## 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. 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.)

• September 16, 2012 at 9:55 pm

Nope. No books, notes or calculators.

3. September 16, 2012 at 9:53 pm

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

4. September 16, 2012 at 10:02 pm

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

5. September 17, 2012 at 9:40 am

how do you find the directional vector for #11

6. September 17, 2012 at 9:42 am

how do you find the directional vector for #11?

• 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. February 4, 2013 at 9:17 pm

Are we allowed to have a cheat sheet?

• February 4, 2013 at 9:31 pm

No

8. 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. February 5, 2013 at 10:01 pm

What did everyone get for 7?

• 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. February 6, 2013 at 1:41 am

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

• February 6, 2013 at 8:04 am

I got the same thing

11. February 6, 2013 at 1:47 am

I got but I may be wrong

12. 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. 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?