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Solve the vector initial-value problem for \(\mathbf{y}(t)\) by integrating and using the initial conditions to find the constants of integration. $$ \mathbf{y}^{\prime \prime}(t)=\mathbf{i}+e^{t} \mathbf{j}, \quad \mathbf{y}(0)=2 \mathbf{i}, \quad \mathbf{y}^{\prime}(0)=\mathbf{j} $$

The triangle with vertices \((0,0),(1,0),\) and \((0,1)\) has three "corners." Discuss whether it is possible to have a smooth vector-valued function whose graph is this triangle. Also discuss whether it is possible to have a differentiable vector-valued function whose graph is this triangle.

Determine whether the statement is true or false. Explain your answer. If a particle moves along a smooth curve \(C\) in 3 -space, then at each point on \(C\) the normal scalar component of acceleration for the particle is the product of the curvature of \(C\) and speed of the particle at the point.

Find the maximum and minimum values of the radius of curvature for the curve \(x=\cos t, y=\sin t, z=\cos t\)

Solve the vector initial-value problem for \(\mathbf{y}(t)\) by integrating and using the initial conditions to find the constants of integration. $$ \mathbf{y}^{\prime \prime}(t)=12 t^{2} \mathbf{i}-2 t \mathbf{j}, \quad \mathbf{y}(0)=2 \mathbf{i}-4 \mathbf{j}, \quad \mathbf{y}^{\prime}(0)=\mathbf{0} $$

(a) Find parametric equations for the curve of intersection of the circular cylinder \(x^{2}+y^{2}=9\) and the parabolic cylinder \(z=x^{2}\) in terms of a parameter \(t\) for which \(x=3 \cos t\) (b) Use a graphing utility to generate the curve of intersection in part (a).

(a) Find the points where the curve $$\mathbf{r}=t \mathbf{i}+t^{2} \mathbf{j}-3 t \mathbf{k}$$ intersects the plane \(2 x-y+z=-2 .\) (b) For the curve and plane in part (a), find, to the nearest degree, the acute angle that the tangent line to the curve makes with a line normal to the plane at each point of intersection.

Determine whether the statement is true or false. Explain your answer. If a particle is moving along a smooth curve \(C\) and passes through a point at which the curvature is zero, then the velocity and acceleration vectors have the same direction at that point.

Find where the tangent line to the curve $$\mathbf{r}=e^{-2 t} \mathbf{i}+\cos t \mathbf{j}+3 \sin t \mathbf{k}$$ at the point \((1,1,0)\) intersects the \(y z\) -plane.

(a) Sketch the graph of $$ \mathbf{r}(t)=\left\langle 2 t, \frac{2}{1+t^{2}}\right\rangle $$ (b) Prove that the curve in part (a) is also the graph of the function \(y=\frac{8}{4+x^{2}}\) [The graphs of \(y=a^{3} /\left(a^{2}+x^{2}\right),\) [The graphs of y = a3/(a2 + x2), where a denotes a constant, were first studied by the French mathemati- cian Pierre de Fermat, and later by the Italian mathe- maticians Guido Grandi and Maria Agnesi. Any such curve is now known as a 鈥渨itch of Agnesi.鈥� There are a number of theories for the origin of this name. Some suggest there was a mistranslation by either Grandi or Agnesi of some less colorful Latin name into Italian. Others lay the blame on a translation into English of Agnesi鈥檚 1748 treatise, Analytical Institutions.]