Final Exam Review
MATH 2204 - Multivariable Calculus
The material in this review is not comprehensive; that is, this review just lists topics we have covered and
examples of problems related to those topics. The exam will cover material from all units.
1. Vectors.
(a) For the vectors ⃗a = ⟨2, 4, −3⟩ and ⃗b = ⟨−1, 2, 2⟩, find |⃗a|, 3⃗a, 3⃗b − 2⃗a.
(b) Find two values of x such that ⃗a = ⟨−2, x, 3x⟩ and ⃗b = ⟨3, 3x, 1⟩ are orthogonal.
(c) Of two vectors, each containing one of the two solutions in the previous question, to which
vector is ⃗c = ⟨6, −3, −9⟩ parallel? How do you know?
(d) Find the angle between vectors ⃗a = ⟨1, 2, 3⟩ and ⃗b = ⟨0, 1, 0⟩.
(e) Find the scalar and vector projections of ⃗b = 5⃗i − ⃗k onto ⃗a = ⃗i + 2⃗j + 3⃗k
(f) Determine whether the given vectors are orthogonal, parallel, or neither.
i. ⃗u = ⟨−5, 4, −2⟩ and ⃗v = ⟨3, 4, −1⟩
ii. ⃗u = ⟨9, −6, 3⟩ and ⃗v = ⟨−6, 4, −2⟩
iii. ⃗u = ⟨c, c, c⟩ and ⃗v = ⟨c, 0, −c⟩
(g) Find a nonzero vector orthogonal to the plane through the points P (−2, 0, 4), Q(1, 3, −2) and
R(0, 3, 5).
2. Lines and Planes in Space
(a) Find an equation of the line through the point (−2, 4, 10) and parallel to the vector ⟨3, 1, −8⟩.
(b) Find an equation of the line through the point (1, 0, 6) and perpendicular to the plane x+3y+z =
5.
(c) Find an equation of the line through the points (3, 1, −1) and (3, 2, −6).
(d) Find an equation of the plane through the point (4, −2, 3) and parallel to the plane 2x−5z = 12.
(e) Find an equation of the plane through the points (0, 1, 1), (1, 0, 1), and (1, 1, 0).
3. Partial Derivatives. Find all derivatives zx , zy , zxx , zyy , zxy and zyy of the functions below.
(a) z = cos(x2 y) + sin(xy 2 )
√
(c) z = 2 xy + lnxy2
(b) z = ln xy 2
(d) z = tan x2 y
4. Tangent Planes. Find the equation of the tangent plane to the a given surface at a specified point.
(a) z = ex ln y at (3, 1, 0).
(b) xy 2 + yz 2 + zx3 = 3 at (1, 1, 1).
5. Linear Approximations.
(a) If f (2, 3) = 5, fx (2, 3) = 4 and fy (2, 3) = 3, approximate f (2.02, 3.1).
p
(b) Find the linearization of z = 20 − x2 − 7y 2 at (2, 1) and use it to approximate the value at
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Final Exam Review
MATH 2204 - Multivariable Calculus
(1.95, 1.08).
6. Limits and Continuity
2
(a) Find lim(x,y)→(2,−1) xxy+xy
2 −y 2
2
(b) Find lim(x,y)→(π,π/2) y sin(x − y)
4
(c) Find lim(x,y)→(0,0) x2xy+y8
(d) Determine where f (x, y) = 1+exyx−y is continuous.
xy
(e) Determine where f (x, y) is continuous if f (x, y) = x2 +xy+y
2 when (x, y) ̸= (0, 0) and f (x, y) = 0
when (x, y) = (0, 0).
7. Multivariable Functions
(a) Find and sketch the domain of f (x, y, z) = ln(16 − 4x2 − 4y 2 − z 2 ).
p
(b) Find the domain and range of f (x, y) = 9 − x2 − y 2 .
y
(c) Find the level curves and draw a contour map of f (x, y) = x2 +y
2.
8. Double Integrals.
(a) Using the midpoint rule with m = 2 and n = 3 to evaluate
˜
2
R (1 − xy )dA over the region
defined by R = [0, 4] × [−1, 2].
ˆ ˆ
(b)
(x + 2y)dxdy where D = {(x, y)|0 ≤ x ≤ 1, 0 ≤ y ≤ x2 }.
ˆ ˆ
D
ˆ ˆ
D
(y 3 )dxdy where D is the triangular region with vertices (0, 2), (1, 1), and (3, 2).
(c)
(xy)dxdy, where D is the region in the first quadrant between the circles x2 + y 2 = 4 and
(d)
D
x2 + y 2 = 25.
ˆ ˆ
1
p
(e)
dxdy, where D is the region inside r = 4 cos θ and outside r = 2.
2
x + y2
D
˜
(f) Sketch the region set up the integrals for both possible orders of integration. D f (x, y)dydx
where D = {(x, y)| arctan x ≤ y ≤ π/4, 0 ≤ x ≤ 1}.
´2´1
(g) Evaluate 0 y/2 y cos(x3 − 1)dxdy.
(h) Find the volume of the solid in the first octant bounded by the surface z = 9 − y 2 and the plane
x = 2.
(i) Find the volume of the solid bounded by the plane x + y + z = 1 in the first octant.
(j) Find the volume of the solid under the paraboloid z = x2 + y 2 and above the disk x2 + y 2 ≤ 9.
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Final Exam Review
MATH 2204 - Multivariable Calculus
(k) Find the volume of the solid above the cone z =
p
x2 + y 2 and below the paraboloid z =
2 − x2 − y 2 .
(l) Find the volume of the solid enclosed by the paraboloids z = x2 + y 2 and z = 36 − 3x2 − 3y 2 .
(m) Find the mass and center of mass of the lamina that occupies the triangular region with vertices
(0, 0), (2, 1), and (0, 3) and has the density function ρ(x, y) = x + y.
p
(n) Find the average value of the function f (x, y) =
x2 + y 2 over the region D bounded by
r = 3 sin(2θ).
9. Triple Integrals.
˝
3 2
2
(a)
E (x y z)dxdydz where E = {(x, y, z)|1 ≤ x ≤ 2, 0 ≤ y ≤ x, 0 ≤ z ≤ y }.
˝
(b)
E (xy)dxdydz where E is the solid tetrahedron with vertices (0, 0, 0), (1, 0, 0), (0, 2, 0), and
(0, 0, 3).
˝ p
(c)
x2 + y 2 dxdydz where E is the region that lines inside the cylinder x2 + y 2 = 16 and
E
between the planes z = −5 and z = 4.
˝
2
2
2
2
(d)
E (2)dxdydz where E is the solid that lies between the cylinders x + y = 1, x + y = 4, and
between the xy-plane and the plane z = x + 2.
˝
2
2
2
2
2
2
(e)
E (xy)dxdydz where E is the region between the spheres x + y + z = 1 and x + y + z = 4
in the first octant.
˝
(f)
E (x + y + z)dV where E is the solid in the first octant that lies under the paraboloid z =
4 − x2 − y 2 .
(g) Find the volume of the solid that lies within the sphere x2 + y 2 + z 2 = 4, above the xy-plane,
p
and below the cone z = x2 + y 2
(h) Find the mass and center of mass of the solid E where E lies under the plane z = x + y + 1 and
√
above the region in the xy-plane bounded by the curves y = x, y = 0, and x = 1. The solid
has density ρ(x, y, z) = 2.
10. Quadric Surfaces
(a) Consider the equation 4x2 + y 2 + 4z 2 − 4y − 40z + 100 = 0
i. Reduce the equation to one of the standard forms for quadric surfaces.
ii. Classify the surface
iii. Write an equation for the cross section at z = 5
iv. Write an equation for the cross section at y = 2.
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Final Exam Review
MATH 2204 - Multivariable Calculus
v. Write an equation for the cross section at x = 0.
vi. Sketch the Surface.
11. Chain Rule.
(a) Find the indicated derivatives.
i. z = x ln (x + 2y), x = cos t, y = sin t. Find z ′ (t) at t = 0.
ii. z = ex y + xy 2 , x = st, y = s2 + t2 . Find zs and zt at (1, 1).
(b) The temperature at a point (x, y) is T (x, y), measured in degrees Celsius. A bug crawls so that
√
its position after t seconds is given by x = 1 + t, y = 2 + 13 t where x and y are measured in
centimeters. The temperature function satisfies Tx (2, 3) = 4 and Ty (2, 3) = 3. Determine how
fast the temperature rises on the bug’s path after 3 seconds.
12. Directional Derivative and the Gradient
(a) Find the directional derivative of f (x, y, z) = x2 y + y 2 z in the direction of v = ⟨2, −1, 2⟩ at the
point (1, 2, 3).
(b) Find the directional derivative of f (x, y) =
√
xy at P (2, 8) in the direction of Q(5, 4).
(c) Find the magnitude and direction of the maximum rate of change of f (x, y, z) = x ln(yz) at
(1, 2, 0.5)..
(d) Find the equation of the tangent plane and the normal line to the surface x4 + y 4 + z 4 = 3x2 y 2 z 2
at (1, 1, 1).
(e) Suppose that over a certain region of space the electrical potential V is given by V (x, y, z) =
5x2 − 3xy + xyz. (i) Find the rate of change of the potential at P (3, 4, 5) in the direction of
⃗v = ⟨1, 1, −1⟩. (ii) In which direction does V change most rapidly at P ? (iii) What is the
maximum rate of change of V at P ?
13. Maximum and Minimum Values. Find and classify the extreme values of the functions below.
(a) z = x2 + y 2 + x2 y + 4.
2 2
−8x+y
(b) z = x y xy
(c) z = xy − 2x − y
(d) f (x, y) = x2 + y 2 + x2 y + 4. Find the absolute extreme values on the region given by
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Final Exam Review
MATH 2204 - Multivariable Calculus
D = {(x, y)||x| ≤ 1, |y| ≤ 1}.
(e) f (x, y) = xy 2 . Find the absolute extreme values on the region given by
D = {(x, y)|x ≥ 0, y ≥ 0, x2 + y 2 ≤ 3}
14. Lagrange Multipliers.
(a) Find the maximum and minimum values of f subject to the given constraint.
i. f (x, y) = x2 − y 2 ; x2 + y 2 = 1.
ii. f (x, y) = x2 y; x2 + 2y 2 = 6.
iii. f (x, y, z) = 2x + 6y + 10z; x2 + y 2 + z 2 = 35.
iv. f (x, y, z) = 3x − y − 3z; x + y − z = 0 and x2 + 2z 2 = 1.
(b) Let ⃗u = ⟨x, y, z⟩ be a unit vector. Let ⃗v = ⟨5, 4, 2⟩. Find ⃗u that maximizes the dot product ⃗u · ⃗v .
15. Space Curves
(a) Sketch the curve given by ⃗r = ⟨t, 2 − t, 2t⟩. Indicate with an arrow the direction of the curve.
(b) Sketch the curve given by ⃗r = ⟨sin(π/t), t, cos(πt)⟩. Indicate with an arrow the direction of the
curve.
(c) Let r⃗1 (t) = ⟨t2 , 7t − 12, t2 ⟩ and r⃗2 (t) = ⟨4t − 3, t2 , 5t − 6⟩ be the trajectories of two different
particles. Do the paths of these two functions intersect? Do the particles collide?
√
2 −t
⃗i + t + 8⃗j + sin πt ⃗k. (i) Describe the domain of ⃗r(t), and (ii)
(d) Consider the function ⃗r(t) = tt−1
ln t
Find limt→1 ⃗r(t).
3
2
(e) Find the unit tangent vector T⃗ (t) of ⃗r(t) = ⟨t2 − 2t, 1 + 3t, t3 + t2 ⟩ when t = 2.
(f) If ⃗r(t) = ⟨t4 , t, t2 ⟩, find ⃗r′ (t) × ⃗r′′ (t).
√
2
(g) Find the equations of the tangent line to the curve ⃗r(t) = ⟨t2 +1, 4 t, et −1 ⟩ at the point (2, 4, 1).
ˆ π/4 (h) Evaluate
sec t tan t⃗i + t cos(2t)⃗k + sin2 (2t) cos(2t)⃗k dt.
0
(i) If r⃗′ (t) = ⟨t, et , tet ⟩ and ⃗r(0) = ⟨1, 1, 1⟩, find ⃗r(t).
(j) Consider the curve defined by x = 2 cos t, y = 2 sin t, and z = 4t. Find an equation of the
tangent line to the curve at the point where t = π.
(k) Let C represent the curve of the intersection of the curve x2 + y 2 = 25 with the plane x + z = 5.
(i) Find the parametric equations representing C (ii) Find the equation of the tangent line to
the curve at the point where t = 0.
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Final Exam Review
MATH 2204 - Multivariable Calculus
(l) Consider the curve C which is the intersection of the surfaces x2 + y 2 = 9 and z = 1 − y 2 . (i)
Find the parametric equations representing C (ii) Find the equation of the tangent line to the
curve C at a point (0, 3, −8).
16. Motion in Space
(a) If the position of an object at time t is given by ⃗r(t) = ⟨2 cos t, 3t, 2 sin t⟩, find the velocity,
acceleration, and speed of the object.
(b) If an object has an acceleration of ⃗a(t) = ⟨sin t, 2 cos t, 6t⟩,an initial velocity of ⃗v (0) = ⟨0, 0, −1⟩,
and an initial position of ⃗r(0) = ⟨0, 1, −4⟩, find the velocity and position vectors for any time t.
(c) A force with a magnitude 20 N acts directly upward form the xy-plane on an object with a mas
sof 4 kg. The object starts at the origin with an initial velocity ⃗v (0) = ⃗i − ⃗j. Find its position
function and its speed at time t.
17. Arc Length and Curvature.
(a) Let C represent the curve of the intersection of the curve x2 + y 2 = 25 with the plane x + z = 5.
(i) Find the parametric equations representing C (ii) Estimate the length of the curve from
t = 0 to t = π.
(b) Consider the curve C which is the intersection of the surfaces x2 + y 2 = 9 and z = 1 − y 2 . (i)
Find the parametric equations representing C ((ii) Find the length of the curve from (3, 0, 1) to
(0, 3, −8).
(c) Find the curvature of y = x4 .
18. Applications
(a) A conveyor belt generates a force of F⃗ = ⟨5, −3, 1⟩ that moves a suitcase from point (1, 1, 1) to
point (9, 4, 7) along a straight line. Find the work done by the conveyor belt. Note that the
distance is measured in meters and the force in Newtons.
(b) Find the area of the parallelogram PQRS with vertices P (1, 1, 0), Q(7, 1, 0), R(9, 4, 2), and
S(3, 4, 2).
(c) Find the area of the triangle PQR with points P (−2, 0, 4), Q(1, 3, −2) and R(0, 3, 5).
(d) The American Pika is a small mammal that prefers high-elevation habitats with cooler temperatures. Due to warming global surface temperatures, the American Pika is becoming locally
threatened in many of its North American habitats. A Pika scurries uphill in search of food in
the Rocky Mountains with a force of F⃗ = 2⃗i + 3⃗j + 2⃗k and travels 15 meters in 5 seconds from
the origin (0, 0, 0) of its trip to the point (10, 5, 10). Find the power output of the pika during
its journey to find food. [Note: Power is defined as P = Wt , where P is power (measured in
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Final Exam Review
MATH 2204 - Multivariable Calculus
Watts), W is work (measured in Joules), and t is the time taken to do the work, (measured in
seconds).]
(e) A boat sails south with the help of wind blowing in the direction S36◦ with magnitude 400 lb.
Find the work done by the wind as the boat moves 120 ft.
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Final Exam Review
MATH 2204 - Multivariable Calculus
Important Formulas
Vectors
Let ⃗a = ⟨a1 , a2 , a3 ⟩ and ⃗b = ⟨b1 , b2 , b3 ⟩.
|⃗a| =
p
a21 + a22 + a23
⃗a · ⃗b = a1 b1 + a2 b2 + a3 b3 .
The normalization of ⃗a is
⃗a
.
|⃗a|
⃗a × ⃗b = ⟨a2 b3 − a3 b2 , a3 b1 − a1 b3 , a1 b2 − a2 b1 ⟩.
⃗a · ⃗b
The angle θ between ⃗a and ⃗b is given by cos θ =
.
|⃗a||⃗b|
The basis vectors in three dimensions are ⃗i = ⟨1, 0, 0⟩, ⃗j = ⟨0, 1, 0⟩, ⃗k = ⟨0, 0, 1⟩.
⃗
The scalar projection of ⃗b onto ⃗a is comp⃗a⃗b = ⃗a|⃗a·b|
⃗
⃗
a·b
u, where ⃗
u = |⃗⃗aa| .
The vector projection of ⃗b onto ⃗a is proj⃗a⃗b = |⃗
⃗a = (comp⃗a⃗b)⃗
a|2
Partial Derivatives and Chain Rule
Chain Rule: For a function z = f (x, y) where x = g(t, s) and y = h(t, s).
1. zt = zx xt + zy yt
2. zs = zx xs + zy ys
fxx
fxy
= fxx fyy − fxy fyx . To classify extreme values (a, b)
fyx fyy
of f (x, y), consider two cases: (i) if D(a, b) > 0, then there is a minimum at (a, b) if fxx (a, b) > 0 and a maximum at (a, b) if
The determinant of the Hessian matrix for f (x, y) is D =
fxx (a, b) < 0. (ii) if D(a, b) < 0, then there is a saddle point at (a, b).
To find extreme values of f (x, y, z) subject to the constraints g(x, y, z) = c and h(x, y) = d, you must find all x, y, and z
values that satisfy the constraints along with the relationships ∇f = λ∇g + µ∇h, where λ and µ are Lagrange multipliers.
Put another way, let H = f (x, y, z) − λ(g(x, y, z) − c) − µ(h(x, y, z) − d). To find extreme values, solve ∇H = 0.
The gradient vector of a function f (x, y, z) is given by ∇f = ⟨fx , fy , fz ⟩.
The derivative of a function f (x, y, z) in the direction of a vector ⃗
u = ⟨a, b, c⟩ is given by Du⃗ f (x, y, z) = ∇f · ⃗
u.
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Final Exam Review
MATH 2204 - Multivariable Calculus
Equations of Lines and Planes
The equation of a line that passes through (x0 , y0 , z0 ) with slope vector ⟨a, b, c⟩ can be defined as ⟨x(t), y(t), z(t)⟩ = ⟨x0 +
at , y0 + bt , z0 + ct ⟩.
The equation of the tangent line of f⃗(x) = ⟨x(t), y(t), z(t)⟩ at f⃗(t0 ) = (x0 , y0 , z0 ) is given by ⟨xT (t), yT (t), zT (t)⟩ = ⟨x0 +
x(t0 )t , ⟨y0 + y(t0 )t , ⟨z0 + z(t0 )t ⟩.
For the function z = f (x, y) the equation of a tangent plane at a point (a, b, c) is fx (a, b)(x − a) + fy (a, b)(y − b) − (z − c) = 0.
For the function f (x, y, z) = 0 the equation of a tangent plane at a point (a, b, c) is fx (a, b, c)(x − a) + fy (a, b, c)(y − b) +
fz (a, b, c)(z − c) = 0. This is also the tangent plane to a level surface f (x, y, z) = k at the point (a, b, c).
= Fyy−b
= Fz z−c
The symmetric equations for the normal line to a surface at a point P (a, b, c) is given by Fxx−a
(a,b,c)
(a,b,c)
(a,b,c)
The osculating plane of a curve C defined by ⃗r(t) that contains the point ⃗r(t0 ) = (x0 , y0 , z0 ) is given by b1 (x − x0 ) + b2 (y −
⃗ 0 ) = ⟨b1 , b2 , b3 ⟩, where B(t)
⃗
y0 ) + b3 (z − z0 ) = 0, where B(t
is the binormal vector.
The normal plane of a curve C defined by ⃗r(t) that contains the point ⃗r(t0 ) = (x0 , y0 , z0 ) is given by n1 (x − x0 ) + n2 (y − y0 ) +
⃗ 0 ) = ⟨n1 , n2 , n3 ⟩, where N
⃗ (t) is the normal vector.
n3 (z − z0 ) = 0, where B(t
Transformations
Cylindrical Coordinates. x = r cos θ, y = r sin θ, z = z. The Jacobian is r.
Spherical Coordinates. x = r cos θ sin ϕ, y = r sin θ sin ϕ, z = r cos ϕ. The Jacobian is r2 sin ϕ.
Polar Coordinates. x = r cos θ, y = r sin θ. The Jacobian is r.
Applications
Average value. The average value of a function f (x, y, z) over the solid region E is given by
fave =
1
V (E)
ˆ ˆ ˆ
f (x, y, z)dxdydz
E
where V (E) is the volume of the solid region E.
Mass and Center of Mass. If a solid object occupies the region E and has density ρ(x, y, z), then the mass is m =
´´´
´´´
´´´
1
1
ρ(x, y, z)dxdydz and the center of mass is (x̄, ȳ, z̄), where x̄ = m
xρ(x, y, z)dxdydz, ȳ = m
yρ(x, y, z)dxdydz,
E
E
E
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Final Exam Review
1
and z̄ = m
´´´
E
MATH 2204 - Multivariable Calculus
zρ(x, y, z)dxdydz.
Arc Length and Curvature
The length of a curve between t = a and t = b is given by L =
The curvature of a curve is given by κ(t) =
κ(t) =
´bp
a
(x′ (t))2 + (y ′ (t))2 + (z ′ (t))2 dt.
|T⃗ ′ (t)|
, where T⃗ is the unit tangent vector. We can also define curvature as
|r⃗′ (t)|
|r⃗′ (t) × r⃗′′ (t)|
.
|r⃗′ (t)|3
⃗′
The unit tangent vector. The unit tangent vector to a curve C defined by ⃗r(t) is given by T⃗ (t) = r⃗′ (t) .
|r (t)|
⃗ (t) = T⃗′ (t) , where T⃗ (t) is the
The unit normal vector. The unit normal vector to a curve C defined by ⃗r(t) is given by N
⃗′
|T (t)|
unit tangent vector of ⃗r(t).
⃗
⃗ (t), where N
⃗ (t) and T⃗ (t) are
The binormal vector. The binormal vector of a curve C defined by ⃗r(t) is B(t)
= T⃗ (t) × N
the unit tangent and unit normal vectors of ⃗r(t), respectively.
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Final Exam Review
MATH 2204 - Multivariable Calculus
Quadric Surfaces
11