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Vector Functions to Partial Derivative

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1. Distance from a a point to a curve: Find the shortest distances between the point (1,2,1) and a point on the curve r(t)= (1/t*i)+(lnt(t)*j)+(sqrt(t)*k)

2. Distance from a point to a curve: Find the maxmium distances from the point (1,2,-1) to a point on the curve of intersection of the plane z=(y/2) and the ellipsoid (X^2/4)+(y^2/9)+(z^2/4)=1.

3.Distance from a point to a curve: Given that two particles moving in 3 space have equations of motion x=2cos(t) , y=3sin(t), z=t (t>=0) and x=t , y=t^2, z=t^3 (t>=0) (A) what is the distance between them at time t and (B) when are they closest? (Assume that x,y,andz are in feet and t in minutes.

4.Speed of a particle: Suppose the equation of motion is given by r(t)=(sin t + cos t)i - (t - sin t)j, where r is in meters t is in seconds

(a)Graph r(t) for 0<= t <= 2pie
(b)find the maximum speed of the particle over the interval 0<= t <=2pie

5. Maximum temperature along a curve: Suppose the temperature T at a point (x,y,z)is given by T=30zexp(-x^2 - 2y^2). Find the maximum value of T along the curve x=tcos(t), y=t, z=exp(-t)

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The vector functions to partial derivatives are examined. The distance from a point to a curve is determined.

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1. Distance from a point to a curve: Find the shortest distances between the point (1,2,1) and a point on the curve

Solution: The normal vector is given as
This is the normal vector, and it passes through (1, 2, 1)
Therefore, vector equation of this line =
And the length is

2. Distance from a point to a curve: Find the maximum distances from the point
(1,2,-1) to a point on the curve of intersection of the plane and the ellipsoid
Solution: The equation of curve of intersection of the plane and the ellipsoid
is given as
or
or
or
This is an ellipse whose minor axis is 2 and major axis is 2.4.

The ...

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