Problem 18 - Thomas Calculus

Question

Find the slopes of the curves at the given points. Sketch the curves along with their tangents at these points. Cardioid \(\quad r=-1+\sin \theta ; \quad \theta=0, \pi\)

Step-by-Step Solution

Answer

Slopes are \(-1\) at \(\theta = 0\) and \(0\) at \(\theta = \pi\).

Step 1: Understanding the Cardioid Equation

The given polar equation is a cardioid: \( r = -1 + \sin \theta \). In polar coordinates, \( r \) is the radius from the origin, and \( \theta \) is the angle. We need to find the slope of the tangent to this curve at \( \theta = 0 \) and \( \theta = \pi \).

Step 2: Convert Polar to Cartesian Coordinates

To find derivatives, we convert the polar form to Cartesian. In polar coordinates: \( x = r \cos \theta = (-1 + \sin \theta) \cos \theta \) and \( y = r \sin \theta = (-1 + \sin \theta) \sin \theta \).

Step 3: Differentiate to Find Slope

To find the slope \( \frac{dy}{dx} \), use the chain rule: \[ \frac{dy}{dx} = \frac{ \frac{dy}{d\theta} }{ \frac{dx}{d\theta} } \]. Compute \( \frac{dx}{d\theta} \) and \( \frac{dy}{d\theta} \): \( \frac{dx}{d\theta} = \cos \theta \cdot d(-1 + \sin \theta)/d\theta + (-1+\sin \theta)(-\sin \theta) \) and \( \frac{dy}{d\theta} = \sin \theta \cdot d(-1 + \sin \theta)/d\theta + (-1+\sin \theta) \cos \theta\).

Step 4: Calculate Slopes for Given \(\theta\) Values

For \( \theta = 0 \):\[\frac{dx}{d\theta} = \cos(0)\cdot \cos(0) + (-1+\sin(0))(-\sin(0)) = 1 \]; \[\frac{dy}{d\theta} = \sin(0)\cdot \cos(0) + (-1+\sin(0))\cos(0) = -1 \]. Slope is \( \frac{dy}{dx} = \frac{-1}{1} = -1 \).For \( \theta = \pi \):\[\frac{dx}{d\theta} = \cos(\pi)\cdot (-1) + (0)(0) = 1 \]; \[\frac{dy}{d\theta} = \sin(\pi)\cdot (-1) + (0)(-1) = 0 \]. Slope is \( \frac{dy}{dx} = \frac{0}{1} = 0 \).

Step 5: Sketch the Curve and the Tangents

Sketch the cardioid shape which generally looks like a heart shape centered at \((0, 0)\). At \( \theta = 0 \), which corresponds to the point \((0, -1)\), draw a tangent line with slope \(-1\). At \( \theta = \pi \), corresponding to the point \((0, 1)\), draw a horizontal tangent line.

Key Concepts

Polar Coordinates

Polar coordinates offer a different way to represent curves, especially useful for shapes like cardioids or spirals. In this system, a point is defined by two values: the radius, \( r \), and the angle, \( \theta \).
Instead of using horizontal and vertical distances as in Cartesian coordinates, polar coordinates describe how far and what direction a point is from the origin.

• Radius \( r \) indicates how far the point is from the origin, which is the center of the coordinate system.
• Angle \( \theta \) tells the direction of this point expressed in radians or degrees.

For the cardioid \( r = -1 + \sin \theta \), the shape depends on the sine of the angle \( \theta \). At certain angles such as \( \theta = 0 \) or \( \theta = \pi \), unique characteristics or shapes can be observed.

Cartesian Conversion

Converting polar coordinates to Cartesian coordinates (also called rectangular coordinates) is essential for analyzing curves with calculus. This transformation allows for the application of familiar derivative techniques.
Using the equations:

• \( x = r \cos \theta \)
• \( y = r \sin \theta \)

you can translate polar expressions into the \( (x, y) \) format that is often easier to work with.
For the cardioid \( r = -1 + \sin \theta \), the conversion yields:

• \( x = (-1 + \sin \theta) \cos \theta \)
• \( y = (-1 + \sin \theta) \sin \theta \)

By analyzing these equations, you can understand how the curve appears on a standard grid and prepare to differentiate them.

Slopes of Curves

The slope of a curve at a point gives insight into the direction the curve is heading. For polar curves, slopes can be found after converting to Cartesian coordinates, allowing us to calculate \( \frac{dy}{dx} \), the derivative of \( y \) with respect to \( x \).
To find the slope for a polar curve, inputting the derivatives, \( \frac{dx}{d\theta} \) and \( \frac{dy}{d\theta} \), into the formula:

• \( \frac{dy}{dx} = \frac{ \frac{dy}{d\theta} }{ \frac{dx}{d\theta} } \)

yields the slope at any given angle.

• At \( \theta = 0 \), the calculated slope describes the line's angle with the horizontal.
• At \( \theta = \pi \), initial calculations reflect a special case, often resulting in horizontal or vertical tangents.

Understanding the slopes of curves in this context helps visualize the direction of tangents, providing deeper insights into the geometry of the cardioid.