Old Test 2 & 3 Answers
1. (a) Graph is a cone centered at the origin along the y-axis. 

   (b) Graph is an elliptic paraboloid with the vertex at the origin opening out along the positive x-axis  

2.(a) Graph is a helix centered along the z-axis. 

t 

x 

y 

z 

0 

1 

0 

0 

1/4*Pi 

0 

1 

`≈`(1/4*Pi, .75) 

1/2*Pi 

-1 

0 

`≈`(1/2*Pi, 1.5) 

3/4*Pi 

0 

-1 

`≈`(3/4*Pi, 2.25) 

 

b*(diff(r(x), x))t = `<,>`(-2*sin(2*t), 2*cos(2*t), 1) 

                   

3. L = 5 

4. (a) T(t) = <  

b*N(t) = `<,>`(-cos(t), -sin(t), 0) 

c*B(0) = `<,>`(0, (-3)/5, 4/5) 

5. kappa = 1/2 

6.  (a) v(t) = `<,>`(1, 2*t, cos(t))b*a(t) = `<,>`(0, 2, -sin(t)) 

 

7. a*t = 30.61*seconds 

   b*max*height = 1147.96*m

8. (a) The level curves are concentric circles which are not evenly spaced apart.

    (b) The level curves are concentric ellipses.
 

9. a*T(Pi) = `<,>`(2, 0, 1)*b*N(Pi)/sqrt(5) and `<,>`(2, 0, 1)*b*N(Pi)/sqrt(5) = `<,>`(0, -1, 0) 

 

 

1. No longer taught in MAT294III - Don't have to know how to do this. 

 

3.  

4. f[xx] = 4*ye^(2*x)+4*xye^(2*x)+2 

   0 

 

5.  

 

-13.20000000 

7. Typesetting:-delayDotProduct(Max, rate) = abs(Typesetting:-delayGradient(f(1, 1))) and abs(Typesetting:-delayGradient(f(1, 1))) = sqrt(5); 1; direction = Typesetting:-delayGradient(f(1, 1)) and Typese... , u = < -1/sqrt(5), -2/sqrt(5) >

8. This will not be on Test 2.  Do not turn in on Tuesday. 

9. a*x+y+sqrt(2)*z = 4       

    b1/2*x-1/2 = 1/2*y-1/2 and 1/2*y-1/2 = 1/2*(z-sqrt(2))/sqrt(2)