Question

Find the length of the curve x = 3t^(2), y = 2t^(3) , 0 ≤ t ≤ 1

Answer #1

**This is the exact length of
curve!**

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.

Find the curvature of the space curve x=(3t^2)-t^3 y=3t^2
z=(3t)+t^3

Determine the length of the curve r(t) = 4i + 2t^2 j + 1/3t^3 k
from the point
(4, 0, 0) to the point (4, 18, 9)

Eliminate the parameter to find a Cartesian equation of the
curves:
a) x=3t+2,y=2t−3
b) x=21sin(t)−3,y=2cos(t)+5

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.

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.

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

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...

Find the exact length of the curve y=(x^3)/3 + 1/(4x) for
2≤x≤3
Conslder the curve deflned by x=t+1 and y=t^2. Find the
corresponding rectangular equation. Produce two graphs: one using
the rectangular equation and one using the parametric equations.
What are the differnce's between the graphs?
Please show work.

Find the exact length of the curve.
x = 9 +
12t2, y
= 2 + 8t3, 0 ≤
t ≤ 5

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