Sketch the curve parametrized by r(t) = 〈|t|+ t, |t|− t〉.

Find the maximum height above the xy-plane of a point on r(t) = 〈e^t, sin t, t (4 − t)〉.

Let be the curve obtained by intersecting a cylinder of radius r and a plane. Insert two spheres of radius r into the cylinder above and below the plane, and let F₁ and F₂ be the points where the plane is tangent to the spheres [Figure 16(A)]. Let K be the vertical distance between the equators of the two spheres. Rediscover Archimedes’s proof that is an ellipse by showing that every point P on satisfies

Hint: If two lines through a point P are tangent to a sphere and intersect the sphere at Q₁ and Q₂ as in Figure 16(B), then the segments and have equal length. Use this to show that PF₁ = PR₁ and PF₂ = PR₂.

Assume that the cylinder in Figure 16 has equation x² + y² = r² and the plane has equation z = ax + by. Find a vector parametrization r(t) of the curve of intersection using the trigonometric functions cos t and sin t.

Now reprove the result of Exercise 43 using vector geomentry. Assume that the cylinder has equation x² + y² = r² and the plane has equation z = ax + by.

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Hint: Show that and have length r and are orthogonal to the plane.

To simplify the algebra, observe that since a and b are arbitrary, it suffices to verify Eq. (2) for the point P = (r, 0, ar).