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1. Assignment for Matlab report
Subject: Calculus 2
Phan Thi Khanh Van

2. 1 Requirement
All the students in the class is divided into 10 groups. Each group has about
5-6 members. The teacher will choose topics (accidentally), and you will have
to solve some of the concrete problems below by using Matlab programm in
the time of the seminar.
2 Assignment
2.1 Topic 1
2.1.1 Find the first order partial derivatives, total differential of
the following funtions:
1. f = x3
+ 3x2
y + exy
2. f =
x
R
y
cos(t2
)dt
3. f = 1
√
x2+y2
4. f = sin( x
1+y
)
5. f = ex
+ ln(x − y2
) at (0, 1)
6. The temperature at a point (x, y) on a flat metal plate is given by
T(x, y) = 60
1+x2+y2 , where T is measured in 0
C, x, y in meters. Find
the rate of change of temperature with respect to distance at the point
(2, 1) in the x− direction and y− direction.
7. The wind-chill index is modeled by the function:
W = 13.12+0.6215T −11.37v0.16
+0.3965Tv0.16
, where T is the temper-
ature (0
C), v - the wind speed (km/h). When T = −150
C, v = 30km/h,
by how much would you expect the apparent temperature to drop if
the actual temperature descreases by 10
C? What if the wind speed
increase by 1km/h?
1

3. 2.1.2 Find the second order partial deravatives of the following
functions
1. z = xexy
2. z = cos(e2y
− 2x)
3. z = ln(y
x
) at (2, 0)
4. z = arctan(x
y
). Find A = f”xx(1, −1) + 2f”yy(−1, 1)
5. Show that the function z = xey
+ yex
is a solution of the equation
∂3z
∂x3 + ∂3z
∂y3 = x ∂3z
∂x∂y2 + y ∂3z
∂x2∂y
6. Show that each of the following functions is a solution of the wave
equation utt = a2
uxx
• u = sin(kx) sin(akt)
• u = t
a2t2−x2
• u = sin(t − at) + ln(x + at)
2.1.3 Verify that the conclusion of Clairaut’s Theorem holds, that
is, uxy = uyx
1. u = x sin(x + 2y)
2. u = ln
√
x2 + y2
3. u = xyey
2.1.4 Use the chain rule to find the first order partial derivatives
of the following composite functions
1. f(x, y) = xy
, x = ln t, y = sin t. Find df
dt
2. f(x, y) = e3x+2y
, x = sin t, y = t2
. Find df
dt
|t=0
3. f(x, y) = yz
x
, x = et
, y = ln t, z = t2
− 1. Find df
dt
|t=1
4. f(u, v) = eu
ln v, u = xy2
, v = x2
y. Find ∂f
∂x
, ∂f
∂y
5. z = x.f(y
x
). Find A = z0
x + y
x
z0
y
2

4. 6. The temperature at a point (x, y) is T(x, y)(0
C). A bug crawls so that
its position after t seconds is given: x =
√
1 + t, y = 2+ 1
3
t(cm). Given
Tx(2, 3) = 4, Ty(2, 3) = 3. How fast is the temperature is rising on the
bug’s path after 3 seconds?
2.1.5 Find the first, second order partial derivatives of the fol-
lowing functions
1. y = y(x) : x3
+ y3
= 6xy. Find dy
dx
2. y = y(x) : 1 + xy − ln(exy
+ e−xy
= 0. Find dy
dx
, d2y
dx2
3. y = y(x) : x − y + arctan y = 0. Find dy
dx
, d2y
dx2
4. z = z(x, y) : z ln(x + z) − xy
z
= 0. Find ∂z
∂x
, ∂z
∂y
5. z = z(x, y) : z3
− 4xz + y2
− 4 = 0. Find ∂z
∂x
(1, 2), ∂z
∂y
(1, 2), given
z(1, 2) = 2
2.1.6 Find the derivative in the direction of vector ~
u of the fol-
lowing functions at the following points.
1. f = yx
at (1, 1), ~
u = (3, 2)
2. f = (x + 2y − 1)3
at (1, 1), ~
u = (1, 3)
3. f = sin(x2
+ y2
) at (0, 0), ~
u = (2, 5)
4. f = ln(1 + x + y) at (0, 0), ~
u = (−2, 4)
5. f = ex
cos y at (0, 0), ~
u = (−1, 3)
6. f = arcsin(x + y2
) at (0, 0), ~
u = (1, 5)
7. Near a buoy, the depth of a lake at the point with coor. (x, y) is
z = 200 + 0.02x2
− 0.001y3
. A fisherman in a small boat starts at the
point (80, 60) and moves toward the buoy, which is located at (0, 0). Is
the water under the boat getting deeper or shallower when he departs?
8. The temperature T in a metal ball is inversely proportional to the
distance from the center of the ball, which we take to be (0, 0, 0).
T(1, 2, 2) = 1200
C.
3

5. • Find the rate of change of T at (1, 2, 2) in the direction towart the
point (2, 1, 3)
• Show that at any point in the ball the direction of greatest increase
in temperature is given by a vector that points toward the origin.
2.1.7 Find the tagent planes of the following surfaces at the fol-
lowing points.
1. (S) : z = 4x2
+ 2y2
at (1, 1, 6)
2. (S) : z = e5−4x2−2y2
at (1, 1, 1
e
)
3. (S) : 4x2
+ 2y2
+ z2
= 4 at (1/2, 1, 1)
4. (S) : 2x2
− y2
+ 3z = 4 at (1, 1, 1)
5. (S) : 4x2
+ 2y2
= z2
at (1, 0, 2)
6. (S) : x2
+ 2y2
− z2
− 2 = 0 at (1, 1, 1)
2.1.8 Find the n- order Taylor expansion of the following func-
tions at the following points.
1. f = yx
at (1, 1), order = 2
2. f = y
x
at (1, 1), order = 3
3. f = (x + 2y − 1)3
at (1, 1), order = 3
4. f = sin(x2
+ y2
) at (0, 0), order = 6
5. f = ex
cos y at (0, 0), order = 5
6. f = arcsin(x + y2
) at (0, 0), order = 5
2.2 Topic 2
2.2.1 Study the extrema of the following functions
1. f = (x − 1)2
+ 2y2
2. f = (x − 1)2
− 2y2
4

6. 3. f = x2
+ y2
+ xy + x − y + 1
4. f = 2x3
+ xy2
+ 5x2
+ y2
5. f = x2
+ xy + y2
− 4 ln x − 10 ln y
6. f = x2
+ 3xy − 8 ln x − 6 ln y
7. f = x4
+ y4
− x2
8. Find the point on the surface z2
= xy +1 that are closest to the origin.
9. Find 3 positive numbers whose sum is 100 and whose product is a
maximum
10. Find the volume of the largest rectangular box in the first octant with 3
faces in the coordinate planes and 1 vertex in the plane: x+2y+3z = 6
2.2.2 Study the extrema of the following functions subject to the
following constraints:
1. f = x2
y: x2
+ 2y2
= 6
2. f = 6 − 5x − 4y: x2
− y2
= 9
3. f = 1 − 4x − 8y: x2
− 8y2
= 8
4. f = x2
+ y2
+ xy: x2
+ 2y2
= 1
5. f = 2x2
+ 12xy + y2
: x2
+ 4y2
= 15
6. f = x2
+ y2
: x/2 + y/3 = 1
2.3 Topic 3
2.3.1 Evaluate the double integral
1. f = x cos y, D : y = 0, y = x2
, x = 1
2. f = x + y, D : y =
√
x, y = x2
3. f = y3
, D : ∆OAB : O(0, 0), A(1, 1), B(2, 0)
4. f = x + 2y, D : y = 2x2
, y = 1 + x2
5

7. 5. f = sin(y2
), D : y = x, y = 1, x = 0, x = 1
6. f = xy, D : y = x − 1, y = 2x + 6
7. f =
√
4 − x2 − y2, D : x2
+ y2
≤ 4, x ≤ y ≤
√
3
8. Electric charge is distributed over the rectangle 1 ≤ x ≤ 3, 0 ≤ y ≤ 2
so that the charge density at (x, y) is σ(x, y) = 2xy+y2
. Find the total
charge on the rectangle
9. Electric charge is distributed over the disk x2
+ y2
≤ 4 so that the
charge density: σ(x, y) = x + y + x2
+ y2
. Find the total charge on the
disk.
2.3.2 Evaluate the area of the following domains on the xy plane
1. D : x + y2
= 1, y − x = 1, x = 0
2. D : y = x2
, y = 2 − x2
3. D : x2
+ y2
= 2x, x2
+ y2
= 4x, y ≤ x
4. D : x2
+ y2
= 2y, x2
+ y2
= 6y, y ≥
√
3x, x ≥ 0
2.3.3 Find the mass and the center of mass of the lamina that
occupies the region D and has the density function ⇢
1. D : y = ex
, y = 0, x = 0, x = 1; ⇢(x, y) = y.
2. D : y =
√
x, y = 0, x = 1; ⇢(x, y) = x.
3. D : x2
+ y2
≤ 1, x ≥ 0, y ≥ 0, density at any point is proportional to
the square of its distance from the origin.
2.3.4 Find the area of the surface
1. The part of the plane 2x + 5y + z = 10 that lies inside the cylinder
x2
+ y2
= 9
2. The part of the parboloid z = 4−y2
−x2
that lies above the xy− plane
3. The part of the sphere x2
+y2
+z2
= 4 that lies above the plane z = 1.
6

8. 2.4 Topic 4: Evaluate the following triple integrals
1. f = x2
, Ω : z = 0, z = x2
+ y2
, x2
+ y2
= 1
2. f = z
x2+y2 , Ω : z = 2⇡, z = 3⇡, x2
+ y2
= 1, x2
+ y2
= 4
3. f = y, Ω : x2
+ y2
= 2y, z = 0, z = 3
4. f = x2
+ y2
, Ω :
√
x2 + y2 = z, z = 2, x2
+ y2
= 4
5. f = yx2
, Ω :
√
x2 + z2 = 2y, y = 2
6. f =
√
x2 + y2, Ω :
√
x2 + z2 = y, y = 4
7. f = z, Ω : x2
+ y2
≤ 2z, x2
+ y2
+ z2
≤ 3
8. f = z
√
x2 + y2, Ω : 0 ≤ z ≤ 1, 0 ≤ y ≤
√
2x − x2
9. f =
√
x2 + y2 + z2, Ω : x2
+ y2
+ z2
≤ z
10. f = e
√
(x2+y2+z2)3
, Ω : x2
+ y2
+ z2
≤ 1
2.4.1 Evaluate the volume of the solids bounded with the follow-
ing surfaces
1. Ω : z = 16 − x2
− 2y2
, y = 2, y = 0, x = 2, x = 0, z = 0
2. Ω : z = x2
+ y2
, y = 2x, y = x2
3. Ω : z = 1 − x2
− y2
, z = 0
4. Ω : z = 4 − x2
− y2
, z = 2
2.4.2 Find the mass and center of mass of the solid E with the
given density function ⇢
1. E is bounded by the parabolic cylinder z = 1 − y2
and the planes
x + z = 1, x = 0, z = 0, ⇢(x, y, z) = 4
2. E is the cube given by 0 ≤ x ≤ 4, 0 ≤ y ≤ 4, 0 ≤ z ≤ 4, ⇢(x, y, z) =
x2
+ y2
+ z2
7

9. 3. E is the tetrahedron bounded by the planes x = 0, y = 0, z = 0, x +
y + z = 1, ⇢(x, y, z) = y
4. Let E be the solid in the first octant bounded by the cylinder y2
+
z2
= 9 and the planes x = 0, y = 3x, z = 0 with the density function
⇢(x, y, z) = x2
+ y2
5. Let E be the solid in the first octant bounded by the cylinder x2
+y2
= 1
and the planes y = z, x = 0, z = 0 with the density function ⇢(x, y, z) =
1 + x + y + z.
6. E- the hemisphere x2
+ y2
+ z2
≤ 1, z ≥ 0, ⇢(x, y, z) =
√
x2 + y2 + z2
2.5 Topic 5: Evaluate the following line integrals
1.
R
C
xds, C - the arc of the parabola y = x2
from (0, 0) to (1, 1)
2.
R
C
x3
zds, C : x = 2 sin t, y = t, z = 2 cos t, 0 ≤ t ≤ π
2
3.
R
C
√
xydx + ey
dy + xzdz, C : r(t) = t4
.i + t2
.j + t3
.k, 0 ≤ t ≤ 1
4.
R
C
x3
ydx − xdy, C - the circle x2
+ y2
= 1 with counterclockwise orien-
tation.
5. Find the work done by the force field F = z.i + x.j + y.k in moving a
particle from the point (3, 0, 0) to the point (0, π
2
, 3) along
• A straight line
• The helix x = 3 cos t, y = t, z = 3 sin t
6. Use Green’s Theorem to evaluate
R
C
√
1 + x3dx+2xydy, C- the triangle
with vertices (0, 0), (1, 0), (1, 3)
2.6 Topic 6: Evaluate the following surface integrals
1. F = (x2
z + y2
z), S - the part of the plane z = 4 + x + y that lies inside
the cylinder x2
+ y2
= 4
8

10. 2. F(x, y, z) = xz.i − 2y.j + 3x.k, S - the sphere x2
+ y2
+ z2
= 4 with
outward orientation.
3. F(x, y, z) = x2
.i+xy.j +z.k, S - the part of the paraboloid z = x2
+y2
below the plane z = 1 with upward orientation
4. Use the Divergence Theorem to calculate the surface integral of F =
x3
.i + y3
.j + z3
.k, S- the surface off the solid bounded by the cylinder
x2
+ y2
= 1 and the planes z = 0, z = 2
5. Compute the outward flux of F(x, y, z) = x.i+y.j+z.k
(x2+y2+z2)
3
2
through the el-
lipsoid 4x2
+ 9y2
+ 6z2
= 36
6. Use Stokes’ theorem to evaluate
R
C
F.dr, F(x, y, z) = (3x2
yz − 3y)i +
(x3
z − 3x)j + (x3
y + 2z)k, C - the polilines with initial point (0, 0, 2),
through (0, 0, 0), (1, 1, 0) and terminal point (0, 3, 0)
2.7 Topic 7: Plot the graphs of these following curves,
surfaces
1. Plane 2x − 4y + 3z − 4 = 0
2. Ellipsoid x2
4
+ (y−1)2
9
+ z2
= 1
3. Sphere x2
+ y2
+ z2
= 9
4. Paraboloid elliptic z = 4x2
+ 9y2
5. Paraboloid hyperbolic z = 4x2
− 9y2
6. Hyperboloid 1 sheet x2
4
− y2
+ z2
4
= 1
7. Hperboloid 2 sheets x2
4
+ y2
− z2
9
= −1
8. Hyperbolic cylinder x2
4
− z2
9
= 1
9. Parabolic cylinder y − 2z2
+ 4z − 4 = 0
10. Elliptic cylinder y2
+ 4z2
= 1
11. Cone x2
+ 4y2
= z2
9

11. 12. Circle:x2
+ (y − 4)2
= 4; z = 4
13. Line: 2x + 4y − z = 4; x − y − z = 4
14. Intersection of 2 surfaces: x2
+ y2
= 4; z = 2x2
+ y2
10