Problem 7 - Calculus Early Transcendentals

Question

In which direction is the curve \(x=-2 \sin t, y=2 \cos t,\) for \(0

Step-by-Step Solution

Answer

Answer: The curve is generated in the clockwise direction.

Step 1: Analyze the parametric equations

First, let's take a look at the parametric equations \(x = -2 \sin t\) and \(y = 2 \cos t\). These equations define the coordinates of a point \((x, y)\) on the curve as a function of the parameter \(t\). As \(t\) varies in the interval \(0 < t < 2 \pi\), the point moves along the curve.

Step 2: Determine the relation between x and y coordinates

Now let's try to eliminate the parameter \(t\) and see if we can find a relation between \(x\) and \(y\). We can use the identity \(\sin^2 t + \cos^2 t = 1\) as follows: Square both equations and sum them: \((-2 \sin t)^2 + (2 \cos t)^2 = 4\sin^2 t + 4\cos^2 t\) Now, we can factor out 4 and use the identity: \(4 (\sin^2 t + \cos^2 t) = 4 (1) = 4\) This gives us \(x^2+y^2=4\), which is the equation for a circle with radius 2 centered at the origin.

Step 3: Determine the behavior of x and y individually

Now let's analyze how \(x\) and \(y\) change as \(t\) goes from \(0\) to \(2 \pi\). \(x = -2 \sin t\) Since the sine function is positive in the first and second quadrants and negative in the third and fourth quadrants, \(-2\sin t\) is negative in the first and second quadrants and positive in the third and fourth quadrants. This means that as \(t\) increases, the \(x\)-coordinate will go from negative to positive, and then back to negative as \(t\) approaches \(2\pi\). \(y = 2 \cos t\) Cosine function is positive in the first and fourth quadrants and negative in the second and third quadrants. Thus, as \(t\) goes from \(0\) to \(2\pi\), the \(y\)-coordinate starts from positive, goes to negative, and then back to positive.

Step 4: Determine the direction of the curve generation

Now, we can combine the analysis from steps 3 and make some conclusions about the direction of the curve. At the beginning of the interval, when \(t=0\), \(x = -2\sin(0) = 0\) and \(y=2\cos(0)=2\). This means that the curve starts at point (0,2). As the curve goes from positive to negative \(y\) values, the \(x\) goes from negative to positive, and then back to negative values. This indicates that the generation of the curve is in the clockwise direction. In conclusion, the curve defined by the parametric equations \(x = -2 \sin t, y = 2 \cos t\) for the interval \(0 < t < 2 \pi\) is generated in the clockwise direction.