Math 152 Week 14 Workshop Problems Write up your solutions to each of these problems on a separate sheet of paper. 4/15/24 1. Several parametric curves are described below. For each of them, find the Cartesian equation of the curve being parametrized by eliminating the parameter. Then graph the parametrized curve (be sure to indicate the direction of travel.) (a) x(t)=√t, y(t) = √t, t≥0 (b) r(t) sin(t), y(t) = cos(t), 0 ≤ts = 2. One well-studied parametrization is called the cycloid, which is the curve defined by x= a(t sin(t)), y = a(1 cos(t)), t≥0 where a is some positive constant. The cycloid traces out the path a fixed point on a wheel makes as the wheel rolls along the ground. (a) Find all values of t for which the cycloid has a cusp. (These will be where dy/dx is not defined.) (b) Find d²y dx² and use it to show that the cycloid is concave down on (0, 2). (c) Find the area underneath the arc of the cycloid from t = 0 to t = 2. (d) Find the arc length of the cycloid from t = 0 to t = 2π. 3. Consider the polar coordinate (5, 1). List three other polar coordinates that determine the same point as this point, satisfying the following properties: One of the points has a negative value for 0. One of the points has a negative value for r. One of the points has an angle that is larger than 10.
Math 152 Week 14 Workshop Problems Write up your solutions to each of these problems on a separate sheet of paper. 4/15/24 1. Several parametric curves are described below. For each of them, find the Cartesian equation of the curve being parametrized by eliminating the parameter. Then graph the parametrized curve (be sure to indicate the direction of travel.) (a) x(t)=√t, y(t) = √t, t≥0 (b) r(t) sin(t), y(t) = cos(t), 0 ≤ts = 2. One well-studied parametrization is called the cycloid, which is the curve defined by x= a(t sin(t)), y = a(1 cos(t)), t≥0 where a is some positive constant. The cycloid traces out the path a fixed point on a wheel makes as the wheel rolls along the ground. (a) Find all values of t for which the cycloid has a cusp. (These will be where dy/dx is not defined.) (b) Find d²y dx² and use it to show that the cycloid is concave down on (0, 2). (c) Find the area underneath the arc of the cycloid from t = 0 to t = 2. (d) Find the arc length of the cycloid from t = 0 to t = 2π. 3. Consider the polar coordinate (5, 1). List three other polar coordinates that determine the same point as this point, satisfying the following properties: One of the points has a negative value for 0. One of the points has a negative value for r. One of the points has an angle that is larger than 10.
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Chapter1: Functions And Models
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Transcribed Image Text:Math 152
Week 14 Workshop Problems
Write up your solutions to each of these problems on a separate sheet of paper.
4/15/24
1. Several parametric curves are described below. For each of them, find the Cartesian
equation of the curve being parametrized by eliminating the parameter. Then graph
the parametrized curve (be sure to indicate the direction of travel.)
(a) x(t)=√t, y(t) = √t, t≥0
(b) r(t) sin(t), y(t) = cos(t), 0 ≤ts
=
2. One well-studied parametrization is called the cycloid, which is the curve defined by
x= a(t sin(t)), y = a(1 cos(t)), t≥0
where a is some positive constant. The cycloid traces out the path a fixed point on a
wheel makes as the wheel rolls along the ground.
(a) Find all values of t for which the cycloid has a cusp. (These will be where dy/dx is
not defined.)
(b) Find
d²y
dx²
and use it to show that the cycloid is concave down on (0, 2).
(c) Find the area underneath the arc of the cycloid from t = 0 to t = 2.
(d) Find the arc length of the cycloid from t = 0 to t = 2π.
3. Consider the polar coordinate (5, 1). List three other polar coordinates that determine
the same point as this point, satisfying the following properties:
One of the points has a negative value for 0.
One of the points has a negative value for r.
One of the points has an angle that is larger than 10.
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