Question

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.

Answer #1

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.

a) Find the arc length parametrization of the line x=-2+4t,
y=3t, z=2+t that has the same direction as the given line and has
reference point (-2,0,2).
Use an arc length S as a parameter. x= y= z= b) Use the
parametric equations obtained in part (a) to find the point on the
line that is 20 units from the reference point in the direction of
increasing parameter.
x=
y=
z=
b) Use the parametric equations obtained in part (a) to...

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

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.

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...

graph f(t)=t^6-4t^4-2t^3+3t^2+2t on the interval [-3/2,5,2]
using matlab

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

Consider the function F(x, y, z) =x2/2−
y3/3 + z6/6 − 1.
(a) Find the gradient vector ∇F.
(b) Find a scalar equation and a vector parametric form for the
tangent plane to the surface F(x, y, z) = 0 at the point (1, −1,
1).
(c) Let x = s + t, y = st and z = et^2 . Use the multivariable
chain rule to find ∂F/∂s . Write your answer in terms of s and
t.

6) please show steps and explanation.
a)Suppose r(t) = < cos(3t), sin(3t),4t
>.
Find the equation of the tangent line to r(t)
at the point (-1, 0, 4pi).
b) Find a vector orthogonal to the plane through the points P
(1, 1, 1), Q(2, 0, 3), and R(1, 1, 2) and the area of the triangle
PQR.

A space curve C is parametrically parametrically defined by
x(t)=e^t^(2) −10,
y(t)=2t^(3/2) +10,
z(t)=−π,
t∈[0,+∞).
(a) What is the vector representation r⃗(t) for C ?
(b) Is C a smooth curve? Justify your answer.
(c) Find a unit tangent vector to C .
(d) Let the vector-valued function v⃗ be defined by
v⃗(t)=dr⃗(t)/dt
Evaluate the following indefinite integral
∫(v⃗(t)×i^)dt. (cross product)

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