## 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!

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 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

• 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.

• October 16, 2011 at 9:59 am

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

2. September 5, 2011 at 3:32 pm

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

• September 5, 2011 at 3:51 pm

Nope

3. September 5, 2011 at 10:34 pm

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

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

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

• September 6, 2011 at 4:55 am

Yes.

6. 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?

• 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. 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?

• September 6, 2011 at 9:47 am

Yes. Good job.

8. September 7, 2011 at 9:45 am

Could you please post the solutions to this practice exam?

• September 7, 2011 at 7:02 pm

Is there a particular question for which you need help?

9. 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?

• September 7, 2011 at 3:55 pm

You need to take the absolute value.

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

Ok thank u!

12. 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. September 7, 2011 at 7:06 pm

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

• September 7, 2011 at 7:10 pm

$L= \displaystyle{\int_{-10}^{10} \lvert \boldsymbol{r}'(t) \rvert\, dt}$

14. September 7, 2011 at 9:24 pm

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

• September 7, 2011 at 9:30 pm

Yup. Good job!

15. September 7, 2011 at 9:27 pm

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

16. September 7, 2011 at 9:48 pm

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

17. September 7, 2011 at 10:10 pm

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

18. September 7, 2011 at 10:18 pm