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Chapter 9: Q. 12 (page 747)
What is the difference between a cardioid and a limacon?
The graph of equation is limacon if
If then the equation represents a cardioid.
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True/False: Determine whether each of the statements that follow is true or false. If a statement is true, explain why. If a statement is false, provide a counterexample.
(a) True or False: Every function can be written in terms of parametric equations.
(b) True or False: Given parametric equations and the parameter can be eliminated to obtain the form y = f (x).
(c) True or False: Every parametric curve passes the vertical line test.
(d) True or False: Every curve in the plane has a unique expression in terms of parametric equations.
(e) True or False: If the functions x = f (t) and y = g(t) are differentiable for every t 鈭� R, then the parametric curve defined by x and y is differentiable for every value of t.
(f) True or False: A curve parametrized by x = x(t), y = y(t) has a horizontal tangent line at (x(t0), y(t0)) if y (t) = 0.
(g) True or False: A curve parametrized by x = x(t), y = y(t) has a horizontal tangent line at (x(t0), y(t0)) if x (t) = 0 and y (t) = 0.
(h) True or False: The cycloid curve associated with a circle of radius r is made up of a series of semicircles of radius r.
An epicycloid is another variation of a cycloid in which the point tracing the path is on the circumference of a wheel, but the wheel is rolling without slipping on the outside of another wheel, instead of along a horizontal track. If the radius of the rolling wheel is k and the radius of the fixed wheel is r, find parametric equations for the epicycloid.
True/False- Determine whether each of the statements that follow is true or false. If a statement is true, explain why. If a statement is false, provide a counterexample.
(a) True or False: Each point in the plane has a unique representation in rectangular coordinates.
(b) True or False: Each point in the plane has a unique representation in polar coordinates.
(c) True or False: If , then the graphs of the polar equations and are different.
(d) True or False: If , then the graphs of the polar equations and are different.
(e) True or False: The graph of for is a horizontal line in a polar coordinate system.
(f) True or False: In a polar coordinate system, the coordinates and represent the same point if and only if .
(g) True or False: When and are nonzero constants, the graph of is a circle in a polar coordinate system.
(h) True or False: Every function in a polar coordinate system can be expressed in terms of rectangular coordinates and .
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