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Chapter 10: Problem 847

curve \(\mathrm{y}=(3 / 2) \sin 2 \theta, \mathrm{x}=\mathrm{e}^{\theta} \cdot \sin \theta, 0<2\) for which value of \(\theta\) tangent is parallel to X-axis? (a) 0 (b) \((\pi / 2)\) (c) \((\pi / 4)\) (d) \((\pi / 6)\)

Short Answer

Expert verified
The value of θ for which the tangent is parallel to the X-axis is \(\frac{\pi}{4}\), which is option (c).

Step by step solution

01

1. Compute derivatives for x and y

First, let's differentiate x and y with respect to θ: Given: \(y = \frac{3}{2} \sin{2\theta}\) and \(x = e^\theta \sin\theta\) Derivative of y with respect to θ: \(\frac{dy}{d\theta} = \frac{3}{2} (2\cos2\theta)\) Derivative of x with respect to θ: \(\frac{dx}{d\theta} = e^\theta\sin{\theta} + e^\theta\cos{\theta}\)
02

2. Compute dy/dx

Now, let's compute dy/dx: \(\frac{dy}{dx} = \frac{\frac{dy}{d\theta}}{\frac{dx}{d\theta}}\) \(\frac{dy}{dx} = \frac{3\cos{2\theta}}{2(e^\theta\sin{\theta} + e^\theta\cos{\theta})}\)
03

3. Set dy/dx to 0 and solve for θ

Since the tangent is parallel to the X-axis, the slope of the tangent should be equal to 0. Hence, we have to set dy/dx equal to 0 and solve for θ: \(\frac{3\cos{2\theta}}{2(e^\theta\sin{\theta} + e^\theta\cos{\theta})} = 0\) To satisfy the equation, the numerator must be equal to 0. Therefore: \(3\cos{2\theta} = 0\)
04

4. Solve for θ and check the given options

Now let's solve for θ and determine which of the given options satisfies the equation: \(\cos{2\theta} = 0\) \(2\theta = \frac{\pi}{2} + n\pi\) where n is an integer. \(\theta = \frac{\pi}{4} + \frac{n\pi}{2}\) Now let's check which of the given options satisfy the above equation. (a) 0 - No (b) \(\frac{\pi}{2}\) - No (c) \(\frac{\pi}{4}\) - Yes (d) \(\frac{\pi}{6}\) - No Hence, the value of θ for which the tangent is parallel to the X-axis is \(\boxed{\frac{\pi}{4}}\), which is option (c).

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Most popular questions from this chapter

If \(\mathrm{y}^{2}=\mathrm{p}(\mathrm{x})\) is polynomial function more than 3 degree then \(2(\mathrm{~d} / \mathrm{dx})\left|\mathrm{y}^{3} \mathrm{y}_{2}\right|=\) (a) \(\mathrm{p}^{\prime}(\mathrm{x}) \cdot \mathrm{p}^{\prime \prime}(\mathrm{x})\) (b) \(\mathrm{p}(\mathrm{x}) \cdot \mathrm{p}^{\prime \prime}(\mathrm{x})\) (c) \(\mathrm{p}^{\prime}(\mathrm{x}) \cdot \mathrm{p}^{\prime \prime \prime}(\mathrm{x})\) (d) \(p(x) \cdot p^{\prime \prime \prime}(x)\)

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