Parabolic coordinates Let T be the transformation x = u 2 – v 2 , y = 2 uv a. Show that the lines u = a in the uv -plane map to parabolas in the xy -plane that open in the negative x -direction with vertices on the positive x -axis. b. Show that the lines v = b in the uv -plane map to parabolas in the xy -plane that open in the positive x -direction with vertices on the negative x -axis. c. Evaluate J ( u, v ). d. Use a change of variables to find the area of the region bounded by x = 4 – y 2 /16 and x = y 2 /4 – 1. e. Use a change of variables to find the area of the curved rectangle above the x -axis bounded by x = 4 – y 2 /16, x = 9 – y 2 /36, x = y 2 /4 – 1, and x = y 2 /64 – 16. f. Describe the effect of the transformation x = 2 uv . y = u 2 – v 2 on horizontal and vertical lines in the uv -plane.
Parabolic coordinates Let T be the transformation x = u 2 – v 2 , y = 2 uv a. Show that the lines u = a in the uv -plane map to parabolas in the xy -plane that open in the negative x -direction with vertices on the positive x -axis. b. Show that the lines v = b in the uv -plane map to parabolas in the xy -plane that open in the positive x -direction with vertices on the negative x -axis. c. Evaluate J ( u, v ). d. Use a change of variables to find the area of the region bounded by x = 4 – y 2 /16 and x = y 2 /4 – 1. e. Use a change of variables to find the area of the curved rectangle above the x -axis bounded by x = 4 – y 2 /16, x = 9 – y 2 /36, x = y 2 /4 – 1, and x = y 2 /64 – 16. f. Describe the effect of the transformation x = 2 uv . y = u 2 – v 2 on horizontal and vertical lines in the uv -plane.
Solution Summary: The author explains that a line u=a maps to the parabola in the xy-plane that open in negative direction with vertices on the positive X -axis.
Parabolic coordinates Let T be the transformation x = u2 – v2, y = 2uv
a. Show that the lines u = a in the uv-plane map to parabolas in the xy-plane that open in the negative x-direction with vertices on the positive x-axis.
b. Show that the lines v = b in the uv-plane map to parabolas in the xy-plane that open in the positive x-direction with vertices on the negative x-axis.
c. Evaluate J(u, v).
d. Use a change of variables to find the area of the region bounded by x = 4 – y2/16 and x = y2/4 – 1.
e. Use a change of variables to find the area of the curved rectangle above the x-axis bounded by x = 4 – y2/16, x = 9 – y2/36, x = y2/4 – 1, and x = y2/64 – 16.
f. Describe the effect of the transformation x = 2uv. y = u2 – v2 on horizontal and vertical lines in the uv-plane.
A region R in the xy-plane is given. Find equations for a transformation T that maps a rectangular region S in the uv-plane onto R, where the sides of S are parallel to the u- and v- axes. (Let u play the role of r and v the role of ?. Enter your answers as a comma-separated list of equations.)
R lies between the circles x2 + y2 = 1 and x2 + y2 = 8 in the first quadrant
A. Plot the circles x2+y2=4 and x2+y2=1 on the same coordinate plane.
B. Find the image of any point on x2+y2=4 under the transformation (x,y)→(12x,12y).
The point (2,0) will go to the ordered pair ( , )
C. What do you notice about x2+y2=4 and x2+y2=1 ?
The small circle is a of the larger one using the origin as a center and a scale factor of
Consider the transformation T, (defined in the uv plane) given by (image)
with u>0 and v>0. We can state that:
there may be more than one correct option
a. T sends horizontal segments in pieces of hyperbolas. b. T sends horizontal segments in line segments passing through the origin. c. T sends vertical segments in pieces of circumferences. d.T sends vertical segments in pieces of parabolas.
Calculus, Single Variable: Early Transcendentals (3rd Edition)
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