Practice Exam for First Midterm

This is a practice exam for the first midterm. Feel free to drop any comment or question below. I will try to answer here as many as possible, until the day of the exam. Good luck!

  1. [5 pts] Find the lengths of the sides of the triangle \triangle PQR, where P=(3,-2,-3), Q=(7,0,1), and R=(1,2,1).
     
     
     
  2. [10 pts] Find an equation of the sphere that passes through the origin and whose center is (1,2,3).
     
     
     
  3. [5 pts] Find a vector that has the same direction as \langle -2, 4, 2 \rangle but has length 6.
     
     
     
  4. [10 pts] For what values of b are the vectors \langle -6, b, 2 \rangle and \langle b, b^2, b \rangle orthogonal?
     
     
     
  5. [10 pts] Compute the length of the curve \boldsymbol{r}(t) = \langle 2\sin t, 5t, 2\cos t \rangle for -10 \leq t \leq 10.
     
     
     
  6. [5 pts] Find the volume of the parallelepiped with adjacent edges PQ, PR and PS, where P=(2,0,-1), Q=(4,1,0), R=(3,-1,1), and S=(2,-2,2).
     
     
     
  7. [5 pts] Find a non-zero vector orthogonal to the plane through the points P=(0,-2,0), Q=(4,1,-2), and R=(5,3,1).
     
     
     
  8. [10 pts] Where does the line through (1,0,1) and (4,-2,2) intersect the plane x+y+z=6?
     
     
     
  9. [10 pts] Find the distance from the point (4,1,-2) to the line x=1+t, y=3-2t, z=4-3t.
     
     
     
  10. [10 pts] Compute the limit \displaystyle{\lim_{t\to 0} \Big\langle \frac{e^t-1}{t}, \frac{\sqrt{1+t}-1}{t}, \frac{3}{1+t} \Big\rangle}
     
     
     
  11. [10 pts] Find parametric equations for the tangent line to the curve \boldsymbol{r}(t) = \langle e^{-t}\cos t, e^{-t}\sin t, e^{-t} \rangle at the point (1,0,1).
     
     
     
  12. [10 pts] Find the curvature of \boldsymbol{r}(t) = \langle t, t^2, t^3 \rangle at the point (1,1,1).
  1. Anonymous
    September 16, 2012 at 1:44 pm

    are we allowed to use calculators on the exam

  2. September 16, 2012 at 1:47 pm

    Will we be allowed to use calculators on the exam? (Sorry if this is a duplicate, but I’m not sure if the first comment went through or not.)

  3. Anonymous
    September 16, 2012 at 9:53 pm

    Is there a way to check if we found the correct answers to these questions?

  4. Gaston
    September 16, 2012 at 10:02 pm

    Did anyone get an answer for number nine, i got the answer to be 7.

  5. Anonymous
    September 17, 2012 at 9:40 am

    how do you find the directional vector for #11

  6. Anonymous
    September 17, 2012 at 9:42 am

    how do you find the directional vector for #11?

    • Anonymous
      February 4, 2013 at 10:28 pm

      you have to take the derivative of r(t) so you get r'(t)=(x’i+y’j+z’k)…. then you plug in the x_0 y_0 and z_0 values of the given point for t in the derivatives that correspond with it… By doing this we are finding the a b and c values we need for our parametric equations. Since we are already given the point we already have our x_0, y_0, and z_0…. now you have all you need to solve if I didnt skip anything…

      x=x_0+at y=y_0+bt z=z_0+ct

  7. Anonymous
    February 4, 2013 at 9:17 pm

    Are we allowed to have a cheat sheet?

  8. John Jackson
    February 5, 2013 at 6:59 pm

    Just gunna throw out my answers for the ones that I have attempted in case any one wants to compare. #1. PQ=6, QR=sqrt(40), RP=6.
    #2. (x-1)^2 + (y-2)^2 + (z-3)^2 = 14.
    #3.
    #4. b=0 and b=2
    #5. ?
    #6. ?
    #7. 13i -14j +5k
    #8. P(7,-4,3)
    #9. 7
    and thats as far as I’ve gone.

    -John Jackson

  9. Anonymous
    February 5, 2013 at 10:01 pm

    What did everyone get for 7?

    • Caleb Padgett
      February 6, 2013 at 3:18 am

      Find the directional vectors for sides PQ and QR. Compute PQ x QR. That should be the answer you’re looking for.

  10. Anonymous
    February 6, 2013 at 1:41 am

    did anyone else get sqrt(76)/(14*sqrt(14)) for 12?

    • Anonymous
      February 6, 2013 at 8:04 am

      I got the same thing

  11. Isaac Wilder
    February 6, 2013 at 1:47 am

    I got but I may be wrong

  12. Isaac Wilder
    February 6, 2013 at 1:49 am

    13, -14 , 5 is what I got for 7. The numbers didn’t go through the first time

  13. Andrew Bjork
    February 6, 2013 at 7:40 am

    I know this is late notice but does anyone know if we have to know the all the different quadric surfaces?

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