First midterm-Practice Test

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 Wednesday Sept 07. Good luck!

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  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).
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  1. Lauretta G.
    September 5, 2011 at 2:12 pm

    Are we allowed to use calculators on the real exam?

    • September 5, 2011 at 2:15 pm

      Nope. You won’t need them

      • Lauretta G.
        September 5, 2011 at 7:20 pm

        I haven’t taken calculus in over 4 years and can’t remember how to do a lot of derivatives and integrals. Do I need to rememorize that for the exam?

        • September 5, 2011 at 8:48 pm

          It might not be that crucial for the first midterm, but yes: Having a good grasp of derivatives and integrals is necessary to pass this course.

        • Anonymous
          October 16, 2011 at 9:59 am

          Anonymous :
          For #3 do you find the unit vector and multiply by 6?

  2. Anonymous
    September 5, 2011 at 3:32 pm

    For number 8 do we just choose a value for t between 0 and 1?

  3. Anonymous
    September 5, 2011 at 10:34 pm

    may we get an answer key to check is we did them correctly?

  4. Anonymous
    September 5, 2011 at 11:01 pm

    may we get an answer key?

    • September 6, 2011 at 4:54 am

      If I find some time on Wednesday, I will. Meanwhile, it will pay off to discuss your answers with other students. You can also stop by my office hours. I will be glad to help.

  5. Anonymous
    September 5, 2011 at 11:05 pm

    For #3 do you find the unit vector and multiply by 6?

  6. Lauretta G.
    September 6, 2011 at 8:01 am

    For number six I keep getting -3. Am I doing something wrong?

    • September 6, 2011 at 8:24 am

      You seem to be very confused. Read the problem carefully, and ask yourself: “What kind of answer are you expecting to obtain?”

      The problem asks for a point P=(x_1, x_2, x_3) in the space. Does your answer of -3 resemble a point? If not, what is the meaning of this value, and how can you use it to obtain an actual point?

      • Lauretta G.
        September 6, 2011 at 11:25 am

        If I am not mistaken, number six is the one that asks for the volume of the parallelopiped. That is where I kept getting -3.

        • September 6, 2011 at 11:29 am

          Oops! I thought it was again for problem #8. My bad!

          What formula are you using? The absolute value of the triple product (page 791) should give you always a positive number.

  7. Lauretta G.
    September 6, 2011 at 8:22 am

    For number 8 do you write the equation of the line in parametric form, then plug the parametic equations into the equation for the plane, solve for t, then plug that t back into the parametric equations to get the coordinates of the point where the line intersects the plane?

  8. Anonymous
    September 7, 2011 at 9:45 am

    Could you please post the solutions to this practice exam?

  9. Drew L.
    September 7, 2011 at 3:48 pm

    I keep getting -3 for number 6 as well and know that this cannot be correct because volume cannot be negative…I am using the Example 5 right under the formula on 791 with the matrices and continue to see -3. What am I doing wrong?

  10. Drew L.
    September 7, 2011 at 3:56 pm

    Is the volume just 3 because that example is only for: a . (bxc) and the volume is the absolute value of that?

  11. Drew L.
    September 7, 2011 at 3:56 pm

    Ok thank u!

  12. Darya
    September 7, 2011 at 5:03 pm

    How do you incorporate the point (1,1,1) into the calculations in number 12? Do you use it to make parametric equations? If so, where do you go from there?

    • September 7, 2011 at 6:59 pm

      You use it to find the parameter t for which \boldsymbol{r}(t) = (1,1,1). After that, you plug that value of t in the expressions \boldsymbol{r}'(t) and \boldsymbol{r}''(t) to obtain actual numbers, and not expressions of t. It makes the computations so much easier and faster.

  13. Anonymous
    September 7, 2011 at 7:06 pm

    I’m not sure where to go on #5 after finding r'(t).

  14. Anonymous
    September 7, 2011 at 9:24 pm

    Is the answer to 8, (7,-4,3)

  15. Anonymous
    September 7, 2011 at 9:27 pm

    Is the answer to number 8 (7,-4,3)

  16. Anonymous
    September 7, 2011 at 9:48 pm

    Is the answer to nine, 25/(squareroot 14)

  17. Anonymous
    September 7, 2011 at 10:10 pm

    is the answer to 11, x=1-t y=t z=1-t

  18. Anonymous
    September 7, 2011 at 10:18 pm

    is the answer to 11,

    x=1-t y=t z=1-t

  19. Anonymous
    September 7, 2011 at 10:21 pm

    Answer to 12 is Squareroot of 76 divided by (squareroot of 14) cubed?

  20. Drew L.
    September 8, 2011 at 8:41 am

    Is # 11: x = 1-t, y = t, z = 1-t ?

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