Stewart Precalc 6e Cumulative Review Test: Chapters 8 and 9

Find two polar coordinate representations of the point \((8, -8)\), one with \(r > 0\) and one with \(r < 0\), and both with \(0 \leq \theta < 2 \pi\).

The graph of the equation \(r = 2 \sin 2 \theta\) is called a *four-leafed rose*. (a) Sketch a graph of this equation. (b) Convert the equation to rectangular coordinates.

Let \(z = \sqrt{3} - i\) and let \(w = 6 \left(\cos \dfrac{5 \pi}{12} + i \sin \dfrac{5 \pi}{12}\right)\). (a) Write \(z\) in polar form. (b) Find \(z w\) and \(\dfrac{z}{w}\). (c) Find \(z^{10}\). (d) Find the three cube roots of \(z\).

(a) Sketch a graph of the parametric equations \(x = 2 - \sin^2 t, y = \cos t\). (b) Eliminate the parameter to obtain an equation for this curve in rectangular coordinates. What type of curve is this?

Let \(\mathbf{u} = \langle 8, 6 \rangle\) and \(\mathbf{v} = 5 \mathbf{i} - 10 \mathbf{j}\). (a) Graph \(\mathbf{u}\) and \(\mathbf{v}\) in the coordinate plane, with initial point \((0, 0)\). (b) Find \(\mathbf{u} + \mathbf{v}\), \(2 \mathbf{u} - \mathbf{v}\), the angle between \(\mathbf{u}\) and \(\mathbf{v}\), and \(\text{proj}_{\mathbf{v}} \mathbf{u}\). (c) Assuming \(\mathbf{u}\) is a force vector, calculate the work done by \(\mathbf{u}\) when a particle moves under its influence from \((2, 0)\) to \((10, 3)\).

Let \(P(1, -1, 3)\) and \(Q(3, -2, 1)\) be two points in three-dimensional space. (a) Find the distance between \(P\) and \(Q\). (b) Find an equation for the sphere that has center \(P\) and for which \(Q\) is a point on its surface. (c) Find parametric equations for the line that contains \(P\) and \(Q\).

Let \(\mathbf{a} = \langle 2, 1, -3 \rangle\) and \(\mathbf{b} = 3 \mathbf{i} + 2 \mathbf{k}\) be two vectors in three-dimensional space. (a) Find \(\mathbf{a} \cdot \mathbf{b}\) and \(\mathbf{a} \times \mathbf{b}\). Are \(\mathbf{a}\) and \(\mathbf{b}\) perpendicular, parallel, or neither? (b) Find an equation for the plane that is parallel to both \(\mathbf{a}\) and \(\mathbf{b}\), and that contains the point \((3, 0, -5)\).