).

Step 2: Compute the Differential Approximation

The differential approximation of a function \( f \) at a point \( c \) for a small change \( \Delta x \) is given by:\[ f(c + \Delta x) \approx f(c) + f'(c) \Delta x\]Here, \( \Delta x = 0.48\pi - 0.5\pi = -0.02\pi \). The derivative of \( \sin(x) \) is \( \cos(x) \). So, \( f'(c) = \cos(0.5\pi) = 0 \). Applying the differential approximation, we get:\[\sin(0.48\pi) \approx 1 + 0 \cdot (-0.02\pi) = 1\]

Step 3: Calculate the Exact Value and Absolute Error

Now, let's calculate the exact value of \( \sin(0.48\pi) \). Using a calculator, \( \sin(0.48\pi) \approx 0.95106 \).The absolute error is given by the absolute difference between the actual value and the approximated value:\[\text{Absolute Error} = |0.95106 - 1| = 0.04894\]

Key Concepts

Sine Function

The sine function, denoted as \( \sin(x) \), is a fundamental component of trigonometry. It describes the ratio of the length of the opposite side to the hypotenuse in a right-angled triangle. This function is periodic, meaning it repeats its values in regular intervals, specifically every \(360^\circ\) or \(2\pi\) radians.
One important property of the sine function is its boundedness between -1 and 1. This range makes it predictably useful in various applications, such as oscillation or wave motion.
In the problem scenario, we were tasked with approximation using the sine function. Finding values of \( \sin(x) \) often involves using known reference points and properties of the sine function to estimate values without a calculator, especially when employing techniques like differential approximation.

Derivative

In calculus, the derivative measures how a function changes as its input changes. When it comes to the sine function, understanding derivatives allows us to predict how the function behaves near specific points.
The derivative of the sine function is the cosine function, denoted as \( \cos(x) \). This derivative tells us the rate at which the sine function is increasing or decreasing. For example, at \( x = 0.5\pi \), \( \cos(x) \) equals 0, indicating that the sine function neither increases nor decreases at this point, which is actually a critical point (a peak in the sine wave).
When applying differential approximation, the derivative helps us see how small changes in \( x \) would linearly affect \( \sin(x) \). In the exercise, we found that around \( x = 0.5\pi \), the derivative is zero, explaining why the approximation value didn’t change much from 1 when calculating \( \sin(0.48\pi) \).

Absolute Error

Absolute error gives us an understanding of how far off an estimate is from the actual or true value. It's calculated as the absolute difference between the measured or estimated value and the actual value.
In mathematical terms, absolute error can be expressed as \( |\text{true value} - \text{estimated value}| \). This calculation is crucial during approximation processes like in our previous exercise, where we approximated \( \sin(0.48\pi) \).
By measuring absolute error, we can determine the reliability of approximation methods. In our example, the absolute error was calculated as \( |0.95106 - 1| = 0.04894 \), indicating how closely the differential approximation aligned with the true value as per computational tools. This approach highlights both the strengths and limitations of linear approximations in mathematics.