Multivariable
Calculus
Practice Final
Chapter 12-15.6
Show your thoughts and work
in a readable way.
NAME ___________________
Date_______
Per________
1. Describe the geometry of the set of points (x,y,z) that satisfy <0,1,1> + t<1,0,-1>
Be sure and describe how this works. (3 points)
2. Derive the distance Formula. (3pts)
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3. Given the two vectors a = <1,1,1> and b = <3,-1,-1> answer the following questions
and describe the geometry of each: (3pts per question)
a. What is a  b ?
b. What is a  b ?
c. What is the scalar and vector projection of a onto b?
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4. Prove that  a and b if a  b = 0 and neither is the zero vector then the vectors are
either parallel or “antiparallel”.
5. Find an equation of a plane containing (1,0,0) (1,2,1) and (0,2,2) (4pts)
6. Find the distance from (1,2,-1) to 2x + y + z = 0 (3pts)
7. Write the equation of a sphere centered at the origin of radius one in both Spherical
and cylindrical coordinates. (3pts)
8. Find r’(t) and T for the curve <𝑡 2 , 𝑒 −𝑡 , 𝑡> (3pts)
9. r’(t) = <𝑡 2 , 𝑡, 1> and the particle started moving at (0,0,1). Find the position function
and the acceleration function. (3pts)
𝑑
10. Prove 𝑑𝑡 (𝑟(𝑡) 𝑥 𝑠(𝑡)) = 𝑟 ′ (𝑡)𝑥 𝑠(𝑡) + 𝑟(𝑡) 𝑥 𝑠 ′ (𝑡) (4pts)
11. What is k =
|𝑟 ′ (𝑡)𝑥𝑟"(𝑡)|
|𝑟 ′ (𝑡)|3
for the function in number 9 and what does it mean (3pts)
12. Find the length of the curve <sint,cost,t> for t ∈ [0,1]. What does this curve look like?
(4pts)
13. For the curve <sint,cost,t> compute 𝑎 𝑇 𝑎𝑛𝑑 𝑎𝑁 and discuss the relevance of these
expressions to the original curve. (5pts)
14. Describe the curve F(x,y) = (𝑥𝑦 3 ) . Use traces! (5pts)
15. 𝑙𝑖𝑚(𝑥,𝑦)→(0,0)
𝑦2𝑥2
√𝑥 2 +𝑦 2
=______ and prove it. (4pts)
16. For f(x,y) = xy + x find the following: (8 pts)
Find 𝑓𝑥 𝑎𝑛𝑑 𝑓𝑦
Find ∇f at (1,1) What does this mean? Use sentences.
𝜋
Find 𝐷𝑢 𝐹, at (1,1) where u is in the direction of 4 . What does this mean? Use
sentences.
17. What is an equation of the tangent plane at the point above (1,1).
18. Suppose the temperature on the surface of a rather bizarre object at (x,y,z) is given by
T(x,y,z) = xyz . If you are a bug on the surface of that object at the point (1,1,1) and
want to go for a stroll but do not want the temperature to vary in what direction should
you travel? (4pts)
want to get as warm as possible in what direction should you travel? (2pts)
19. An open cylindrical box is to be manufactured having a fixed volume of 4 ft cubed.
What dimensions would minimize the manufacturing cost? (5pts)
20. Find the average value of 𝑓(𝑥, 𝑦) =
𝑦√𝑥 + 𝑦 2 𝑜𝑛 𝑡ℎ𝑒 𝑟𝑒𝑐𝑡𝑎𝑛𝑔𝑙𝑒 𝑤𝑖𝑡ℎ 𝑣𝑒𝑟𝑡𝑖𝑐𝑒𝑠(0,0), (1,0), (1,1), 𝑎𝑛𝑑 (0,1). (6pts).
21. Write an integral which expresses the volume below z = 𝑟 and above the x-y plane
and inside r = sin𝜃. (4pts)
22. Write the integrals for the centroid of one loop of the three leaf rose
𝑟 = 2Sin(3𝜃). Don’t compute these! Write your answers as coordinates involving
integrals and then explain what the centroid is.
23. Find the center of mass of a two dimensional plate that occupies the triangle
[0,1]x[0,x] and has density function xy.
24. Find the Surface Area of a cone of base radius 1 and height 1 using integrals. (5pts)
25. Calculate ∭ 𝑥𝑦𝑧𝑑𝑣 where R is the region inside the hemisphere of radius 1. What is
one interpretation of this integral? (6pts)
26. Compute the total mass of the region within the cylinder of radius 4, where 0≤ 𝑧 ≤ 𝑦
and the density function is given by p(x,y,z) = z. (6pts)
What to Study:
Chapter 12-15.6
Read through your notes and the book. Then try and explain the “Big Ideas” to
someone, either in or not in our class. Then try problems that use these ideas. These
big ideas include:
-Distance, curves, planes(what defines a plane?), lines, points and coordinate systems
in R3
-Why are vectors so important in R3? What are they used for? How are they used?
What does it mean to project one vector onto another? What operations and properties
do these vectors follow? What strange anomalies occur in computation?
- Calculus in R3 including derivatives, antiderivatives, arc length, curvature, tangential
and normal acceleration
What geometry is tied to what operation?
-You will be asked to graph something using traces. Curvature, derivatives and
extreme values
-Be able to calculate and discuss your calculation for maxima and minima
-What is the difference and relevance of the directional derivative and the gradient?
-How are they related and why are they different sometimes?
-Know the chain rule and its applications
-understand the computation process and relevance of partial derivatives.
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Average value of a function
Double and triple integrals
Moment, Centroids and Center of Mass
Surface Area
Double and triple integrals in Cylindrical and polar coordinates
Details about the exam:
About half of your time will be spent on Chapter 15 material on the exam.
You may use a calculator, though I suspect you will not use it.
There will be no multiple choice questions.
There will be 1 proof which will have appeared on a precious exam or quiz, but you
will be asked to explain some of your calculations as above.