Many of the following exercises are modified versions of the exercises from your text book Mn1!!vD/!Dq1? OD1on1ns by James Stewart. Ou1K so仙? oJ !U? bn?s!!ous ou !U!s Dup o!U?/ Uo仙?wo/斗 s?!s w!11 q? 8/Dp?p.

1. Do exercises 25–30 of §13. 1 of Stewart’s Mn1!!vD/!Dq1? OD1on1ns.4

2.    (a) Draw and parametrize the circle of radius two centered at (1, −1, 1) and lying on the plane x + y + z = 1. H!u!: find two orthogonal unit vectors which are parallel to the plane.

(b) Find parametric equations for two circles C1 and C2 in space such that

•  C1 and C2 have the same radius;

•  C1 and C2 intersect at the points P(2, −1, 3) and Q(2, 1, 3) and nowhere else.

3.  Show that the space curve with parametric equation r(t) = ht2 − 1, −t+2t2 , 4+6ti lies in a plane, and find an equation of this plane. H!u!: try a plane ax + by + cz + d = 0.

4. Find a parametric equation which describes the curve defined by intersecting the cylinder x2 +y2 =

25 with the revolutionary paraboloid 9z +(x2 + y2 ) = 0. Sketch the surfaces and the curve.

5. Galadriel is chasing Annatar.   Suppose their trajectories are given by the parametric equations rG (t) =  h−2t,−1 + 2t,2 + 4t2 i and rA (t) =  h2t − 3, 2 − 2t,−1 + 4t2 i for 0 < t < 3.  Let CG and CA denote the corresponding space curves.

(a)  Show that CG and CA intersect in exactly one point, and find the angle between the tangent lines to CG and CA at that point (in radians, to two decimal places).

(b) Does Galadriel ever catch Annatar?  In other words, is there a time at which Galadriel and

Annatar are at the same position?

6. Find all points at which the curve with parametrization r(t) = ht − 1,t2 +1,t3 − 1i intersects the plane y = 2. Also find the angle between the plane and the (tangent line to the) curve at each such point (in radians, to two decimal places).

7. A particle’s velocity is described by v~(t) = het sin(t),et,et cos(t)i for t 2 [0, 5].

(a) If the particle’s initial position is (−  , 1,  ), describe its position r(t) and acceleration a(t) as functions of t. H!u!:  (et(sint + cost)) = 2et cost and  (et(sint − cost)) = 2et sint.

(b) Verify that the particle’s path lies on a circular cone of the form y2 = a2 (x2 + z2 ) for some a (which?) and sketch the path.

(c) What is the length of the particle’s path from t = 0 to t = 5?

(d) Parameterize the particle’s position in terms of arc length.

(e) Compute the curvature K(t) of the space curve traced by the particle, and show that the curva- ture is a decreasing function of time. [The on/vD!n/? of a space curve with vector function r is given by

K(t) = |r\ (t) ⇥ r\\ (t)| ;

and is a measurement of how much the curve is bending at a given time t.]

4 The exercise numbering among different editions of the text is probably incompatible. You can find the book exercises here: https://drive.google.com/drive/folders/1hAm2QVBEQRzvPQCzLLztt3ogh6zH4_Da?usp=sharing

8. The hyperbolic cosine and sine functions are defined by

sinh(x) =                and cosh(x) =

(a) Compute the derivatives and antiderivatives of cosh and sinh.

(b) Let C be the plane curve with parametric equation r(t) = hcosh(t), sinh(t)i for t 2 R. Show that C is the right branch of the hyperbola with equation x2 − y2 = 1.