Problem 22 - Calculus - AP Edition

Question

(a) Use the local linear approximation of \(\tan x\) at \(x_{0}=0\) to approximate tan \(2^{\circ},\) and compare the approximation to the result produced directly by your calculating device. (b) How would you choose \(x_{0}\) to approximate \(\tan 61^{\circ} ?\) (c) Approximate \(\tan 61^{\circ} ;\) compare the approximation to the result produced directly by your calculating device.

Step-by-Step Solution

Answer

(a) Approx. tan(2°) is 0.03491 (real: 0.03492). (b) Choose x_0 = 60° for tan(61°). (c) Approx. tan(61°) is 1.738 (real: 1.8040).

Step 1: Define Local Linear Approximation Formula

The local linear approximation of a function \(f(x)\) at \(x_0\) is given by: \[ L(x) = f(x_0) + f'(x_0)(x - x_0) \] For \(f(x) = \tan(x)\), we need \(f(x_0)\) and \(f'(x_0)\).

Step 2: Compute Derivatives at x_0 = 0

Calculate the function and derivative values: \[ f(0) = \tan(0) = 0 \]\[ f'(x) = \sec^2(x) \quad \text{so} \quad f'(0) = \sec^2(0) = 1 \]

Step 3: Linear Approximation for tan(2°)

Since \(2° = \frac{\pi}{90}\) radians, use local linear approximation:\[ L\left(\frac{\pi}{90}\right) = f(0) + f'(0)\left(\frac{\pi}{90} - 0\right) = 0 + 1 \times \frac{\pi}{90} \rightarrow \frac{\pi}{90} \approx 0.03491 \]

Step 4: Compare With Calculator

Use a calculator to find \( \tan(2°) \approx 0.03492 \). The local linear approximation \(0.03491\) is very close.

Step 5: Choose x_0 for tan(61°)

Since \(61°\) is closer to \(60°\), choose \(x_0 = 60°\) for better approximation. Convert to radians: \(x_0 = \frac{\pi}{3}\).

Step 6: Compute tan and Derivative at x_0 = 60°

Calculate at \(x_0 = \frac{\pi}{3}\):\[ f\left(\frac{\pi}{3}\right) = \tan\left(\frac{\pi}{3}\right) = \sqrt{3} \]\[ f'\left(\frac{\pi}{3}\right) = \sec^2\left(\frac{\pi}{3}\right) = 4 \]

Step 7: Linear Approximation for tan(61°)

Since \(61° = \frac{61\pi}{180}\) radians, use local linear approximation:\[ L\left(\frac{61\pi}{180}\right) = \sqrt{3} + 4\left(\frac{61\pi}{180} - \frac{\pi}{3}\right) \approx \sqrt{3} + 4\left(\frac{\pi}{180}\right) \approx 1.738 \]

Step 8: Compare tan(61°) with Calculator

Using a calculator, \( \tan(61°) \approx 1.8040 \). The linear approximation \(1.738\) is fairly close.

Key Concepts

Tangent Function

The tangent function is a fundamental trigonometric function that provides the ratio of the opposite side to the adjacent side in a right triangle. While \(\tan(x)\) can be expressed in terms of sine and cosine: \(\tan(x) = \frac{\sin(x)}{\cos(x)}\), it is particularly significant in relation to angles and various applications, such as slopes of lines in calculus.
The tangent function has periodicity, which means it repeats its values in a regular cycle. This period for tan is \(\pi\), which is why you'll see the same values repeat when adding or subtracting multiples of \(\pi\) to the input angle.
In the context of calculus and the given exercise, the tangent function serves as the primary function being approximated locally around specific points, using its behavior and the derivative to achieve this approximation.

Derivative

The derivative is a powerful tool in calculus used to find the rate at which a function changes. For a given point on a function, the derivative provides information on the slope of the tangent line at that point.
For the tangent function, the derivative is given by the square of the secant function, \(\sec^2(x)\). This is crucial because it tells us how steep the tangent function gets at different points. In the exercise, this derivative is used to calculate the local linear approximation of tan(x) near specific values. Since \(\tan(0) = 0\) and \(\sec^2(0) = 1\), these values drive the approximation at \(x = 0\).
Similarly, at \(x = 60^\circ\) or \(\pi/3\) in radians, the derivative again provides a slope, \(4\), which is pivotal for approximating tan(61°). This slope is the gradient part of the local linear approximation formula applied in the step-by-step solution.

Radian Conversion

In mathematics, angles can be measured in degrees or radians, but for calculus, radians are preferred. One radian is equivalent to the angle formed when the arc length is equal to the radius of the circle. To convert degrees to radians, use the relation \(\theta = \frac{\pi}{180} imes \text{degrees}\).
In the exercise, converting degrees into radians is necessary for accurate calculations. For example, \(2\circ\) becomes \(\frac{\pi}{90}\) radians, allowing for use in derivative and local linear approximation formulas. Redefining angles in radians ensures proper application of calculus principles where units cancel and integration and differentiation become consistent.
Remember, when calculating trigonometric functions using calculus, radian conversion is not just a numerical step, but a crucial compatibility requirement for the consistency and accuracy of results. It's always good practice to make these conversions early in your problem-solving process to avoid errors with angle misinterpretation.