At what points does the curve r(t) = ti + (3t − t2)k intersect the paraboloid z = x2 + y2? (if an answer does not exist, enter dne.) (x, y, z) = (smaller t-value) (x, y, z) = (larger t-value)

We have given curve and paraboloid as

,

And along the given curve , we have , so the equation of the paraboloid is

⇔

⇒

⇒

⇒

Therefore at the point the curve intersect the paraboloid .

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Along the curve , we have , so the equation of the paraboloid reduces to

the direct variation says that:

then the equation is of the form is given by:

            where k is the constant variation.

from the given table:

consider any value of x and m(x)=y

x = 2 and m(x)=y = 3

substitute these value in [1]; we have;

divide both sides by 2 we get;

⇒

check:

put x = -1 and m(x) = -1.5

substitute in the equation:

-1.5 = -1.5                   true

therefore, the function m(x) represents a direct variation

graph of this function as shown below: