/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 18 Find the points at which the fol... [FREE SOLUTION] | 91影视

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Chapter 11: Problem 18

Find the points at which the following polar curves have a horizontal or a vertical tangent line. $$r=3+6 \sin \theta$$

Short Answer

Expert verified
Answer: The polar curve has a horizontal tangent at the point (r, 胃) = (9, 蟺/2). There are no valid points on the curve with a vertical tangent.

Step by step solution

01

Calculate the derivative of r with respect to 胃

To find the tangent lines, we need to first find the rate of change of the polar curve. To do that, let's calculate the derivative of r with respect to 胃: $$\frac{dr}{d\theta} = \frac{d}{d\theta}(3+6\sin\theta)$$ Applying the chain rule, we have: $$\frac{dr}{d\theta} = 6\cdot\frac{d}{d\theta}\sin\theta = 6\cos\theta$$
02

Identify the points where the tangent is horizontal or vertical

A polar tangent is horizontal when the rate of change of radius is 0. Therefore, for a horizontal tangent, we can set dr/d胃 to 0 and solve for 胃. $$6\cos\theta = 0$$ $$\cos\theta = 0$$ From this equation, we can find two solutions for 胃, where the tangent is horizontal: $$\theta = \frac{\pi}{2}, \frac{3\pi}{2}$$ A polar tangent is vertical when the rate of change of radius is undefined or infinite. For vertical tangent, this condition occurs when the denominator, cos(胃), is 0. However, we already found the 胃-values for horizontal tangents, which are the same as the vertical tangent condition, but there is no need to check for vertical tangents, because the polar curve has a slope at each point.
03

Find the corresponding r-values

Now we need to find the points on the curve corresponding to the 胃-values we found. To do that, simply substitute the 胃-values into the original equation and find the corresponding r-values: For 胃 = 蟺/2: $$r = 3 + 6\sin\frac{\pi}{2} = 3 + 6\cdot 1 = 9$$ So at 胃 = 蟺/2, r = 9. For 胃 = 3蟺/2: $$r = 3 + 6\sin\frac{3\pi}{2} = 3 + 6\cdot(-1) = -3$$ However, since r cannot be negative in polar coordinates, this value of 胃 does not provide a valid point on the curve.
04

Identify the points on the polar curve

With the r and 胃-values we found, we can now identify the points on the polar curve where the tangent is horizontal: For 胃 = 蟺/2, r = 9: The point (r, 胃) = (9, 蟺/2) has a horizontal tangent. In conclusion, the polar curve $$r = 3+6\sin\theta$$ has a horizontal tangent at the point (r, 胃) = (9, 蟺/2). There are no valid points on the curve with a vertical tangent.

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