/*! 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 19 Find the points at which the fol... [FREE SOLUTION] | 91影视

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Chapter 11: Problem 19

Find the points at which the following polar curves have a horizontal or a vertical tangent line. $$r=1-\sin \theta$$

Short Answer

Expert verified
Answer: The curve has horizontal tangents at the points \((1, 0)\) and \((0, \frac{\pi}{2})\), and a vertical tangent at the point \((\frac{1+\sqrt{2}}{2}, \frac{3\pi}{4})\).

Step by step solution

01

Find the slope of the tangent line in polar coordinates

In polar coordinates, the slope of the tangent line can be given by the formula: $$\frac{dy}{dx} = \frac{r'(胃)\sin(胃) + r(胃)\cos(胃)}{r'(胃)\cos(胃) - r(胃)\sin(胃)}$$ Let's first find the derivative of the given polar function: $$r(胃) = 1 - \sin(胃)$$
02

Calculate the derivative \(r'(胃)\)

To find the derivative of r with respect to 胃, we differentiate the given function with respect to 胃: $$r'(胃) = \frac{d(1-\sin(胃))}{d胃} = -\cos(胃)$$
03

Plug \(r(胃)\) and \(r'(胃)\) into the slope formula

We now substitute both r(胃) and r'(胃) into the slope formula: $$\frac{dy}{dx} = \frac{(-\cos(胃))\sin(胃) + (1-\sin(胃))\cos(胃)}{(-\cos(胃))\cos(胃) - (1-\sin(胃))\sin(胃)}$$
04

Find conditions for horizontal and vertical tangents

For horizontal tangent, the slope should be zero: $$0 = \frac{(-\cos(胃))\sin(胃) + (1-\sin(胃))\cos(胃)}{(-\cos(胃))\cos(胃) - (1-\sin(胃))\sin(胃)}$$ For vertical tangent, the slope should be undefined: $$\frac{dy}{dx} =\infty$$
05

Solve for 胃 values satisfying the conditions

Solve for 胃 in both cases: Horizontal tangent (numerator should be zero): $$(-\cos(胃))\sin(胃) + (1-\sin(胃))\cos(胃) = 0$$ Solutions: $$胃_1 = 0, 胃_2 = \frac{\pi}{2}$$ Vertical tangent (denominator should be zero): $$(-\cos(胃))\cos(胃) - (1-\sin(胃))\sin(胃) = 0$$ Solutions: $$胃_3 = \frac{3\pi}{4}$$
06

Find the corresponding points on the curve

Now, we find the points on the curve associated with these values of 胃: $$r(胃_1) = 1 - \sin(0) = 1$$ Point with horizontal tangent: \((1, 0)\) $$r(胃_2) = 1 - \sin(\frac{\pi}{2}) = 0$$ Point with horizontal tangent: \((0, \frac{\pi}{2})\) $$r(胃_3) = 1 - \sin(\frac{3\pi}{4}) = \frac{1+\sqrt{2}}{2}$$ Point with vertical tangent: \((\frac{1+\sqrt{2}}{2}, \frac{3\pi}{4})\) Thus, the curve has horizontal tangents at the points \((1, 0)\) and \((0, \frac{\pi}{2})\), and a vertical tangent at the point \((\frac{1+\sqrt{2}}{2}, \frac{3\pi}{4})\).

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Most popular questions from this chapter

Let \(H\) be the right branch of the hyperbola \(x^{2}-y^{2}=1\) and let \(\ell\) be the line \(y=m(x-2)\) that passes through the point (2,0) with slope \(m,\) where \(-\inftyu(m)>1 .\) For what values of \(m\) are there two intersection points? b. Evaluate \(\lim _{m \rightarrow 1^{+}} u(m)\) and \(\lim _{m \rightarrow 1^{+}} v(m)\) c. Evaluate \(\lim _{m \rightarrow \infty} u(m)\) and \(\lim _{m \rightarrow \infty} v(m)\) d. Evaluate and interpret \(\lim _{m \rightarrow \infty} A(m)\)

Graph the following equations. Then use arrows and labeled points to indicate how the curve is generated as \(\theta\) increases from 0 to \(2 \pi\). $$r=\frac{1}{1+2 \cos \theta}$$

Explain and carry out a method for graphing the curve \(x=1+\cos ^{2} y-\sin ^{2} y\) using parametric equations and a graphing utility.

Without using a graphing utility, determine the symmetries (if any) of the curve \(r=4-\sin (\theta / 2)\)

Give the property that defines all hyperbolas.
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