Problem 51 - Calculus Early Transcendentals

Question

Use a graphing utility to determine the first three points with \(\theta \geq 0\) at which the spiral \(r=2 \theta\) has a horizontal tangent line. Find the first three points with \(\theta \geq 0\) at which the spiral \(r=2 \theta\) has a vertical tangent line.

Step-by-Step Solution

Answer

Answer: The first three points with horizontal tangent lines are approximately (0.463, 1.040), (-1.802, -2.923), and (3.029, -4.813); the first three points with vertical tangent lines are approximately (-1.178, -3.378), (3.348, -0.299), and (-5.497, 1.996).

Step 1: Convert the polar equation to Cartesian coordinates

To find the equations of tangent lines, we need the expressions for \(x\) and \(y\) in terms of \(\textit{polar coordinates}\): \[x = r\cos{\theta}\] \[y = r\sin{\theta}\] Using the given equation \(r = 2\theta\), we can write \(x\) and \(y\) as functions of \(\theta\): \[x(\theta) = 2\theta \cos{\theta}\] \[y(\theta) = 2\theta \sin{\theta}\]

Step 2: Compute derivatives of x and y with respect to \(\theta\)

To find the tangent lines, we need expressions for \(dx/d{\theta}\) and \(dy/d{\theta}\). Applying the product rule, we find these derivatives: \[\frac{dx}{d\theta}=2\cos{\theta}-4\theta\sin{\theta}\] \[\frac{dy}{d\theta}=2\sin{\theta}+4\theta\cos{\theta}\]

Step 3: Find points with horizontal tangent lines

A horizontal tangent line occurs when the derivative \(dy/dx = 0\). To find these points, we compute: \[\frac{dy/dx}=\frac{dy/d\theta}{dx/d\theta}=\frac{2\sin{\theta}+4\theta\cos{\theta}}{2\cos{\theta}-4\theta\sin{\theta}}.\] We can now equate the fraction to zero and solve for \(\theta\): \[2\sin{\theta}+4\theta\cos{\theta}=0\] To find horizontal tangents, we are looking for points with \(\theta\geq 0\). Set \(\theta = k\pi\) (where \(k\) is a positive integer), and use a graphing utility to find that the first three points occur at \(\theta_1=0.235\pi, \theta_2=1.235\pi\), and \(\theta_3=2.235\pi\). Compute their \(\textbf{polar coordinates}\) \((r,\theta)\), and convert them to \(\textbf{Cartesian coordinates}\) \((x,y)\).

Step 4: Find points with vertical tangent lines

A vertical tangent line occurs when the derivative \(dx/dy = 0\). To find these points, we compute: \[\frac{dx/dy}=\frac{dx/d\theta}{dy/d\theta}=\frac{2\cos{\theta}-4\theta\sin{\theta}}{2\sin{\theta}+4\theta\cos{\theta}}.\] We can now equate the fraction to zero and solve for \(\theta\): \[2\cos{\theta}-4\theta\sin{\theta}=0\] To find vertical tangents, we are looking for points with \(\theta\geq 0\). Set \(\theta = k\pi\) (where \(k\) is a positive integer), and use a graphing utility to find that the first three points occur at \(\theta_1=0.765 \pi, \theta_2=1.765 \pi\), and \(\theta_3=2.765\pi\). Compute their \(\textbf{polar coordinates}\) \((r,\theta)\), and convert them to \(\textbf{Cartesian coordinates}\) \((x,y)\).