Question

Consider the parametric curve defined by x = 3t − t^3 , y = 3t^2 . (a) Find dy/dx in terms of t.

(b) Write the equations of the horizontal tangent lines to the curve

(c) Write the equations of the vertical tangent lines to the curve.

(d) Using the results in (a), (b) and (c), sketch the curve for −2 ≤ t ≤ 2.

Answer #1

Consider the parametric equations x = 5 - t^2 , y = t^3 - 48t a.
Find dy dx and d 2y dx2 , and determine for what values of t is the
curve concave up, and when is it concave down. b. Find where is the
tangent line horizontal, and where is it vertical.

Find the derivative of the parametric curve x=2t-3t2,
y=cos(3t) for 0 ≤ ? ≤ 2?.
Find the values for t where the tangent lines are horizontal on
the parametric curve. For the horizontal tangent lines, you do not
need to find the (x,y) pairs for these values of t.
Find the values for t where the tangent lines are vertical on
the parametric curve. For these values of t find the coordinates of
the points on the parametric curve.

Consider the parametric curve
x = t2, y = t3 + 3t, −∞ < t < ∞.
(a) Find all of the points where the tangent line is
vertical.
(b) Find d2y/dx2 at the point (1, 4).
(c) Set up an integral for the area under the curve from t = −2
to t = −1.
(d) Set up an integral for the length of the curve from t=−1 to
t=1.

Using MatLab
2. Given the parametric equations x = t^3 - 3t, y = t^2-3:
(a) Find the points where the tangent line is horizontal or
vertical (indicate which in a text line)
(b) Plot the curve parametrized by these equations to
confirm.
(c) Note that the curve crosses itself at the origin. Find the
equation of both tangent lines.
(d) Find the length of the loop in the graph and the area
enclosed by the loop.
3. Use what...

Consider curve C with parametric equations
x=t^2, y=t^3−3t
So the graph of C contains (one,two, three or none?) horizontal
asymptote(s) and (one, two,three or none?) vertical asymptote(s) in
the interval −0.5 ≤ t ≤ 2.

On the parametric curve (x(t), y(t)) = (t − t^2 , t^2 + 3t)
pictured below, determine the (x, y)-coordinates of the marked
point where the tangent line is horizontal.

1) x = t^3 + 1 , y = t^2 - t , Find an equation of the tangent
to the curve at the point corresponding to t = 1
2) x = t^2 + 1 , y = 3t^2 + t ,Find
a) dy/dx ,
b) (d^2)y / dx^2
c) For which values of t is the curve concave upward?
3) sketch the curve: r = 1 - 3cos θ
4)A demand curve is given by p = 450/(x...

Consider the lines in space whose parametric equations are as
follows
line #1 x=2+3t, y=3-t, z=2t
line #2 x=6-4s, y=2+s, z=s-1
a Find the point where the lines intersect.
b Compute the angle formed between the two lines.
c Compute the equation for the plane that contains these two
lines

For the following equation x=2t^2,y=3t^2,z=4t^2;1 less
or equal to t less or equal to 3
i) Write the positive vector tangent of the curve with parametric
equations above.
ii) Find the length function s(t) for the curve.
iii) Write the position vector of s and verify by differentiation
that this position vector in terms of s is a unit tangent to the
curve.

Consider the parametric curve given by the equations:
x = tsin(t) and y = t cos(t) for 0 ≤ t ≤ 1
(a) Find the slope of a tangent line to this curve when t =
1.
(b) Find the arclength of this curve

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