Problem 14 - Calculus Early Transcendentals

Question

Find the image \(R\) in the \(x y\) -plane of the region \(S\) using the given transformation \(T\). Sketch both \(R\) and \(S\). $$S=\left\\{(u, v): u^{2}+v^{2} \leq 1\right\\} ; T: x=2 u, y=4 v$$

Step-by-Step Solution

Answer

The image of the region S under the transformation T is a filled ellipse in the xy-plane with semi-major axis of length 4 along the y-axis, semi-minor axis of length 2 along the x-axis, and the center at the origin. The inequality representing the region R is \(\left(\frac{x}{2}\right)^2 + \left(\frac{y}{4}\right)^2 \leq 1\).

Step 1: Identify Region S

Region S is defined by the inequality \(u^2 + v^2 \leq 1\). This represents a filled circle in the uv-plane with radius 1 and origin as its center.

Step 2: Applying the Transformation T

We apply the transformation T to obtain the relationship between the coordinates \((u, v)\) and \((x, y)\). The transformation equations are given by \(x=2u\) and \(y=4v\). Substitute these equations in the inequality for S to find the equation for R.

Step 3: Obtain the Inequality for Region R

After substituting the transformation equations, we get: $$\left(\frac{x}{2}\right)^2 + \left(\frac{y}{4}\right)^2 \leq 1$$ This inequality defines the image R in the xy-plane.

Step 4: Identify Region R

The inequality \(\left(\frac{x}{2}\right)^2 + \left(\frac{y}{4}\right)^2 \leq 1\) represents a filled ellipse in the xy-plane with semi-axes of lengths equal to 2 and 4 along the x and y axes, respectively, and with its center at the origin.

Step 5: Sketching Region S and Region R

Now we can sketch region S, region R and the transformation: 1. In the uv-plane, sketch a filled circle with radius 1 and origin as its center. This is Region S. 2. In the xy-plane, sketch a filled ellipse with semi-major axis of length 4 along the y-axis, semi-minor axis of length 2 along the x-axis, and the center at the origin. This is Region R. 3. Mark the transformation T applied on Region S to obtain Region R, showing that the circle from S transforms into the ellipse of R using the equations \(x = 2u\), \(y = 4v\).