Problem 7 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

Use a tangent line approximation to approximate the following. In each case, use concavity to determine whether the approximation is larger or smaller than the actual value. Then compare your results with the approximations given by a calculator or computer. (a) \(\sin 0.2\) (b) \(\sin 0.1\) (c) \(\sin 0.01\) (d) \(\sin (-0.1)\)

Step-by-Step Solution

Answer

The tangent line approximations for \(\sin 0.2\), \(\sin 0.1\), and \(\sin 0.01\) are 0.2, 0.1, and 0.01 respectively, which are slightly less than their actual values due to the concave up nature of the sine function. However, the approximation for \(\sin -0.1\) is -0.1, slightly greater than the actual value due to the concave down nature of the sine function for negative values.

Step 1: Determine Concavity of Sine Function

The sine function is concave up on the interval from 0 to \(\pi\) and is concave down on the interval from \(\pi\) to \(2\pi\). Since we are approximating for values close to 0, and all positive, the function is concave up.

Step 2: Apply Tangent Line Approximation for \(\sin 0.2\)

The tangent line approximation of \(\sin x\) at a point \(a\) is \(f(a) + f'(a)(x - a)\). For the \(\sin\) function, this is \(\sin(a) + \cos(a)(x - a)\). Using a point close to 0.2, say 0 (because \(\sin(0)\) and \(\cos(0)\) are known), this approximation is calculated as follows: \(\sin(0) + \cos(0)(0.2 - 0) = 0 + 1 \cdot 0.2 = 0.2\). Since the function is concave up, the tangent line lies below the curve for \(x > a\). Therefore, the approximation for \(\sin 0.2\) is less than the actual value.

Step 3: Apply Tangent Line Approximation for \(\sin 0.1\)

Using a similar approach to Step 2, the approximation for \(\sin 0.1\) is \(\sin(0) + \cos(0)(0.1 - 0) = 0 + 1 \cdot 0.1 = 0.1\). Again as the function is concave up, the tangent line lies below the curve for \(x > a\). Therefore, the approximation for \(\sin 0.1\) is less than the actual value.

Step 4: Apply Tangent Line Approximation for \(\sin 0.01\)

With the same reasoning as in Step 2 and Step 3, the approximation for \(\sin 0.01\) is \(\sin(0) + \cos(0)(0.01 - 0) = 0 + 1 \cdot 0.01 = 0.01\). Because the function is concave up, the tangent line lies below the curve for \(x > a\), so our approximation for \(\sin 0.01\) is less than the actual value.

Step 5: Apply Tangent Line Approximation for \(\sin (-0.1)\)

The approximation for \(\sin -0.1\) is \(\sin(0) + \cos(0)(-0.1 - 0) = 0 + 1 \cdot -0.1 = -0.1\). As the function is concave down for \(x < a\), the tangent line lies below the curve for \(x < a\). So the approximation for \(\sin -0.1\) is greater than the actual value.

Key Concepts

Sine Function

The sine function is one of the fundamental trigonometric functions that describes a smooth, periodic wave. It is particularly known for its properties of oscillating between -1 and 1 as it cycles through its period, which is typically denoted in radians. The function is defined by the formula \( \sin(\theta) \), where \( \theta \) is an angle. Hence, it is closely associated with the unit circle, where it represents the y-coordinate of a point at a given angle from the positive x-axis.

Some key properties of the sine function include:

• It has a periodic nature with a period of \(2\pi\), meaning it repeats its values every \(2\pi\) radians.
• The sine function is odd, which means \( \sin(-x) = -\sin(x) \).
• It is smooth and continuous and has derivatives of all orders.

Understanding these fundamental properties is essential for approximating and predicting its behavior using techniques like tangent line approximation.

Concavity

Concavity refers to the way a curve bends and is a concept that helps us understand the nature of a function's graph. When dealing with the sine function, concavity gives us insights into how our approximations might deviate from the actual values. Concavity can be identified by observing the second derivative of the function.

• If the second derivative \( f''(x) > 0 \) on an interval, the function is concave up on that interval, resembling a "U" shape.
• If \( f''(x) < 0 \) on an interval, the function is concave down, resembling an upside-down "U".

For the sine function, which is \(f(x) = \sin x\), the second derivative is \(-\sin(x)\). Therefore:

• From 0 to \(\pi\), the second derivative is negative, making the sine function concave down.
• From \(\pi\) to \(2\pi\), \(-\sin(x)\) is positive, meaning it is concave up on this interval.

Concavity directly affects tangent line approximations. For example, when the sine function is concave down, as in the interval from 0 to \(\pi\), tangent lines approximate below the curve. This knowledge helps refine our estimates.

Calculator Comparison

Now that we understand how tangent line approximations are used, it's practical to compare these estimates against more accurate tools like calculators or computers. These digital tools use more complex algorithms to deliver precise values for the sine function within moments, benefiting from floating-point arithmetic and advanced algorithms.

With functions such as the sine, especially for very small angles, calculators rely on mathematical expansions like Taylor or Fourier series to provide values that are accurate to multiple decimal places. For example:

• To compute \(\sin 0.2\), a calculator might use the angle's radian format and compute \(\sin\) to return a value like \(0.198669\,...\)
• As opposed to manual tangent line approximations, calculators incorporate more terms from a series expansion to minimize error.

Comparing your manual calculations with a calculator's precise results can help understand the limitations and strengths of simplified models like tangent line approximations.

Concave Up and Down Intervals

Identifying intervals where a function is concave up or down is crucial when working with approximations. This knowledge informs us about the nature of error in our approximations. For the sine function, this is particularly applicable in predicting and refining the outcome of tangent line approximations.

For \(\sin x\):

• It is concave up on intervals like \((\pi, 2\pi)\), where the second derivative \(-\cos(x)\) is greater than zero. This means the curve bends upwards.
• Conversely, from 0 to \(\pi\), the function is concave down because \(-\cos(x) < 0\).

Understanding these intervals helps in predicting the nature of errors in tangent line approximations. For small values of \(x\) near zero, where the sine function is concave up, the approximation tends to underpredict the actual sine value. Recognizing these patterns is essential for refining mathematical strategies and achieving better approximation accuracy.