Sketch the curve represented by the vector-valued function and give the orientation of the curve.

Question

Sketch the curve represented by the vector-valued function  and give the orientation of the curve.

Quizplus · Accepted Answer

The curve represented by the vector-valued function r(t) = ⟨11e^t, e^(-t), 11e^t⟩ is a curve in three-dimensional space. We can see that all the components of the vector-valued function are positive, so the vector is pointing in the positive direction in all three axes. Therefore, the orientation of the curve is such that it is moving in the positive direction along all three axes simultaneously.

To sketch the curve, we can consider the behavior of the coordinate functions separately. The x-component of the vector is 11e^t, which is an exponential function that grows very quickly as t increases. The y-component of the vector is e^(-t), which is also an exponential function, but it decreases very quickly as t increases. Finally, the z-component of the vector is also 11e^t, which grows very quickly as t increases.

Taking all this information into account, we can sketch the curve as a sort of spiral that is rapidly increasing in the x and z directions, while decreasing very rapidly in the y direction.

Sketch the Curve Represented by the Vector-Valued Function and Give