Question

1. Sketch the following surfaces: (a) z-+y2/9 (b) a2 =y2 +22 (c) 2/4+(y-1)2+(z+1)/9 1 (d) r2+y-22+1 0 (e) -y2+-1 0. 2. Find an equation for the surface consisting of all points that are- point (1,-3, 5) and the plane r = 3. 3. Sketch the curve F(t)<t cos(t), t sin (t), t > 4. Find a vector equation that represents the curve of the intersec r2y =9 and the plane y + z = 2. 5. Find a vector function the represents the curve of intersection 53-2y and y= r3- 5 cos(3t)i+ sin(3t)7+ tk. 6. Let F(t) (a) Sketch the curve F(t) (b) Find the derivative of F(t). (c) Find the arc length of F(t) on the interval [0, 3]. 7. Show that T(t) T ( t ) = 0 for all t. 8. Show that the following limit does not exist: 3ar2 5y2 lim (z,y)-(0,0) r2 + 1

question #6

Answer 1

PART(a):

The given curve is

r(t cos 3t, sin 3t, t>

we can plot the curve as follows

2+ 1 x N

PART(b):

r(t cos 3t, sin 3t, t>

diff wrt t

r(t) 3 sin 3t, 3 cos 3t, 1>

PART(c):

r(t) 3 sin 3t, 3 cos 3t, 1>

Let us find the magnitude

-3 sin 3t)2 (3 cos 3t)212 r(t)

(t) 9 sin2 3t9cos2 3t1

( 9 (sin2 3t cos2 3)1

(t 9 (1)1

10 r(t) T

The arc length is given by

b L (t) dt p |(1),4

L= VI0.d

integrate wrt t

L = V10t

L=V10(3) 10(0)

L 3 10

I hope this answer helps,
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