91Ó°ÊÓ

Sketch the graph of the plane curve given by the vector-valued function, and, at the point on the curve determined by \(\mathbf{r}\left(t_{0}\right)\), sketch the vectors \(\mathbf{T}\) and \(\mathbf{N}\). Note that \(\mathbf{N}\) points toward the concave side of the curve. $$ \mathbf{r}(t)=t \mathbf{i}+\frac{1}{t} \mathbf{j} \quad t_{0}=2 $$

(a) find the point on the curve at which the curvature \(K\) is a maximum and (b) find the limit of \(K\) as \(x \rightarrow \infty\). $$ y=\ln x $$

Define the unit tangent vector, the principal unit normal vector, and the tangential and normal components of acceleration.

How is the unit tangent vector related to the orientation of a curve? Explain.

Sketch the space curve represented by the intersection of the surfaces. Then represent the curve by a vector-valued function using the given parameter. $$ 4 x^{2}+4 y^{2}+z^{2}=16, \quad x=z^{2} \quad z=t $$

Given a twice-differentiable function \(y=f(x)\), determine its curvature at a relative extremum. Can the curvature ever be greater than it is at a relative extremum? Why or why not?

Show that the curvature is greatest at the endpoints of the major axis, and is least at the endpoints of the minor axis, for the ellipse given by \(x^{2}+4 y^{2}=4\)

Find all \(a\) and \(b\) such that the two curves given by \(y_{1}=a x(b-x) \quad\) and \(\quad y_{2}=\frac{x}{x+2}\) intersect at only one point and have a common tangent line and equal curvature at that point. Sketch a graph for each set of values of \(a\) and \(b\).

A sphere of radius 4 is dropped into the paraboloid given by \(z=x^{2}+y^{2}\) (a) How close will the sphere come to the vertex of the paraboloid? (b) What is the radius of the largest sphere that will touch the vertex?

Projectile Motion Find the tangential and normal components of acceleration for a projectile fired at an angle \(\theta\) with the horizontal at an initial speed of \(v_{0}\). What are the components when the projectile is at its maximum height?