## 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!

[Math Cheat Sheet T-shirt] | [Engineer Cheat Sheet T-shirt] |

**[5 pts]**Find the lengths of the sides of the triangle where and

**[10 pts]**Find an equation of the sphere that passes through the origin and whose center is

**[5 pts]**Find a vector that has the same direction as but has length 6.

**[10 pts]**For what values of are the vectors and orthogonal?

**[10 pts]**Compute the length of the curve for

**[5 pts]**Find the volume of the parallelepiped with adjacent edges and where and

**[5 pts]**Find a non-zero vector orthogonal to the plane through the points and

**[10 pts]**Where does the line through and intersect the plane

**[10 pts]**Find the distance from the point to the line

**[10 pts]**Compute the limit

**[10 pts]**Find parametric equations for the tangent line to the curve at the point

**[10 pts]**Find the curvature of at the point

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### sagemath

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Are we allowed to use calculators on the real exam?

Nope. You won’t need them

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?

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.

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

Nope

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

may we get an answer key?

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.

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

Yes.

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

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

pointin the space. Does your answer of resemble a point? If not, what is the meaning of this value, and how can you use it to obtain an actual point?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.

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.

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?

Yes. Good job.

Could you please post the solutions to this practice exam?

Is there a particular question for which you need help?

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?

You need to take the absolute value.

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

Ok thank u!

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?

You use it to find the parameter for which After that, you plug that value of in the expressions and to obtain actual numbers, and not expressions of It makes the computations so much easier and faster.

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

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

Yup. Good job!

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

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

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

is the answer to 11,

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

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

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