91Ó°ÊÓ

Find the indicated limit, if it exists. $$ \mathbf{R}(t)=e^{t+1} \mathbf{i}+|t+1| \mathbf{j} ; \lim _{t \rightarrow-1} \mathbf{R}(t) $$

Draw the position representation of the given vector \(\mathrm{A}\) and also the particular representation through the given point \(P\); find the magnitude of \(\mathrm{A}\). $$ \mathbf{A}=\langle 0,-2\rangle ; P=(-3,4) $$

A particle is moving along the curve having the given vector equation. In each problem, find the vectors \(\mathbf{V}(t), \mathbf{A}(t), \mathbf{T}(t)\), and \(\mathbf{N}(t)\), and the following scalars for an arbitrary value of \(t:|\mathbf{V}(t)|, A_{T}, A_{N}, K(t) .\) Also find the particular values when \(t=t_{1} .\) At \(t=t_{1}\), draw a sketch of a portion of the curve and representations of the vectors \(\mathbf{V}\left(t_{1}\right)\), \(\mathbf{A}\left(t_{1}\right), A_{T} \mathbf{T}\left(t_{1}\right)\), and \(A_{N} \mathbf{N}\left(t_{1}\right)\). $$ \mathbf{R}(t)=e^{t} \mathbf{i}+e^{-t} \mathbf{j} ; t_{1}=0 $$

For the given curve, find \(\mathrm{T}(t)\) and \(\mathbf{N}(t)\), and at \(t=t_{1}\) draw a sketch of a portion of the curve and draw the representations of \(\mathbf{T}\left(t_{1}\right)\) and \(\mathbf{N}\left(t_{1}\right)\) having initial point at \(t=t_{1}\). $$ \mathbf{R}(t)=\ln \cos t \mathbf{i}+\ln \sin t \mathbf{j}, 0

Find the curvature \(K\) and the radius of curvature \(\rho\) at the given point. Draw a sketch showing a portion of the curve, a piece of the tangent line, and the circle of curvature at the given point. $$ 4 x^{2}+9 y^{2}=36 ;(0,2) $$

Prove that parametric equations of the catenary \(y=a \cosh (x / a)\) where the parameter \(s\) is the number of units in the length of the arc from the point \((0, a)\) to the point \((x, y)\) and \(s \geq 0\) when \(x \geq 0\) and \(s<0\) when \(x<0\), are $$ x=a \sinh ^{-1} \frac{s}{a} \text { and } y=\sqrt{a^{2}+s^{2}} $$

Find the radius of curvature at any point on the given curve. The cycloid \(x=a(t-\sin t), y=a(1-\cos t)\)

A hypocycloid is the curve traced by a point \(P\) on a circle of radius \(b\) which is rolling inside a fixed circle of radius \(a\), \(a>b\). If the origin is at the center of the fixed circle, \(A(a, 0)\) is one of the points at which the point \(P\) comes in contact with the fixed circle, \(B\) is the moving point of tangency of the two circles, and the parameter \(t\) is the number of radians in the angle \(A O B\), prove that parametric equations of the hypocycloid are $$ x=(a-b) \cos t+b \cos \frac{a-b}{b} t $$ and $$ y=(a-b) \sin t-b \sin \frac{a-b}{b} t $$

Prove by vector analysis that the line segment joining the midpoints of the nonparallel sides of a trapezoid is parallel to the parallel sides and its length is one-half the sum of the lengths of the parallel sides.

Prove that two nonzero vectors are parallel if and only if the radian measure of the angle between them is 0 or \(\pi\).