MVC Review topics
Multivariable Calculus
10.1 – 10.3 Quiz
Name ________________________
Hr. ___________
For questions 1 – 5, consider the points 𝑨(0, 4, 1) 𝐚𝐧𝐝 𝑩(3, −5, −2).
1. Plot A and B on the graph.
2. Find the distance between A and B.
3. Let the directed line segment ⃑⃑⃑⃑⃑
𝐴𝐵 represent
vector v. Find the component form of v.
4. Write v in standard unit vector form.
5. Write the equation of the sphere containing B with
point A as its center.
Use the following vectors for questions 6 – 9.
𝐮 = ⟨1, 3, −1⟩
𝐯 = 3𝐢 + 𝐣 − 5𝐤
w= ⟨6, 2, −10⟩
6. Find the dot product of u and v.
7. Evaluate 2u – w.
8. Which of the vectors are parallel?
9. Find a unit vector in the direction of u. (formula:
𝒖
)
‖𝐮‖
10. Use vectors w and z to sketch each new vector below (you don’t need to sketch a plane)
(a) −𝐰
(b)
1
2
𝐳
(c) w + z
MVC
Practice Quiz 10.4 to 10.6
Name_________________________
1. Given u = i – j + 2k and v = 3i – 3j + k, Find each of the following:
(a) 𝐯 × 𝐮
(b) 𝐮 × 𝐯
(c) 𝐮 × 𝐮
2. Find a unit vector orthogonal to both v = i – k and w = −2𝐢 + j + 2k
3. Find parametric equations for the line through the point P(2, -1, 4)
and parallel to the vector i + j + k (I changed this)
4. In the graph at right, sketch the line represented
by the equations in question 3.
5. Find the equation of the plane with normal vector 𝐧 = ⟨1, 2, 3⟩,
containing the point (2, -1, 5)
6. Find the distance from the point T(2, -3, 4) to the plane x + 2y + 2z = 13
7. Classify and sketch the surface given by 𝑧 2 = 1 + 𝑦 2 − 𝑥 2
3
8. Sketch the surface in space given by 𝑦 = − √𝑥
9. Sketch the surface given by 𝑦 = −(𝑥 2 + 𝑧 2 )
MVC
Chapter 1 Test (10.4 to 10.7)
Name _________________________
Hr. _______
1. For vectors 𝐯 = −4𝐢 + 3𝐣 + 𝐤 and u = 2i + j + k, sketch and label 𝐮 × 𝐯 and 𝐯 × 𝐮
2. Show that 𝐮 × (𝐯 + 𝐰) = (𝐮 × 𝐯) + (𝐮 × 𝐰), using :
𝐮 = ⟨3, −2, 1⟩
𝐯 = ⟨2, −4, −3⟩
𝐰 = ⟨−1, 2, 2⟩
Type equation here.
3. Find the equation of the plane containing points 𝑃(1, 1, −1), 𝑄(2, 0, 2), and 𝑅(0, −2, 1)
4. Find the equation in parametric form of the line parallel to the plane in question #3 and
passing through (-3, 1, 4).
5. Consider the plane given by the equation 2(𝑥 − 2) − 2𝑦 + 6(𝑧 + 2) = 8.
(a) Find the distance from the plane to the origin.
(b) Find the angle (in degrees) between the plane above and the plane that has
normal vector 𝐧 = ⟨1, 3, 4⟩ and passes through (−1, 82, 0)
6. Classify and sketch the surface in space given by 4𝑥 2 + 4𝑦 2 + 𝑧 2 = 16
7. Classify and sketch the surface in space given by 𝑥 2 + 𝑦 2 = 𝑧 2
______8. Match the equation for the surface in space given by 2𝑦 2 − 2𝑧 2 − 18𝑥 = 0 to its
graph below.
(a)
(b)
(c)
(d)
9. Give the equation for the yz trace for question #8.
10. Convert the rectangular coordinates (1, √3, 2) to cylindrical AND spherical coordinates.
𝜋 3𝜋
11. Convert the spherical coordinates (2, ,
6
2
) to cylindrical AND rectangular coordinates.
12. Give the equation for the cone given by 𝑧 = 2√𝑥 2 + 𝑦 2 in cylindrical coordinates AND in
spherical coordinates.
13. Write the equation 𝑥 2 − 𝑦 2 = 1 in cylindrical form.
Bonus: sketch the cylindrical surface given by 𝜃 =
2𝜋
3
MVC
Practice Quiz 11.1-11.3
Name ________________________
1.) Represent the plane curve by a vector-valued function.
2𝑥 − 4𝑦 + 6 = 0
2.) Assume 𝐫(𝑡) in question 1 is the derivative of some vector-valued function 𝐑(𝑡) that
satisfies the initial condition 𝐑(0) = 4𝐢 − 3𝐣. Find R(t)
3.) The position vector 𝐫(𝑡) = 𝑒 −𝑡 𝐢 + 𝑒 𝑡 𝐣 describes the path of an object moving in the xyplane. Sketch the graph of r and the velocity and acceleration vectors at the point (1,1).
1
4.) An object’s path in space is modeled by the position vector 𝐫(𝑡) = 3𝑡𝐢 + 𝑡𝐣 + 𝑡 2 𝐤.
4
Find the object’s speed, and write the velocity as the product of speed and direction.
For each “picture function” below, sketch your representation of the integral.
𝑦=
𝑦=
𝑦=
𝑦=
MVC
Quiz 11.1/11.2
Name ______________________
1. Represent 𝑦 =
1
𝑥
as a vector-valued function 𝐫(𝑡).
2. Find the derivative of 𝐫(𝑡) in question 1.
3. Given the vector valued function 𝐫(𝑡) = (sin 𝑡)𝐢 + (3 cos 𝑡) 𝐣 + 𝑡𝐤,
0≤𝑡≤𝜋
(a)
Sketch the curve represented by r(t)
(Hint: it’s in your DNA!)
(b)
Find 𝐫 ′ (𝑡)
_________________________________
4. Consider the real valued functions given by 𝑥 = 𝑒 𝑡 , 𝑦 = −√𝑡, and 𝑧 = cos 𝑡.
(a)
Represent the function as a vector valued function r
(b)
Find the domain of r
(c)
Find
(d)
𝐝𝐫
𝐝𝐭
.
What does r’ represent?
5. Given vector valued functions 𝒓(𝑡) = 𝑡 2 𝐢 − 𝑡𝐤 𝑎𝑛𝑑 𝐮(𝑡) = 3𝑡𝐢 − 𝐣 + 4𝑡 2 𝐤,
find 𝐷𝑡 [𝐫(𝑡) ∙ 𝐮(𝑡)]
(advice: make sure to show your process just in case your derivatives are incorrect!)
MVC
Chapter 11 Test
Name _______________________________________
1. Use a vector-valued function to represent the space curve which marks the intersection
of the surfaces
𝑥2 + 𝑦2 = 4
𝑧 = −𝑥 2
Use the parameter 𝑥 = 2 sin 𝑡.
2. Evaluate the indefinite integral.
3
∫(sec 2 𝑡 𝐢 + sin 𝑡 𝐣 − 𝑡 2 𝐤) 𝑑𝑡
3. A moving particle starts at an initial position 𝐫(0) = ⟨1,0,0⟩ with initial velocity 𝐯(0) =
𝐢 − 𝐣 + 𝐤. It’s acceleration is 𝐚(𝑡) = 4𝑡𝐢 + 6𝑡𝐣 + 𝐤. Find its velocity and position at time t.
4. Let 𝑟(𝑡) = ⟨𝑡, 𝑡 2 , 3⟩.
(𝑎) Find parametric equations for the tangent line to the curve at 𝑡 = 1.
(𝑏) Find the unit tangent vector 𝑇(𝑡) at time 𝑡 = 1
(𝑐) principal unit normal vector for 𝑁(1)
(a decimal answer is fine, just round to the nearest hundredth)
(𝑑) sketch 𝐫(𝑡), 𝐓(1), and 𝐍(1)on the graph below.
5. A child slides down a slide at a playground that is exactly and conveniently modeled by
the vector-valued function 𝒓(𝑡) = (4 cos t)𝐢 + (4 sin 𝑡)𝐣 + 3𝑡𝐤. Find the length, in feet, of
the path the child will travel in one ride down. (this slide, like most playground slides,
makes one complete turn)
As long as you show me your work, you may use your calculator to find the value. Or, you
1
may use the fact that ∫ √𝑢2 ± 𝑎2 𝑑𝑢 = (𝑢√𝑢2 ± 𝑎2 ± 𝑎2 ln |𝑢 + √𝑢2 ± 𝑎2 |) + C
2
6. Is it always true that 𝐷𝑡 [𝑟(𝑡) × 𝑟 ′ (𝑡)] = 𝑟(𝑡) × 𝑟 ′′ (𝑡)? Can you prove it?
(Yes or no doesn’t cut it here)
**Bonus: The image below represents a model train moving at constant speed on a straight horizontal
track. As the engine moves along, it fires a marble into the air by a spring in the engine’s smokestack.
The marble, which continues to move with the same forward speed as the engine, rejoins the engine 1
second after it was fired.
Measure the angle the marble’s path makes with the horizontal and use the information to find how high
the marble went and how fast the engine was moving.
Name _________________________
12.1 and 12.3 Quiz
1. Find and describe the domain of each function below.
(𝑎) 𝑓(𝑥, 𝑦) = √𝑦 − 𝑥
___________________________
√𝑥+𝑦+6
___________________________
(𝑏) 𝑓(𝑥, 𝑦) =
(𝑐) 𝑧 =
𝑥
1
___________________________
√𝑥 2 +𝑦 2 −4
___________________________
(𝑑) 𝑓(𝑥, 𝑦) = ln(𝑥 + 𝑦)
2. Find 𝑓𝑥 and 𝑓𝑦 for 𝑓(𝑥, 𝑦) = 𝑥 4 + 6√𝑦 − 10𝑥𝑦
3. Find
𝜕𝑤
𝜕𝑥
for 𝑤 = 𝑥 2 𝑦 − 10𝑦 2 𝑧 3 + 43𝑥 − 7 tan(4𝑦)
4. Find all second-order partial derivatives of the function 𝑓(𝑥, 𝑦) = 𝑥𝑒 𝑦 + 𝑦 + 1
5. For 𝑓(𝑥, 𝑦, 𝑧) = 𝑦 2 𝑥 4 𝑧 3 + 2𝑒 𝑥𝑧 , find 𝑓𝑥𝑦𝑧
6. Sketch and label the level curves for each function at the given values of c.
(𝑎) 𝑓(𝑥, 𝑦) = 4𝑥 2 + 𝑦 2 ,
(𝑏) 𝑧 = 6 − 2𝑥 − 3𝑦,
𝑐 = 0, 1, 4, 16
𝑐 = 0, 2, 4, 6, 8
Name ___________________________
12.3/12.5/12.6 Review Quiz
1. Remember how to find and describe the domain of a function 𝑓(𝑥, 𝑦)
2. Remember how to find 𝑓𝑥 , 𝑓𝑦 , 𝑓𝑥𝑥 , 𝑓𝑥𝑦 , 𝑓𝑦𝑦 , 𝑓𝑦𝑥𝑦 , 𝑒𝑡𝑐.
3. Remember how to draw and label level curves.
Know how to draw a bullfighter (will not be a bonus!)
Review how to find extrema (12.8)
1. What is the slope of 𝑓(𝑥, 𝑦) = (2𝑥 − 3𝑦)3 at (1, 0) in the direction of x?
2. What is the slope of 𝑓(𝑥, 𝑦) = 𝑥 2 + 2𝑦 2 − 3𝑧 2 at (1, 1, 1)in the direction of 𝐮 = 𝐢 + 𝐣 + 𝐤
3. Find
𝜕3 𝑧
𝜕𝑦𝜕𝑥 2
4. Evaluate
𝜕𝑤
if 𝑧 = 𝑒 𝑥𝑦
𝜕𝑤
𝜕𝑡
for the function 𝑤 = 𝑧 − sin 𝑥𝑦, with 𝑥 = 𝑡,
𝑦 = ln 𝑡,
𝑧 = 𝑒 𝑡−1 , and
𝑡=1
𝜕𝑤
5. Find
and for 𝑤 = (𝑥 + 𝑦 + 𝑧)2 when 𝑥 = 𝑟 − 𝑠, 𝑦 = cos(𝑟 + 𝑠) , 𝑧 = sin(𝑟 + 𝑠),
𝜕𝑟
𝜕𝑠
𝑎𝑛𝑑 𝑟 = 1, 𝑠 = −1.
6. Use the gradient to find 𝐷𝑢 for 𝑓(𝑥, 𝑦) = 𝑥 2 𝑦 in the direction of 𝐯 = 𝐢 + 𝐣 at (2, 1).
Name _____________________________
MVC Ch. 12 Test
Second Partials Test:
2
(𝑎) 𝑓 has a 𝐥𝐨𝐜𝐚𝐥 𝐦𝐚𝐱𝐢𝐦𝐮𝐦 at (𝑎, 𝑏) if 𝑓𝑥𝑥 < 0 AND 𝑓𝑥𝑥 𝑓𝑦𝑦 − (𝑓𝑥𝑦 ) > 0 𝑎𝑡 (𝑎, 𝑏)
2
(𝑏) 𝑓 has a 𝐥𝐨𝐜𝐚𝐥 𝐦𝐢𝐧𝐢𝐦𝐮𝐦 at (𝑎, 𝑏) if 𝑓𝑥𝑥 > 0 AND 𝑓𝑥𝑥 𝑓𝑦𝑦 − (𝑓𝑥𝑦 ) > 0 𝑎𝑡 (𝑎, 𝑏
2
(𝑐) 𝑓 has a 𝐬𝐚𝐝𝐝𝐥𝐞 𝐩𝐨𝐢𝐧𝐭 at (𝑎, 𝑏) if 𝑓𝑥𝑥 𝑓𝑦𝑦 − (𝑓𝑥𝑦 ) < 0 𝑎𝑡 (𝑎, 𝑏)
The test fails if d=0
1. For the function 𝑓(𝑥, 𝑦) = √100 − 𝑥 2 − 𝑦 2 ,
(𝑎) Find and describe the domain of 𝑓
(𝑏) find 𝑓𝑥𝑦
(𝑐) Sketch a contour map of 𝑓 using level curves at 𝑐 = 0, 3, 6, 8, 9.
(d) Give a rough sketch of the surface of f based on your
level curves.
𝑦
2. Find the directional derivative of the function 𝑓(𝑥, 𝑦) = 3𝑥 2 + in the direction of
2
𝐯 = 𝐢 − 2𝐣 at the point (1, 2).
3. Find and classify the local extreme value(s) or saddle points for the function
𝑓(𝑥, 𝑦) = 𝑥 2 − 𝑦 2 − 2𝑥 + 4𝑦 + 6
4. Find and classify the local extreme value(s) or saddle points for the function
𝑓(𝑥, 𝑦) = 2𝑥𝑦 − 𝑥 2 − 2𝑦 2 + 4𝑥 + 4𝑦 − 4
𝑑𝑓
5. Using the function from #4, find , the derivative of 𝑓, at 𝑡 = 1, if 𝑥 and 𝑦 are
𝑑𝑡
represented by the following functions of 𝑡:
𝑥=𝑡
𝑦 = 3𝑡 2
6. Draw a bullfighter
**Bonus: Which topographic map shows contour lines for the peak of the Matterhorn, in
Switzerland?
a)
b)
Matterhorn, Switzerland
Name ___________________________
MVC Quiz 13.1 to 13.2
1. Evaluate the integral.
4
4
𝑥
∫ ∫ ( + √𝑦) 𝑑𝑥 𝑑𝑦
1 0 2
2. Use an iterated integral to represent each region R shown below: (you do not need to
evaluate the integral)

y




(a)
(b)








y

𝑦 = 3𝑥
𝑦 = 2𝑥
𝑥=3

𝑦 = 𝑥2



x


x








__________________________________________
__________________________________________
3. Sketch the region of integration and write an equivalent double integral with the order
of integration reversed.
𝑒
ln 𝑥
∫ ∫
1
4. Evaluate the integral
1
√1−𝑦 2
∫ ∫
0
0
(𝑥 − 𝑦) 𝑑𝑥 𝑑𝑦
0
𝑥𝑦 𝑑𝑦 𝑑𝑥
5. Find the volume of the solid bounded above by 𝑧 = 4 − 𝑦 2 and below by the region R:
𝑦 = 𝑥, 𝑦 = 2, 𝑥 = 0
6. Set up (but do not evaluate) a double integral in polar coordinates representing the
area of the region R given below.
16
9
Bonus: Evaluate the improper integral.
∞
1
𝑥
∫ ∫ 𝑦 𝑑𝑦 𝑑𝑥
1
0
Name ____________________________
MVC Quiz 13.3 to 13.6
𝑥 = 𝑟 cos 𝜃
and
𝑟2 = 𝑥2 + 𝑦2
and
Speed of light: about 299,792,458 m/s
𝑦 = 𝑟 sin 𝜃
𝑦
tan 𝜃 =
𝑥
“mathematics” in Swahili is hisabati
Just in case these come up…
None of the integrals on this quiz require cylindrical or spherical coordinates, though you
are welcome to use either if possible.
2
0
1. Sketch the region of integration given by ∫ ∫
0
6𝑥 𝑑𝑦𝑑𝑥
(𝑑𝑜𝑛′ 𝑡 𝑒𝑣𝑎𝑙𝑢𝑎𝑡𝑒)
−√4−𝑥 2
2. For the double integral below, reverse the order of integration and rewrite the integral.
(don’t evaluate)
2
∫ ∫
1
ln 𝑦 (sin−1
0
𝑥) ( 3√𝑦)𝑥 −2
𝑑𝑥𝑑𝑦
𝑒𝑥
3. There are MANY errors in the integral given for the volume of a surface below. Mark all
blunders and describe each mistake.
−2
𝑉=∫
0
𝜋
2
√4−𝑥 2 −𝑦 2
∫ ∫
0
√4 − 𝑥 2 − 𝑦 2 𝑑𝑥𝑑𝑧𝑑𝑦
0
4. Convert the integral below from polar to rectangular form. (do not evaluate)
𝜋
2
1
∫ ∫ 𝑟 3 sin 𝜃 cos 𝜃 𝑑𝑟𝑑𝜃
0
0
5. Convert the integral in rectangular form below into an equivalent polar integral. Then
evaluate the polar integral.
√9−𝑦 2
3
(𝑥 2 + 𝑦 2 ) 𝑑𝑥𝑑𝑦
∫ ∫
0
0
𝑑
6. Use polar coordinates to set up and evaluate a double integral ∬𝑅 𝑓(𝑥, 𝑦)𝑑𝐴
𝑓(𝑥, 𝑦) = 𝑒 (𝑥
2 +𝑦 2 )
,
𝑅: 𝑥 2 + 𝑦 2 ≤ 25, 𝑥 ≥ 0
7. Use a triple integral to find the volume of the region in the first octant bounded by the
coordinate planes and the surface 𝑧 = 4 − 𝑥 2 − 𝑦.
8. Use a triple integral to find the volume of the region between the planes 𝑥 + 𝑦 + 2𝑧 = 2
and 2𝑥 + 2𝑦 + 𝑧 = 4 in the first octant.
9. The following integral is difficult to evaluate using the current order of integration.
Evaluate it by changing the order in the most appropriate way. (make sure you change
enough…)
1
1
1
2
∫ ∫ ∫ 12𝑥𝑧𝑒 𝑧𝑦 𝑑𝑦𝑑𝑥𝑑𝑧
0
0
𝑥2
Name _______________________________
MVC Chapter 13 Test
The following question comes from a Harvard Multivariable Calculus test from the Fall of
last year.
For questions #1-6, match each graph of R to the appropriate
𝑑
double integral ∫ ∫𝑅 𝑓(𝑥, 𝑦)𝑑𝐴
1
𝑥2
______1. ∫ ∫ 𝑓(𝑥, 𝑦)𝑑𝑦𝑑𝑥
−1 −𝑥 2
1
𝑓(𝑥, 𝑦)𝑑𝑥𝑑𝑦
−1 𝑥 2 −1
______4. ∫ ∫
𝑓(𝑥, 𝑦)𝑑𝑥𝑑𝑦
−1 0
1−𝑥 2
______5. ∫ ∫
√1−𝑦 2
1
−1 𝑦 2 −1
1
______2. ∫ ∫ 𝑓(𝑥, 𝑦)𝑑𝑥𝑑𝑦
−1 −𝑦 2
1−𝑦 2
______3. ∫ ∫
𝑦2
1
1
𝑓(𝑥, 𝑦)𝑑𝑦𝑑𝑥
0
______6. ∫ ∫
−1 −√1−𝑥 2
𝑓(𝑥, 𝑦)𝑑𝑦𝑑𝑥
𝑑
7. Evaluate ∫ ∫𝑅 𝑓(𝑥, 𝑦) 𝑑𝐴 for 𝑓(𝑥, 𝑦) = 4 cos(𝑦 2 ) over the region shown below.
𝑒
𝑥
𝑧
8. Evaluate the triple integral ∫ ∫ ∫
1
0
1
√1−𝑥 2
9. Convert the triple integral ∫ ∫
0
0
2𝑦
𝑑𝑦 𝑑𝑧 𝑑𝑥
𝑧4
(𝑥 2 +𝑦 2 )
∫
21𝑥𝑦 2 𝑑𝑧𝑑𝑦𝑑𝑥
−√1−𝑥 2 −(𝑥 2 +𝑦 2 )
to cylindrical coordinates.
10. Consider the solid that is bounded above by the plane 𝑧 = 4 − 𝑥 and on the sides by
the cylinder 𝑥 2 + 𝑦 2 = 4, and below by the xy-plane.
(a) Set up a double integral AND a triple integral for finding the volume of the solid.
You may use any coordinate system
(b) Find the volume of the sold by evaluating ONE of your integrals from part (a)
(feel free to confirm by evaluating the second integral).
11. A water reservoir in Burlington, MA (the map to the right is centered there) is
bounded by a solid cylinder 𝑥 2 + 𝑦 2 ≤ 1. It has as the roof the cone 𝑥 2 + 𝑦 2 = (𝑧 − 6)2
and is bounded from below by the xy-plane 𝑧 = 0.
(a) What is the volume of the reservoir?
(b) What is the surface area of the roof of the reservoir?
(Hint: it’s the same as the surface area for 𝑥 2 + 𝑦 2 = 𝑧 2 )
12. Consider the solid that lies inside the sphere 𝑥 2 + 𝑦 2 + 𝑧 2 = 4, above the xy-plane, and
below the cone 𝑧 = √𝑥 2 + 𝑦 2 .
(a) set up an integral in spherical coordinates for the volume of the solid
(b) evaluate the integral.
2𝜋
√2
√4−𝑟 2
13. Consider the triple integral ∫
∫
∫
0
0
𝑟
3𝑟 𝑑𝑧𝑑𝑟𝑑𝜃.
(a) Convert to rectangular coordinates
(b) Convert to spherical coordinates
(c) Evaluate ONE of the integrals (any form).
14. Set up an integral in rectangular coordinates equivalent to the integral (DO NOT SOLVE)
𝜋
√3
∫ ∫
0
0
√4−𝑟 2
∫
𝑟 2 (sin 𝜃 − cos 𝜃)𝑧 2 𝑑𝑧𝑑𝑟𝑑𝜃
1
**Bonus:
I. Demonstrate the futility of your hard work in #10 by finding a geometric formula for
the volume of the solid.
Formulas and Conversions:
Rectangular to Polar:
𝑥 = 𝑟 cos 𝜃
𝑟2 = 𝑥2 + 𝑦2
𝑦 = 𝑟 sin 𝜃
𝑦
tan 𝜃 =
𝑥
Rectangular to cylindrical:
𝑟2 = 𝑥 2 + 𝑦2,
𝑡𝑎𝑛𝜃 =
𝑦
,
𝑥
𝑧=𝑧
Spherical to rectangular:
𝑥 = 𝜌 sin 𝜑 cos 𝜃 ,
𝑦 = 𝜌 sin 𝜑 sin 𝜃 ,
𝑧 = 𝜌 cos 𝜑
Rectangular to spherical:
𝜌2 = 𝑥 2 + 𝑦 2 + 𝑧 2 ,
tan 𝜃 =
𝑦
,
𝑥
𝑧
𝜑 = arccos (
)
√𝑥 2 + 𝑦 2 + 𝑧 2
Spherical to cylindrical (𝐫 ≥ 𝟎):
𝑟 2 = 𝜌2 sin2 𝜑 ,
𝜃 = 𝜃,
𝑧 = 𝜌 cos 𝜑
Cylindrical to spherical (𝐫 ≥ 𝟎):
𝜌 = √𝑟 2 + 𝑧 2 ,
𝜃 = 𝜃,
𝜑 = arccos (
𝑧
√𝑟 2 + 𝑧 2
)