MATH 222 Class Exercise on Parametric Curves with Solutions
I.
Let
x(t) = sin(2t) and y(t) = sin(t) for 
.
a)
Fill
in the following table of x and y values for the given values of t and then
plot those points in the graphical region below.
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b)
Now graph the curve
with the calculator and check that the points you plotted were correct.
c)
Find
in terms of the
variable t. _________________
d)
Find
all points (in (x,y) coordinates) at which the curve has horizontal tangent
lines.
e)
Find
all points (in (x,y) coordinates) at which the curve has vertical tangent
lines.
f)
Find
where the curve crosses it self. Find
the equations of both tangent lines at that point.
g)
Find
the area enclosed by the curve as t goes from 0 to 
. Recall the formula
for area : 
. Use a calculator to
evaluate the integral.
A = ____________
h)
Find
the length of the whole curve. The
formula for the length of a parametric curve is 
. We will cover this
formula on Friday.
L = _________________
II.
Let
x(t) = t3 – 2t and y(t) = t2 – t for 
a)
Find 
in terms of the
variable t. _________________
b) Find all points at which the curve has
a horizontal tangent line.
c) Find all points at which the curve has
a vertical tangent line.
d) Find the x-y coordinates of the point
where the curve crosses itself. (Hint:
the point has integer coordinates.)
(x,y) = _________
What
are the t-values that correspond to this point?
t1 = ____________, t2 = ____________
e) Find the area enclosed in that loop
formed by the self-crossing of the curve.
A = ____________
Solutions.
I.
a)
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t |
0 |
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x |
0 |
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0 |
-1 |
0 |
0 |
-1 |
0 |
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y |
0 |
1/2 |
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1 |
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0 |
-1 |
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0 |
b)
c) 
d) Find
the values of t at which 
. 
The x-y points
corresponding to these values of t are (0,1) and (0,-1).
e)
Find the values of t at which 
. 
The
corresponding points are 
f)
The curve crosses itself at (0,0). The
corresponding t-values are t = 0 and t = p. The values of the derivative at this point are ½ and –1/2. Thus
the tangent lines are 
g) 
h) 
II.
a) 
b)
2t – 1 = 0. t = ½. The point corresponding to this t is 
.
c) 
the points corresponding to these
t-values are 
.
d) The curve crosses itself at (1,1). Thus y = 1.
To find the t-values we solve 
which yields 
. 
e) 