Exercises in Calculus IV
Spring 2024 Edition
by
Jason Terry
Louisiana Tech University
March 18, 2024
Table of Contents
Preface
v
1 Multivariable Calculus
1.1
Partial Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.1.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.1.2 Answers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2
The Multivariable Chain Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2.2 Answers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.3
Directional Derivatives and Gradients . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.3.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.3.2 Answers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.4
Tangent Planes and Linearization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.4.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.4.2 Answers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.5
Extreme Values and Lagrange Multipliers . . . . . . . . . . . . . . . . . . . . . . . . . .
1.5.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.5.2 Answers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.6
Multi-Integral Substitution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.6.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.6.2 Answers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1
2
2
4
6
6
9
11
11
13
14
14
15
16
16
17
18
18
19
2 Vector Calculus
2.1
Path Integrals Over Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.1.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.1.2 Answers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2
Path Integrals in Vector Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.2 Answers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3
Green’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3.2 Answers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
21
22
22
23
24
24
26
27
27
28
iv
Table of Contents
2.4
Surface Integrals Over Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.4.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.4.2 Answers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Surface Integrals in Vector Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.5.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.5.2 Answers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
The Divergence Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.6.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.6.2 Answers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
29
29
30
31
31
32
33
33
34
3 Infinite Sums
3.1
Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.1.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.1.2 Answers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2
Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2.2 Answers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3
The Comparison Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3.2 Answers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.4
The Ratio and Alternating Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.4.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.4.2 Answers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.5
Power Series and Taylor Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.5.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.5.2 Answers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.6
Application of Taylor Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.6.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.6.2 Answers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
35
36
36
38
39
39
41
42
42
44
45
45
47
48
48
49
50
50
52
2.5
2.6
Preface
Math textbooks can be very expensive and hard to understand. Students might pay upwards of $100-$200
for a standard textbook, and then discover that it is written by people who are too smart, use big confusing
words, and fill the book with tons of impractical exercises that are never used. So I (hopefully not suffering
from these faults) came up with a solution to help ease the financial and mental burden. The goal of
this exercise book is to provide a comprehensive and useful set of homework exercises with the required
breadth and depth for a full Calculus IV course.
I would like this book to allow students to choose whatever resource is best for them. You may still buy a
standard textbook if needed. But you may also seek out less expensive/free alternatives, such as purchasing a Dover book, searching online for free material, or just using the notes provided during class lectures.
This approach may force students to take a more proactive role in learning, which may not be a bad thing.
At the very least, it allows students to avoid buying a potentially expensive and useless textbook.
If you have any questions or comments, feel free to conact me at jterry@latech.edu.
Take THAT you expensive textbooks! I’m coming for you.
Jason Terry
12 March 2024
vi
Preface
Chapter 1
Multivariable Calculus
2
Multivariable Calculus
1.1
Partial Derivatives
1.1.1
Exercises
In Exercises 1 - 4, use the limit definition to compute the first partial derivative of the given function with
respect to each variable.
1. f (x, y ) = x 2 + 3xy + y − 4
2. f (x, y ) =
x
y
3. f (x, y ) =
p
2x + 3y − 1
4. f (x, y , z) = xy 2 z 3
In Exercises 5 - 14, compute the first partial derivative of the given function with respect to each variable.
3 4
2
5. f (x, y ) = y 2 ex y
10. f (x, y) = π 6xy arcsin (ln (y ))
6. f (x, y ) = sin3 3x − y 2
√ 7. f (x, y ) = ln x y cot ( y )
11. f (s, t) = sec sin (s2 t 2 )
12. f (x, y , z) = log4 (9z) cosh (x e + y π )
λ
x 3 + sinh (y 2 )
8. f (x, y ) =
tan (x)
13. f (λ, ϕ, ρ) = csc (ϕ) arctan
p
14. f (x, y, z) = y xz + z tan (xy )
9. f (x, y ) =
x arctan (xy )
2
ρ3
In Exercises 15 - 18, compute the indicated partial derivatives of the given function.
15. f (x, y ) = x 2 + 3xy + y − 1, fx (4, −5), fy (4, −5)
17. f (x, y) = xy 2 sin (x 2 y ); fxx , fyy , fxy , fyx
16. f (x, y ) = x 3 + x 2 y 3 − 2y 2 , fx (2, 1), fy (2, 1)
18. f (x, y , z) = 1 − 2xy 2 z + x 2 y , fyxyz (−3, 2)
In Exercises 19 - 22, use implicit differentiation to compute ∂∂ xz and ∂∂ yz for the given equation.
19. yz − ln (z) = x + y
21. ez = xyz
20. x 3 + y 3 + z 3 + 6xyz + 4 = 0
22. x 2 + y 2 + z 2 = sin (yz)
In Exercises 23 - 28, prove that the given function is a solution to the given partial differential equation.
23. u(x, y ) = ebx −ay , aux + buy = 0
26. u(x, t) = sin (kt) cos (kx), utt = uxx
24. u(x, y ) = sin (e−x y ), ux + yuy = 0
27. u(x, y) = sinh (ky ) cos (kx), uxx + uyy = 0
2
25. u(x, t) = e−k t cos (kx), ut = uxx
28. u(r , θ ) = r n sin (nθ ), urr + 1r ur + r12 uθθ = 0
29. Prove that if f (x, y ) = eax+by and a2 + b2 = 1, then fxx + fyy = f .
30. The gas law for a fixed mass m at absolute temperature T , pressure P, and volume V is PV = mRT ,
where R is the gas constant. Prove that ∂∂VP ∂∂VT ∂∂ TP = −1.
1.1 Partial Derivatives
3
31. For the gas law in the previous problem, prove that T ∂∂ TP ∂∂VT = mR.
2
K ∂ K
= K.
32. The kinetic energy of an object with mass m and velocity v is K = 12 mv 2 . Prove that ∂∂ m
∂2v
33. The temperature at a point (x, y) on a flat metal plate is given by T (x, y ) = 1+x60
2 +y 2 , where x and y
are in meters and T is in °C. Compute the rate at which the temperature of the plate is changing
along the x- and y-direction at the point (2, 1).
34. A computer manufacturer has production P(x, y ) = 0.1xy 2 ln (2x + 3y + 2), where x is the size of
the labor force measured in work-hours per week and y is the amount of capital measured in units
of $1000. When 50 hours of labor and $20,000 of capital are used, compute the rate at which
production changes if capital is held at $20,000.
35. The body mass index (BMI) is defined as B(m, h) = hm2 , where m is mass in kg and h is height in
meters. Suppose a young man is still growing while his weight is unchanged. Compute the rate at
which his BMI is change when he is 64 kg and 1.68 m tall. How much is his BMI expected to change
if he grows an additional cm?
36. When three resistors R1 , R2 , R3 (measured in ohms) are connected in parallel, their equivalent
resistance R is governed by the equation R1 = R11 + R12 + R13 . When the three resistors have values
30, 45, and 90 ohms respectively, compute the rate at which the equivalent resistance is changing
with repsect to the second resistor if the first and third resistor are held at 30 and 90 ohms.
4
Multivariable Calculus
1.1.2
Answers
1. fx = 2x + 3y , fy = 3x + 1
1
3
3. fx = √2x+3y
, f = 2√2x+3y
−1 y
−1
2. fx = y1 , fy = − yx2
4. fx = y 2 z 3 , fy = 2xyz 3 , fz = 3xy 2 z 2
3 4
3 4
3 4
5. fx = 3x 2 y 6 ex y , fy = 2yex y + 4x 3 y 5 ex y
6. fx = 9 sin2 3x − y 2 cos 3x − y 2 , fy = −6y sin2 3x − y 2 cos 3x − y 2
7. fx = yx , fy =
8. fx =
√
√
ln (x) cot ( y)− 21 csc2 ( y )y −1/2
√
cot ( y)
3x 2 tan (x)−[(]x 3 +sinh (y 2 )] sec2 (x)
cosh (y 2 )
, fy = 2y tan
(x)
tan2 (x)
h
i
2
9. fx = 21 [x arctan (xy )]−1/2 arctan (xy ) + 1+xxy2 y 2 , fy = 12 [x arctan (xy )]−1/2 1+xx 2 y 2
2
2
10. fx = 6y 2 ln (π )π 6xy arcsin (ln (y )), fy = 12xy ln (π )π 6xy arcsin (ln (y )) + √π
y
6xy 2
1−ln2 (y)
11. fs = 2st 2 sec sin (s2 t 2 ) tan sin (s2 t 2 ) cos (s2 t 2 ), ft = 2s2 t sec sin (s2 t 2 ) tan sin (s2 t 2 ) cos (s2 t 2 )
e
π
12. fx = ex e−1 log4 (9z) sinh (x e + y π ), fy = π y π−1 log4 (9z) sinh (x e + y π ), fz = coshln(x(4)z+y )
λ
(ϕ) ln (2)2
13. fλ = csc
, fϕ = − csc (ϕ) cot (ϕ) arctan
ρ3 +22λ ρ−3
λ
2
ρ3
λ
(ϕ)2
, fρ = − ρ34csc
+22λ ρ−2
14. fx = z ln (y)y xz + y ln (z)z tan (xy ) sec2 (xy ), fy = xzy xz −1 + x ln (z)z tan (xy) sec2 (xy ), fz = x ln (y )y xz +
tan (xy )z tan (xy )−1
15. fx (4, −5) = −7, fy (4, −5) = 13
16. fx (2, 1) = 16, fy (2, 1) = 8
17. fxx = 6xy 3 cos (x 2 y) − 4x 3 y 4 sin (x 2 y ), fyy = (2x − x 5 y 2 ) sin (x 2 y ) + 4x 3 y cos (x 2 y ), fxy = fyx =
(2y − 2x 4 y 3 ) sin (x 2 y) + 7x 2 y 2 cos (x 2 y )
18. fyxyz (−3, 2) = −4
2
19. ∂∂ xz = yzz−1 , ∂∂ yz = zyz−−z1
2
∂z
xz
21. ∂∂ xz = ez yz
−xy , ∂ y = ez −xy
2
x +2yz ∂ z
y +2xz
20. ∂∂ xz = − 2xy+z
2 , ∂ y = − 2xy+z 2
z cos (yz)−2y
2x
∂z
22. ∂∂ xz = y cos (yz)
−2z , ∂ y = 2z −y cos (yz)
1.1 Partial Derivatives
5
23. Good luck.
26. Good luck.
24. Good luck.
27. Good luck.
25. Good luck.
28. Good luck.
29. Good luck.
30. Good luck.
31. Good luck.
32. Good luck.
33. Tx (2, 1) = − 20
°C/m and Ty (2, 1) = − 10
°C/m
3
3
34. Px (50, 20) ≈ 228 computers/hour
35. Bh (64, 1.68) ≈ −27 (kg/m2 )/m. His BMI will decrease by about 0.27.
36. ∂∂RR2 |(30,45,90) = 19
6
Multivariable Calculus
1.2
The Multivariable Chain Rule
1.2.1
Exercises
In Exercises 1 - 4, use the multivariable chain rule to compute the indicated partial derivatives of the given
function.
1. z =
p
1 + x 2 + y 2 , x = ln (t), y = cos (t); dz
dt
2. u = x 4 y + y 2 z 3 , x = rset , y = rs2 e−t , z = r 2 s sin (t); ∂∂us |(r ,s,t)=(2,1,0)
3. z = ex sin (y ), x = st 2 , y = s2 t; ∂∂ zs and ∂∂zt
4. u = er cos (θ ), r = st, θ =
√
s2 + t 2 ; ∂∂us and ∂∂ut
In Exercises 5 - 8, draw a tree diagram and write the chain rule for all derivatives of the given function.
5. w = w(x, y), x = x(s), y = y (s, t)
6. w = w(x, y , z), x = x(r , s, t), y = y (r , s), z = z(s, t)
7. w = w(x, y , z, t), x = x(t), y = y (t), z = z(t)
8. w = w(x), x = x(u, t), u = u(t)
9. Suppose that W (s, t) = F (u(s, t), v (s, t)) where u(1, 0) = 4, us (1, 0) = −2, ut (1, 0) = −7, v (1, 0) = 2,
vs (1, 0) = −2, vt (1, 0) = −2, Fu (4, 2) = 6, Fv (4, 2) = −3. Compute Ws (1, 0) and Wt (1, 0).
10. Suppose that z = f (x, y ), x = x(u, v ), y = y (u, v ) where x(1, 2) = 2, y(1, 2) = 1, fx (1, 2) = a,
fy (1, 2) = b, xu (1, 2) = p, yu (1, 2) = s, fx (2, 1) = c, fy (2, 1) = 3, xv (1, 2) = −4, yv (1, 2) = r . Compute
zv (1, 2).
11. Prove that if g(s, t) = f (s2 − t 2 , t 2 − s2 ) given f (x, y ), then:
t
∂g
∂g
+s
=0
∂s
∂t
12. Prove that if z = f (x + at) + g(x − at) given f (x) and g(x), then:
2
∂2z
2∂ z
=
a
∂t 2
∂x 2
13. Prove that if z = y1 [f (x + y ) + g(x − y )] given f (t) and g(t), then:
∂2z
1 ∂
2 ∂z
=
y
∂x 2 y 2 ∂y
∂y
1.2 The Multivariable Chain Rule
7
14. Prove that if y = y (x) and F (x, y ) = 0, then:
dy
Fx
=−
dx
Fy
15. Prove that if z = z(x, y ) and F (x, y , z) = 0, then:
Fy
∂z
∂z
Fx
and
=−
=−
∂x
Fz
∂y
Fz
16. Prove that if z = z(x, y ), x = r cos (θ ), and y = r sin (θ ), then:
∂z
∂x
2
+
∂z
∂y
2
=
∂z
∂r
2
1
+ 2
r
∂z
∂θ
2
17. Prove that if u = f (es cos (t), es sin (t)) given f (x, y ), then:
2
∂2u ∂2u
∂2u
2s ∂ u
+
=e
+
∂ s2 ∂ t 2
∂x 2 ∂y 2
18. Prove that if I(x, u, v ) =
v
u
h(x, t)dt, u = u(x), and v = v (x), then:
v
dI
dv
du
=
hx (x, t)dt + h(x, v )
− h(x, u)
dx
dx
dx
u
In Exercises 19 - 24, use Exercises 14 and 15 to compute the indicated derivatives for the given equation.
19. x 3 + y 3 = 6xy ; dy
dx
22. x 2 + 2y 2 + 3z 2 = 1; ∂∂ xz and ∂∂ yz
20. arctan (x 2 y ) = x + xy 2 ; dy
dx
23. ez = xyz; ∂∂ xz and ∂∂ yz
21. yz − ln (z) = x + y ; ∂∂ xz and ∂∂ yz
24. x 2 + y 2 + z 2 = sin (yz); ∂∂ xz and ∂∂ yz
In Exercises 25 - 30, use Exercise 18 to compute the given quantity.
∞ −t
2
e − e−tx
d
cos (tx)
28.
dt, x > 0
dt
25.
t
dx 1
t
0
x
∞ −xt
d
sin (tx)
e
sin (t)
26.
dt
29.
dt, x > 0
dx √x
t
t
0
x
1 p
d
x −1
27.
(x − t) arctan (t)dt
30.
dx, p > −1
dx 0
ln (x)
0
31. The length, width, and height of a rectangular box are changing with time. At the moment its dimensions are 1 m, 2 m, and 3 m, they are changing at a rate of 1 m/s, 1 m/s, and -3 m/s, respectively.
Compute the rate at which the volume of the box is changing at this moment.
8
Multivariable Calculus
32. The radius and height of a circular cone are changing with time. At the moment its radius is 2 cm
and its height is 3 cm, its radius is changing at a rate of 5 cm/s and its volume is change at a rate of
28π cm3 /s. Compute the rate at which its height is changing.
33. The pressure, volume, and temperature of a mole of an ideal gas satisfies the equation PV = 8.31T ,
where P is in kilopascals, V is in liters, and T is in kelvins. Compute the rate at which the pressure is
changing when the temperature is 300 K and increasing at a rate of 0.1 K/s, and the volume is 100
L and increasing at a rate of 0.2 L/s.
34. The voltage in a circuit satisfies the equation V = IR, where I is current in amps, R is resistance
in ohms, and V is voltage in volts. Compute the rate at which the current is changing when the
resistance is 600 ohms and changing at a rate of 0.5 ohms/sec, the current is 0.04 amps, and the
voltage is changing at a rate of -0.01 volts/sec.
35. The function T (x, y ) models the temperature at a√point (x, y), where x and y are in cm. The position
of an object after t seconds is given by x =
t + 1 and y = 31 t + 2. If Tx (2, 3) = 4◦ /cm and
◦
Ty (2, 3) = 3 /cm, then compute how fast the object’s temperature is changing after 3 s.
36. The function T (x, y ) models the temperature at a point (x, y ) on the circle x = cos (t) and y = sin (t)
for 0 ≤ t ≤ 2π . If Tx = 8x − 4y and Ty = 8y − 4x, then compute all the coordinates where the
maximum and all the coordinates where the minimum temperatures occur.
1.2 The Multivariable Chain Rule
1.2.2
9
Answers
−1
sin (t) cos (t)
1. dz
= t √ln (t)−
dt
2
2
1+ln (t)+cos (t)
2. ∂∂us |(r ,s,t)=(2,1,0) = 192
2
2
2
2
3. ∂∂ zs = t 2 est sin (s2 t) + 2stest cos (s2 t), ∂∂zt = 2stest sin (s2 t) + s2 est cos (s2 t)
h
√
h
√
4. ∂∂us = est t cos
∂u
st
∂t = e
s cos
s2 + t 2 − √ s2
s2 + t 2
sin
2
√
s2 + t 2
i
,
s +t
√
i
s2 + t 2
− √ 2t 2 sin
s +t
+ ∂∂wy ∂∂ys , ∂∂wt = ∂∂wy ∂∂yt
5. ∂∂ws = ∂∂wx dx
ds
6. ∂∂wr = ∂∂wx ∂∂xr + ∂∂wy ∂∂yr , ∂∂ws = ∂∂wx ∂∂xs + ∂∂wy ∂∂ys + ∂∂wz ∂∂ zs , ∂∂wt = ∂∂wx ∂∂xt + ∂∂wz ∂∂zt
7. dw
= ∂∂wx dx
+ ∂∂wy dy
+ ∂∂wz dz
+ ∂∂wt
dt
dt
dt
dt
∂ x du
∂x
= dw
+ dw
8. dw
dt
dx ∂ u dt
dx ∂ t
9. Ws (1, 0) = −6, Wt (1, 0) = −36
10. zv (1, 2) = 3r − 4c
11. Good luck.
12. Good luck.
13. Good luck.
14. Good luck.
15. Good luck.
16. Good luck.
17. Good luck.
18. Good luck.
2
2
−x
19. dy
= y2y2 −
dx
2x
4 2
z −z
21. ∂∂ xz = yzz−1 , ∂∂ yz = yz
−1
2
4 4
+x y −2xy
20. dy
= 1+xx 2y−+y
dx
2xy −2x 5 y 3
x ∂z
22. ∂∂ xz = − 3z
, ∂ y = − 2y
3z
10
Multivariable Calculus
∂z
xz
23. ∂∂ xz = ez yz
−xy , ∂ y = ez −xy
z cos (yz)−2y
2x
∂z
24. ∂∂ xz = y cos (yz)
−2z , ∂ y = 2z −y cos (yz)
25. cos (2x)x−cos (x)
28. ln (x)
2
(x
26. 2 sinx(x ) − 3 sin2x
3/2
)
29. arccot(x)
27. x arctan (x) − 12 ln (1 + x 2 )
30. ln (p + 1)
31. 3 m3 /s
32. 6 cm/s
33. −0.04155 kPa/s
34. −5 × 10−5 amps/s
35. 2◦ C/s
√
√ 36. Maximums at − 22 , 22
and
√
√ 2
, − 22
2
. Minimums at
√
√ 2
, 22
2
√
√ and − 22 , − 22 .
1.3 Directional Derivatives and Gradients
11
1.3
Directional Derivatives and Gradients
1.3.1
Exercises
In Exercises 1 - 2, use the limit definition to compute the directional derivative for the given function,
direction, and point.
1. f (x, y ) = x 2 + xy , û = ⟨ √1 , √1 ⟩, P(1, 2)
2
2. f (x, y ) = 2x 2 + y 2 , ⃗v = ⟨3, −4⟩, P(−1, 1)
2
3. Prove that the directional derivative of f (x, y) in the direction of û at the point (a, b) is given by the
⃗ f (a, b) · û. [Hint: Consider g(h) = f (a + hu1 , b + hu2 ). Then compute g ′ (0) using
formula Dû f (a, b) = ∇
the limit definition from Calculus I and the multivariable chain rule from Calculus IV.]
4. Prove that the maximum value of the directional derivative of f (x, y) at point (a, b) occurs when û is
⃗ f (a, b) and that the value is |∇
⃗ f (a, b)|. Prove the corresponding results for
in the same direction as ∇
the minimum value and 0 value of the directional derivative.
In Exercises 5 - 10, compute the indicated directional derivative for the given function, direction, and point.
5. f (x, y ) = x 2 + xy , û = ⟨ √1 , √1 ⟩, P(1, 2)
2
2
6. f (x, y ) = 2x 2 + y 2 , ⃗v = ⟨3, −4⟩, P(−1, 1)
7. f (x, y ) = x 2 sin (2y ), ⃗v = ⟨3, −4⟩, P(1, π2 )
8. f (x, y ) = x 3 − 3xy + 4y 2 , û is at angle π6 , P(1, 2)
9. f (x, y , z) = x 2 + 2y 2 − 3z 2 , ⃗v = ⟨4, 2, −5⟩, P(1, −1, 3)
10. f (x, y , z) = x sin (yz), ⃗v = ⟨1, 2, −1⟩, P(1, 3, 0)
In Exercises 11 - 16, compute the direction and maximum rate of change for the given function and point.
11. f (x, y ) = xey , P(2, 0)
14. f (x, y ) = sin (xy ), P(1, 0)
12. f (x, y ) = x 2 y + exy sin (y ), P(1, 0)
15. f (x, y , z) = x ln (yz), P(1, 2, 12 )
13. f (x, y ) = 5xy 2 , P(3, −2)
x
, P(8, 1, 3)
16. f (x, y , z) = y+z
17. Complete and prove the following gradient rules for all functions f (x, y ) and g(x, y ).
⃗ (f + g) =
(a) ∇
⃗ (kf ) =
(b) ∇
⃗ (fg) =
(c) ∇
⃗ (f ) =
(d) ∇
g
18. Compute the rate of change of f (x, y) =
√
xy at point P(2, 8) in the direction from P(2, 8) to Q(5, 4).
12
Multivariable Calculus
19. Compute the rate of change of f (x, y ) = xey at point P(2, 0) in the direction from P(2, 0) to Q( 12 , 2).
20. Compute the rate of change of f (x, y , z) = xy − xy 2 z 2 at point P(2, −1, 1) in the direction from
P(2, −1, 1) to Q(5, 1, 7).
80
21. The temperature at a point in space is T (x, y , z) = 1+x 2 +2y
2 +3z 2 , where x, y , and z are in meters and
T is in °C. Compute the direction where the temperature increases the fastest at point (1, 1, −2) and
compute this rate of change.
22. Consider the temperature function T (x, y , z) = 2xy − yz, where x, y , and z are in meters and T
is in °C. Show whether or not there is a direction where the rate of change of the temperature at
P(1, −1, 1) is -3°C/m.
1.3 Directional Derivatives and Gradients
1.3.2
13
Answers
1. √5
2. −4
3. Good luck.
4. Good luck.
5. √5
− 323
8. 13
2
6. −4
9. 6 5
7. 85
10. − 26
2
√
2
√
√
11. ⟨1, 2⟩;
√
14. ⟨0, 1⟩; 1
5
√
12. ⟨0, 2⟩; 2
15. ⟨0, 1, 4⟩;
√
17
2
16. ⟨1, −2, −2⟩; 34
13. ⟨1, −3⟩; 20 10
17. The proofs are left to you.
⃗ (f + g) = ∇
⃗f +∇
⃗g
(a) ∇
⃗ (kf ) = k ∇
⃗f
(b) ∇
⃗ (fg) = f ∇
⃗ g + g∇
⃗f
(c) ∇
⃗
⃗
⃗ ( f ) = g ∇f −2f ∇g
(d) ∇
g
g
18. 52
19. 1
20. − 18
7
√
21. ⟨−1, −2, 6⟩, 5 841 °C/m
√
√
⃗ f (1, −1, 1)| = 6, the minimum rate of change is − 6 ≈ −2.45. So there is no direction
22. Since |∇
where the rate of change is -3°C/m.
14
Multivariable Calculus
1.4
Tangent Planes and Linearization
1.4.1
Exercises
In Exercises 1 - 4, use a gradient to compute the equation of the tangent line for the given level curve at
point P. Then sketch the level curve, the tangent line at P, and the gradient vector at P in the xy-plane.
1. f (x, y ) = xy , P(3, 2)
3. x 2 − xy + y 2 = 7, P(−1, 2)
2. f (x, y ) = y − sin (x), P(π , 0)
4. x 2 − y = 1, P( 2, 1)
√
In Exercises 5 - 10, use a gradient to compute the equation of the tangent plane for the given surface at
point P.
2
5. x4 + y 2 + z9 = 3, P(−2, 1, −3)
2
8. f (x, y ) = 2x 2 + y 2 , P(1, 1)
6. x 2 + y 2 − z 2 = 4, P(2, 1, 1)
9. x + y + z = exyz , P(0, 0, 1)
7. f (x, y ) = 2 − x 2 − y 2 , P( 12 , 12 )
10. z + 1 = xey cos (z), P(1, 0, 0)
11. Prove that if ⃗r (t) = ⟨g(t), h(t), k(t)⟩ is a curve on the level surface f (x, y , z) = c, then the gradient of f
is normal to the curve.
12. Compute the linearization of f (x, y ) = 2x 2 + y 2 at point (1, 1). Then use it to approximate the value
of f (1.01, 1.01).
13. Compute the linearization of f (x, y ) =
of f (3.04, 3.98).
p
14. Compute the linearization of f (x, y , z) =
the value of f (3.02, 1.97, 5.99).
x 2 + y 2 at point (3, 4). Then use it to approximate the value
p
x 2 + y 2 + z 2 at point (3, 2, 6). Then use it to approximate
15. Compute the equation of the tangent line to the curve of intersection of the paraboloid f (x, y ) = x 2 +y 2
and the ellipsoid 4x 2 + y 2 + z 2 = 9 at the point (−1, 1, 2).
16. Compute the points on the ellipsoid 2x 2 + 2y 2 + 3z 2 = 1 where its tangent plane is parallel to the
plane −4x − 4y + 3z = 3.
1.4 Tangent Planes and Linearization
1.4.2
15
Answers
1. 2x + 3y = 12
3. 4x − 5y = −14
2. y = −x + π
4. y = 2 2x − 3
5. 3x − 6y + 2z = −18
8. 4x + 2y − z = 3
6. 2x + y − z = 4
9. x + y + z = 1
7. x + y + z = 25
10. x + y − z = 1
√
11. Good luck.
12. L(x, y) = 4x + 2y − 3; L(1.01, 1.01) = 3.06
13. L(x, y) = 5 + 53 (x − 3) + 45 (y − 4); L(3.04, 3.98) = 5.008
14. L(x, y, z) = 7 + 37 (x − 3) + 27 (y − 2) + 67 (z − 6); L(3.02, 1.97, 5.99) = 4894
≈ 6.9914
700
15. ⃗r (t) = ⟨−1 − 10t, 1 − 16t, 2 − 12t ⟩
16.
√2
19
, √2 , − √1
19
19
, − √2 , − √2 , √1
19
19
19
16
Multivariable Calculus
1.5
Extreme Values and Lagrange Multipliers
1.5.1
Exercises
In Exercises 1 - 8, compute all the critical points for the given function. Then use the second derivative test
to classify each point as a local maximum, local minimum, a saddle point, or state if the test fails.
1. f (x, y ) = x 2 + y 2 − 4y + 9
5. f (x, y ) = 2 ln (x) + ln (y ) − 4x − y
2. f (x, y ) = xy − x 2 − y 2 − 2x − 2y + 4
6. f (x, y) = y sin (x)
3. f (x, y ) = x 4 + y 4 − 4xy + 1
7. f (x, y) = x 2 + 4y 2 − 4xy + 2
4. f (x, y ) = 3y 2 − 2y 3 − 3x 2 + 6xy
8. f (x, y) = (x 2 + y 2 )ey −x
2
2
In Exercises 9 - 12, compute the coordinates for all absolute maxima and absolute minima for the given
function on the given closed and bounded set.
9. f (x, y ) = x 2 − 2xy + 2y on {(x, y ) : 0 ≤ x ≤ 3, 0 ≤ y ≤ 2}.
10. f (x, y ) = x 2 + y 2 + x 2 y + 4 on {(x, y ) : |x | ≤ 1, |y | ≤ 1}.
11. f (x, y ) = 3 + xy − x − 2y on the triangular region with vertices (1, 0), (5, 0), (1, 4).
12. f (x, y) = 2 + 2x + 4y − x 2 − y 2 on the region enclosed by the curves x = 0, y = 0, y = 9 − x.
In Exercises 13 - 16, use Lagrange multipliers to compute the coordinates of the indicated extreme value
of the given function subject to the given constraint.
2
2
y
13. Maximum of f (x, y ) = 4xy with x > 0 and y > 0, subject to x9 + 16
= 1.
2
2
14. Maximum and minimum of f (x, y ) = xy on the ellipse x8 + y2 = 1.
15. Maximum and minimum of f (x, y ) = 3x + 4y on the circle x 2 + y 2 = 1.
16. Minimum of f (x, y , z) = 2x 2 + y 2 + 3z 2 subject to 2x − 3y − 4z = 49.
17. A certain post office will only accept rectangular boxes whose length plus perimeter of its cross
section is no more than 108 inches. Compute the dimensions of the box with the largest volume that
the post office will accept.
18. A rectangular box has one side lying on the xy-plane with one vertex at the origin and the opposite
vertex lying on the plane 6x + 4y + 3z = 24. Compute the maximum possible volume of the box.
19. Compute the point on the plane 2x + y − z = 5 that is closest to the origin and then compute this
distance.
20. Compute the coordinates of the absolute maximum and absolute minimum for the function f (x, y) =
x 2 + 2y 2 − 2x + 3 on the set {(x, y) : x 2 + y 2 ≤ 10}.
1.5 Extreme Values and Lagrange Multipliers
1.5.2
17
Answers
1. (0, 2) is a local minimum.
2. (−2, −2) is a local maximum.
3. (0, 0) is a saddle point. (1, 1) and (−1, −1) are local minimums.
4. (0, 0) is a saddle point. (2, 2) is a local maximum.
5. ( 21 , 1) is a local maximum.
6. (k π , 0) are saddle points for every integer k .
7. The test fails for points (2a, a) for every real number a.
8. (±1, 0) are saddle points. (0, 0) is a local minimum.
9. The candidate points are (1, 1), (0, 0), (3, 0), (3, 2), (2, 2), (0, 2). Absolute maximum at (3, 0, 9).
Absolute minimums at (0, 0, 0) and (2, 2, 0).
10. The candidate points are (0, 0), (±1, ±1), (±1, − 21 ), (0, 1). Absolute maximums at (±1, 1, 7). Absolute minimum at (0, 0, 4).
11. The candidate points are (2, 1), (1, 0), (1, 4), (5, 0), (3, 2). Absolute maximums at (1, 0, 2) and (3, 2, 2).
Absolute minimums at (1, 4, −2) and (5, 0, −2).
12. The candidate points are (1, 2), (0, 0), (9, 0), (1, 0), (0, 9), (0, 2), (4, 5). Absolute maxmimum at
(1, 2, 7). Absolute minimum at (9, 0, −61).
√
13. ( √3 , 2 2, 24)
2
14. Maximums at (2, 1, 2) and (−2, −1, 2). Minimums at (2, −1, −2) and (−2, 1, −2).
15. Maximum at ( 53 , 45 , 5). Minimum at (− 35 , − 54 , −5).
16. (3, −9, −4, 147)
17. 36 in. x 18 in. x 18 in.
18. 64
9
19. ( 53 , 56 , − 56 ), √5
6
20. Absolute maximum at (−1, ±3, 24) and absolute minimum at (1, 0, 2).
18
Multivariable Calculus
1.6
Multi-Integral Substitution
1.6.1
Exercises
In Exercises 1 - 4, use the given transformation to compute given multi-integral. Sketch the region of
integration in both the xy-plane and uv-plane.
4 (y /2)+1 2x − y
1.
dxdy using the transformation u = 2x2−y and v = y2 .
2
0
y /2
r
y √
2.
+ xy dxdy, where R is the region in the first quadrant bounded by the curvs xy = 1,
x
R
xy = 9, y = x, and y = 4x, using the transformation x = vu and y = uv .
2/3 2−2y
3.
0
(x + 2y )ey −x dxdy using the transformation u = x + 2y and v = x − y.
y
ydA, where R is the region above the x-axis bounded by the parabolas y 2 = 4 − 4x and y 2 =
4.
R
4 + 4x, using the transformation x = u 2 − v 2 and y = 2uv for u ≥ 0 and v ≥ 0.
In Exercises 5 - 9, determine a transformation to compute the given multi-integral. Sketch the region of
integration in both the xy-plane and uv-plane.
1 1−x
√
5.
x + y (y − 2x)2 dydx
0
0
2 y r
6.
1
1/y
y √xy
e
dxdy
x
(2x 2 − xy − y 2 )dxdy, where R is the region bounded by the lines y = −2x + 4, y = −2x + 7,
7.
R
y = x − 2, y = x + 1.
8.
(3x 2 + 14xy + 8y 2 )dxdy, where R is the region bounded by the lines y = − 32 x + 1, y = − 23 x + 3,
R
y = − 14 x, y = − 14 x + 1.
y −x
9.
cos
dxdy, where R is the trapezoidal region with vertices (1, 0), (2, 0), (0, 2), (0, 1).
y +x
R
10. Compute R (x − 3y )dA, where R is the triangular region with vertices (0, 0), (2, 1), (1, 2) using the
transformation x = 2u + v and y = u + 2v .
11. Compute R e(x+y)/(x −y) dA, where R is the trapezoidal region with vertices (1, 0), (2, 0), (0, −2),
(0, −1).
12. Compute R cos (4x 2 + 9y 2 )dA, where R is the ellipital region 4x 2 + 9y 2 ≤ 1.
1.6 Multi-Integral Substitution
1.6.2
1.
2.
3.
4.
19
Answers
21
0
0
32
1
1
2u
0
0
11
0
0
(u)(2)dudv = 2
(v + u)
2u
v
dvdu = 8 + 52
ln (2)
3
ue−v | − 13 |dvdu = e−2 + 13
(2uv )4(u 2 + v 2 )dudv = 2
1u
5. u = x + y , v = −2x + y ;
6. u =
√
xy , v =
q
y
;
x
2 2/u
1
72
10.
1 1− u
0
0
−1 uv
4
8. u = 3x + 2y, v = x + 4y ;
9. u = x + y , v = −x + y ;
uv 2
2u
v
veu
1
7. u = x − y , v = 2x + y ;
√
−2u
0
64
1
dvdu = 29
dvdu = 2e2 − 4e
1
3
dudv = 33
4
1
10
−u cos
2u
uv
0
2
1
3
dvdu = 64
5
v
u
1
2
1
2
dudv = 34 (e − e−1 )
dvdu = 23 sin (1)
(−u − 5v )(3)dvdu = −3
11. u = x + y , v = x − y ;
2v
1
u /v
−v e
12. x = 12 u cos (v ), y = 13 u sin (v );
2π 1
0
0
cos (u 2 )
u
6
dudv = π6 sin (1)
20
Multivariable Calculus
Chapter 2
Vector Calculus
22
Vector Calculus
2.1
Path Integrals Over Curves
2.1.1
Exercises
In Exercises 1 - 14, compute the given path integral over the given curve.
1. C x ds and C is the segment from (0, 0) to (2, 4).
2. C x ds and C is the parabola y = x 2 from (0, 0) to (2, 4).
√
3. C x + 2y ds and C is the segment from (0, 0) to (1, 4).
4.
5.
6.
2
C
yex ds and C is the curve ⃗r (t) = ⟨4t, −3t ⟩, −1 ≤ t ≤ 2.
C
(x − y + 3)ds and C is the unit circle traversed counterclockwise from (1, 0).
C
(2 + x 2 y ) ds and C is the upper half of the unit circle from (1, 0) to (−1, 0).
2x ds and C consists of the parabola y = x 2 from (0, 0) to (1, 1), followed by the vertical line
segment from (1, 1) to (1, 2).
8. C x 2 +y1 2 +1 ds and C is the segment from (0, 0) to (1, 0).
7.
9.
10.
11.
12.
13.
14.
C
C
(x + y ) ds and C is the segment from (0, 1, 0) to (1, 0, 0).
C
(x − y + z − 2) ds and C is the segment from (0, 1, 1) to (1, 0, 1).
C
y sin (z) ds and C is the circular helix given by x = cos (t), y = sin (t), z = t, 0 ≤ t ≤ 2π .
p
x 2 + y 2 ds and C is the curve ⃗r (t) = ⟨4 cos t, 4 sin (t), 3t ⟩, −2π ≤ t ≤ 2π .
C
√
3
ds and C is the infinite curve ⃗r (t) = ⟨t, t, t ⟩, t ≥ 1.
C x 2 +y 2 +z 2
√
(x + y − z 2 ) ds and C consists of the parabola y = x 2 from (0, 0, 0) to (1, 1, 0), followed by the
vertical line segment from (1, 1, 0) to (1, 1, 1).
C
15. Compute the area of the wall under the surface f (x, y ) = x +
√
y over the curve y = x 2 , 0 ≤ x ≤ 2.
16. Compute the area of the wall under the surface f (x, y) = 4 + 3x + 2y over the curve 2x + 3y = 6,
0 ≤ x ≤ 6.
17. Compute
√ √ the mass of a wire with constant density δ (x, y , z) = 1 that lies along the curve ⃗r (t) =
⟨ 2t, 2t, 4 − t 2 ⟩, 0 ≤ t ≤ 1.
√
18. Compute the mass of a wire with varying density δ (x, y , z) = 15 y + 2 that lies along the curve
⃗r (t) = ⟨0, t 2 − 1, 2t ⟩, −1 ≤ t ≤ 1.
2.1 Path Integrals Over Curves
2.1.2
Answers
√
1. 2 5
√
17−1
2. 17 12
√
3. 2 17
15 16
4. 32
(e − e64 )
5. 6π
6. 2π + 23
7. 5
√
5+11
6
8. π4
9.
√
2
√
10. − 2
√
11. π 2
12. 80π
13. 1
√
14. 5 65+9
√
15. 17 617−1
√
16. 26 13
17.
√
2 + ln (1 +
18. 80
√
2)
23
24
Vector Calculus
2.2
Path Integrals in Vector Fields
2.2.1
Exercises
In Exercises 1 - 8, use a parameterization to compute the given path integral in the given vector field.
⃗ · d⃗r , where F
⃗ = ⟨y 2 , x ⟩ and C is the sement from (−5, −3) to (0, 2).
1. C F
⃗ · d⃗r , where F
⃗ = ⟨x 2 , −y ⟩ and C is the parabola x = y 2 from (4, 2) to (1, −1).
2. C F
3. C xydx + 3y 2 dy , where C is the curve ⃗r (t) = ⟨11t 4 , t 3 ⟩, 0 ≤ t ≤ 1.
⃗ · d⃗r , where F
⃗ = ⟨y , −x ⟩ and C is the quarter circle from (1, 0) to (0, 1).
4. C F
5. C (x − y )dx + (x + y )dy , where C is the triangle traversed counterclockwise with vertices (0, 0),
(1, 0), (0, 1).
⃗ · d⃗r , where F
⃗ = ⟨xy , yz, xz ⟩ and C is the curve ⃗r (t) = ⟨t, t 2 , t 3 ⟩, 0 ≤ t ≤ 1.
6. C F
⃗ · d⃗r , where F
⃗ = 21 ĵ and C is the segment from (0, 0, 0) to (1, 1, 1).
7. C F
x +1
8. C ydx + zdy + xdz, where C consists of the segment from (2, 0, 0) to (3, 4, 5), followed by the segment from (3, 4, 5) to (3, 4, 0).
In Exercises 9 - 14, use the Fundamental Theorem of Path Integrals to compute the given path integral in
the given vector field.
⃗ · d⃗r , where F
⃗ = ⟨e−y , −xe−y ⟩ and C is the sement from (0, 1) to (2, 0).
9. C F
⃗ · d⃗r , where F
⃗ = ⟨2y 3/2 , 3x √y ⟩ and C is the curve ⃗r (t) = ⟨t + 1, 3t + 1⟩, 0 ≤ t ≤ 1.
10. C F
11.
12.
13.
14.
(2,1)
(1,0)
C
2xe−y dx + (2y − x 2 e−y )dy
⃗ · d⃗r , where F
⃗ = ⟨2xz + y 2 , 2xy , x 2 + 3z 2 ⟩ and C is the curve ⃗r (t) = ⟨t 2 , t + 1, 2t − 1⟩, 0 ≤ t ≤ 1.
F
⃗ · d⃗r , where F
⃗ = ⟨2 cos (y),
F
C
(0,1,1)
(1,0,0)
1
− 2x sin (y )
y
, z1 ⟩ and C is any path from (0, 2, 1) to (1, π2 , 2).
cos (x) sin (y )dx + sin (x) cos (y )dy + dz
15. Prove that the path integral
⃗ · ⃗r ′ (t)dt.
F
C
C
⃗ ·T
⃗ ds for a curve C parameterized by ⃗r (t) can be computed by
F
⃗
⃗
16. Prove that if f (x, y , z) is a scalar function
where ∇f (x, y, z) = F (x, y , z) and C is a curve that begins
⃗
⃗
at point A and ends at point B, then C F · d r = f (B) − f (A) (assuming the favorable conditions of
this chapter).
⃗f =F
⃗ , then
17. Prove that if f is a scalar function where ∇
the favorable conditions of this chapter).
C
⃗ · d⃗r = 0 for every closed path C (assuming
F
2.2 Path Integrals in Vector Fields
25
⃗f = F
⃗ and F
⃗ (x, y ) = ⟨P(x, y ), Q(x, y )⟩, then Py = Qx
18. Prove that if f is a scalar function where ∇
(assuming the favorable conditions of this chapter).
⃗ = ⟨x 2 , −xy ⟩ in moving a particle along the quarter circle
19. Compute the work done by the force field F
from (1, 0) to (0, 1).
⃗ = ⟨eyz , xzeyz + z cos (y ), xyeyz + sin (y )⟩ in moving a
20. Compute the work done by the force field F
particle along the line segment from (1, 0, 1) to (1, π2 , 0).
⃗ = ⟨− 2 y 2 , 2 x 2 ⟩.
21. Consider the vector field F
x +y
x +y
(a) Show whether or not ∂∂Py = ∂∂Qx .
(b) Compute
C
⃗ · d⃗r , where C is the unit circle.
F
⃗ a conservative field? Why or why not?
(c) Is F
⃗ (xy 2 z 3 ) along the segment from (1, 1, 1) to (2, 1, −1).
⃗ =∇
22. Compute the flow of the vector field F
⃗ (x, y ) = ⟨x, y ⟩ around the unit circle.
23. Compute the counterclockwise circulation of the vector field F
⃗ (x, y ) = ⟨−y, x ⟩ around the unit circle.
24. Compute the counterclockwise circulation of the vector field F
⃗ (x, y , z) = ⟨x, y , z ⟩ around the ellipse
25. Compute the counterclockwise circulation of the vector field F
2
where the plane 2x + 3y − z = 0 intersects the cylinder x + y 2 = 12.
⃗ · N̂ ds for a vector field F
⃗ (x, y ) = ⟨P(x, y ), Q(x, y )⟩ across a curve C
F
26. Prove that the flux integral
C
can be computed by C −Q dx + P dy .
In Exercises 27 - 30, compute the outward flux of the given vector field across the given curve.
⃗ (x, y ) = ⟨x, y ⟩ and C is the unit circle.
27. F
⃗ (x, y ) = ⟨−y , x ⟩ and C is the unit circle.
28. F
2
⃗ (x, y ) = ⟨x, y ⟩ and C is the ellipse x 2 + y = 1.
29. F
4
⃗ (x, y ) = ⟨2x, −3y ⟩ and C is the circle of radius 3 centered at the origin.
30. F
26
Vector Calculus
2.2.2
Answers
1. − 56
5. 1
2. − 39
2
6. 27
28
3. 45
7. π4
4. − π2
8. 19
2
9. 2
12. 7
10. 30
13. ln ( π2 )
11. e4
14. 1
⃗′
and ds = |⃗v (t)|dt.
15. Hint: Use the facts T̂ = |⃗rr ′ (t)
(t)|
16. Hint: Use the Multivariable Chain Rule.
17. Hint: Use the result of Exercise 16.
18. Hint: Use the fact fxy = fyx .
19. − 23
20. 0
21.
(a) They are equal.
(b) 2π
(c) No. A conservative field would have a closed integral value of zero.
22. −3
23. 0
24. 2π
25. 0
, dy ⟩.
26. Hint: Use the facts N̂ = T̂ × k̂ and dds⃗r = ⟨ dx
ds ds
27. 2π
28. 0
29. 4π
30. −9π
2.3 Green’s Theorem
2.3
Green’s Theorem
2.3.1
Exercises
27
In Exercises 1 - 6, use Green’s Theorem to compute the given path integral.
1.
C
⃗ · d⃗r , where F
⃗ = ⟨x − y, x ⟩ and C is the unit circle.
F
2.
C
−y 2 dx + xy dy and C is the square in the first quadrant between the lines x = 1 and y = 1.
3.
C
y 2 dx + x 2 dy and C is the triangle bounded by the lines x = 0 x + y = 1, and y = 0.
4.
5.
6.
4x 3 y dx + x 4 dy and C consists of the segment from (−1, 0) to (1, 0), followed by upper half-circle
traversed counterclockwise from (1, 0) to (−1, 0).
C
C
⃗ · d⃗r , where F
⃗ = ⟨x + y , −x 2 − y 2 ⟩ and C is triangle bouned by the lines y = 0, x = 1, and y = x.
F
⃗ · d⃗r , where F
⃗ = ⟨xy + y 2 , x − y ⟩ and C consists of the curve y = x 2 from (0, 0) to (1, 1), followed
F
by the curve x = y 2 from (1, 1) to (0, 0).
C
In Exercises 7 - 10, use Green’s Theorem to compute both the counterclockwise circulation and outward
flux for the given vector field and curve.
⃗ = ⟨x − y , y − x ⟩ and C is the unit square in the first quadrant.
7. F
⃗ = ⟨x 2 + 4y , x + y 2 ⟩ and C is the unit square in the first quadrant.
8. F
⃗ = ⟨y 2 − x 2 , x 2 + y 2 ⟩ and C is the triangle with boundaries y = 0, x = 3, y = x.
9. F
⃗ = ⟨arctan ( y ), ln (x 2 + y 2 )⟩ and C is the boundary of the polar region 1 ≤ r ≤ 2, 0 ≤ θ ≤ π .
10. F
x
⃗ (x, y ) = ⟨P(x, y ), Q(x, y)⟩ and a tiny rectangle C with vertices (x, y ), (x +
11. Consider a vector field F
∆x, y ), (x + ∆x, y + ∆y ), and (x, y + ∆y). Prove that
C
⃗ · d⃗r ≈
F
∂Q
∂P
∂x − ∂y
∆x ∆y .
⃗ (x, y ) = ⟨P(x, y ), Q(x, y)⟩ and a tiny rectangle C with vertices (x, y ), (x +
12. Consider a vector field F
∆x, y ), (x + ∆x, y + ∆y ), and (x, y + ∆y). Prove that
C
⃗ · N̂ds ≈
F
∂P
∂Q
∂x + ∂y
∆x ∆y .
⃗ (x, y ) = ⟨4x − 2y , 2x − 4y ⟩ in
13. Use Green’s Theorem to compute the work done by the vector field F
2
2
moving a particle once around the circle (x − 2) + (y − 2) = 4 traversed counterclockwise.
⃗ (x, y ) = ⟨3xy − x 2 , ex +
14. Use Green’s Theorem to compute the outward flux of the vector field F
1+y
arctan (y)⟩ across the polar graph r = 1 + cos (θ ).
15. Use Green’s Theorem to compute the area enclosed by ⃗r (t) = ⟨k cos (t), k sin (t)⟩, 0 ≤ t ≤ 2π .
16. Use Green’s Theorem to compute the area enclosed by ⃗r (t) = ⟨cos3 (t), sin3 (t)⟩, 0 ≤ t ≤ 2π .
28
Vector Calculus
2.3.2
Answers
1. 2π
2. 32
3. 0
4. 0
5. − 76
7
6. − 60
7.
C
⃗ · d⃗r = 0 and
F
8.
C
⃗ · d⃗r = −3 and
F
C
9.
C
⃗ · d⃗r = 9 and
F
C
⃗ · N̂ds = −9
F
10.
C
⃗ · d⃗r = 0 and
F
C
⃗ · N̂ds = 2
F
11. Good luck.
12. Good luck.
13. 16π
14. 0
15. π k 2
16. 38π
C
⃗ · N̂ds = 2
F
⃗ · N̂ds = 2
F
2.4 Surface Integrals Over Surfaces
29
2.4
Surface Integrals Over Surfaces
2.4.1
Exercises
In Exercises 1 - 8, compute the area of the given surface with a surface integral.
1. The cone z =
p
x 2 + y 2 for 0 ≤ z ≤ 1.
2. The top half of the sphere x 2 + y 2 + z 2 = 4.
3. The part of the plane z = −x inside the cylinder x 2 + y 2 = 4.
4. The part of the plane y + 2z = 2 inside the cylinder x 2 + y 2 = 1.
p
5. The part of the cone z = 2
x 2 + y 2 between the planes z = 2 and z = 6.
6. The part of the cylinder x 2 + y 2 = 1 between the planes z = 1 and z = 4.
7. The cap of the paraboloid z = 2 − x 2 − y 2 cut by the cone z =
p
x 2 + y 2.
8. The band of the sphere x 2 + y 2 + z 2 = 4 cut between the planes z = −1 and z =
√
3.
In Exercises 9 - 14, compute the given surface integral over the given surface.
9. S xyz d σ , where S is the square cut from the plane z = 1 for 0 ≤ x ≤ 1, 0 ≤ y ≤ 1.
10.
11.
12.
p
S
x 2 d σ , where S is the cone z =
S
x 2 d σ , where S is the sphere x 2 + y 2 + z 2 = 1.
S
y d σ , where S is the surface z = x + y 2 , 0 ≤ x ≤ 1, 0 ≤ y ≤ 2.
x 2 + y 2 , 0 ≤ z ≤ 1.
z d σ , where S consists of three surfaces S1 , S2 , S3 . S1 is the side of the cylinder x 2 + y 2 = 1, S2
is the enclosed unit circle x 2 + y 2 ≤ 1, z = 0, and S3 is the plane z = x + 1 that lies above S2 .
p
14. S 1 − x 2 − y 2 d σ , where S is formed by rotating the curve x = cos (z), y = 0, − π2 ≤ z ≤ π2
around the z-axis.
13.
S
15. Compute the mass of the thin hemispherical shell x 2 + y 2 + z 2 = 4, z ≥ 0 with constant density
δ (x, y , z) = 3.
16. Compute the mass of the thin cone shell z =
density δ (x, y, z) = z12 .
p
x 2 + y 2 between the planes z = 1 and z = 2 with
30
Vector Calculus
2.4.2
Answers
√
1. π 2
2. 8π
√
3. 4π 2
√
4. π 2 5
√
5. 8π 5
6. 6π
√
7. π6 5 5 − 1
√ 8. π 4 + 4 3
9.
10.
11.
12.
13.
14.
11
0
0
uvdudv = 14
√
√
2π 1 2
π 2
2
r
cos
(
θ
)(r
2)drd
θ
=
4
0
0
2π π
0
0
sin2 (ϕ) cos2 (θ ) sin (ϕ)d ϕd θ = 43π
√
12 √
2 dvdu = 13 2
v
2
+
4v
3
0 0
2π 1+cos (θ)
0
0
2π π/2
zdzd θ + 0 +
2π 1
p
−π/2 | sin (u)| cos (u)
0
15. 24π
√
16. 2π 2 ln (2)
0
0
[1 + r cos (θ )]r
√
√
2drd θ = 32π + π 2
√
1 + sin2 (u)dudv = 43π 2 2 − 1
2.5 Surface Integrals in Vector Fields
2.5
Surface Integrals in Vector Fields
2.5.1
Exercises
31
In Exercises 1 - 4, compute the given outward flux integral of the given vector field across the given surface.
⃗ · N̂ d σ , where F
⃗ = ⟨yz, x, −z 2 ⟩ and S is the surface y = x 2 , 0 ≤ x ≤ 1, 0 ≤ z ≤ 4.
1. S F
⃗ · N̂ d σ , where F
⃗ = ⟨0, yz, z 2 ⟩ and S is the surface cut from the cylinder y 2 + z 2 = 1, z ≥ 0
F
between the planes x = 0 and x = 1.
⃗ · N̂ d σ , where F
⃗ = ⟨z, y, x ⟩ and S is the sphere x 2 + y 2 + z 2 = 1.
3. S F
2.
4.
S
S
⃗ · N̂ d σ , where F
⃗ = ⟨y , x, z ⟩ and S is the paraboloid z = 1 − x 2 − y 2 above the xy-plane.
F
In Exercises 5 - 8, use Stokes’ Theorem to compute the counterclockwise circulation of the given vector
field around the given curve.
⃗ = ⟨y, −x, 0⟩ and C is the curve x 2 + y 2 = 9, z = 0.
5. F
⃗ = ⟨−y 2 , x, z 2 ⟩ and C is the curve of intersection of the plane y + z = 2 and the cylinder x 2 + y 2 = 1.
6. F
⃗ = ⟨x 2 − y , 4z, x 2 ⟩ and C is the curve of intersection of the cone z =
7. F
p
x 2 + y 2 and the plane z = 2.
⃗ = ⟨xz, xy , 3xz ⟩ and C is the boundary of the plane 2x + y + z = 2 that lies in the first octant.
8. F
⃗ = ⟨y , −x, 1⟩, the surface S formed when the hyperboloid z = y 2 − x 2 is
9. Consider the vector field F
cut by the cylinder x 2 + y 2 = 1, and the counterclockwise curve C which is the boundary of S.
(a) Compute
C
⃗ · d⃗r by parameterizing C.
F
(b) Compute
C
⃗ · d⃗r using Stokes’ Theorem.
F
⃗ = ⟨−y , x, 2⟩, the surface S consisting of the cone z 2 = x 2 + y 2 for
10. Consider the vector field F
0 ≤ z ≤ 4, and the counterclockwise curve C which is the boundary of S.
(a) Compute
C
⃗ · d⃗r by parameterizing C.
F
(b) Compute
C
⃗ · d⃗r using Stokes’ Theorem.
F
⃗ = ⟨y , −xz, xz 2 ⟩ and the surface S consisting of the elliptical paraboloid
11. Consider the vector field F
2
2
⃗ ×F
⃗ across the surface
z = x + 4y that lies below the plane z = 1. Compute the outward flux of ∇
of S.
⃗ = ⟨zey , x cos (y ), xz sin (y)⟩ and the surface S consisting of the hemi12. Consider the vector field F
⃗ ×F
⃗ across the surface
sphere x 2 + y 2 + z 2 = 16 for y ≥ 0. Compute the inward (or leftward) flux of ∇
of S.
32
Vector Calculus
2.5.2
1.
2.
3.
4.
5.
6.
7.
8.
Answers
41
0
0
0
[sin (v ) cos2 (v ) + sin3 (v )]dudv = 2
2π π
0
0
2π 1
0
0
2π 3
0
0
2π 1
0
0
[2 sin2 (ϕ) cos (ϕ) cos (θ ) + sin3 (ϕ) sin2 (θ )]d ϕd θ = 43π
[1 + 4r 2 cos (θ ) sin (θ ) − r 2 ]r drd θ = π2
−2rdrd θ = −18π .
[1 + 2r sin (θ )]rdrd θ = π .
2π 2 1
0
0
0
√
√ [4 cos (θ ) + r sin (2θ )](
1 2−2u
0
2
2r )drd θ = 4π .
(7u + 4v − 6)dvdu = −1.
2π
[− sin2 (t) − cos2 (t) + 4 sin (t) cos (t)]dt = −2π
0
2π 1
(b) 0 0 (−2r )drd θ = −2π
(a)
10.
(a)
12.
(2u 3 v − u)dudv = 2
π1
9.
11.
0
2π
[16 sin2 (t) + 16 cos2 (t)]dt = 32π
0
2π 4
(b) 0 0 (2u)dudv = 32π
2π
0
2π
0
[− 12 sin2 (t) − 21 cos2 (t)]dt = −π
−16 sin2 (t)dt = −16π
2.6 The Divergence Theorem
2.6
The Divergence Theorem
2.6.1
Exercises
33
In Exercises 1 - 10, use the Divergence Theorem to compute the outward flux of the given vector field
across the given surface.
⃗ = ⟨y, −x, 0⟩ and S is the sphere x 2 + y 2 + z 2 = 1.
1. F
⃗ = ⟨y − x, z − y , y − x ⟩ and S surronds the cube bounded by the planes x = ±1, y = ±1, z = ±1.
2. F
⃗ = ⟨x 2 , y 2 , z 2 ⟩ and S surrounds the unit cube in the first octant.
3. F
⃗ = ⟨xy , yz, xz ⟩ and S surrounds the unit cube in the first octant.
4. F
2
⃗ = ⟨xy , y 2 + exz , sin (xy )⟩ and S surrounds the region enclosed by the surface z = 1 − x 2 and the
5. F
planes z = 0, y = 0, y + z = 2.
⃗ = ⟨x 2 , 4xyz, zex ⟩ and S surrounds the box 0 ≤ x ≤ 3, 0 ≤ y ≤ 2, 0 ≤ z ≤ 1.
6. F
⃗ = ⟨x 2 , y 2 , z 2 ⟩ and S encloses region bounded by the cylinder x 2 + y 2 = 4 and the planes z = 0
7. F
and z = 1.
⃗ = ⟨y , xy , −z ⟩ and S encloses region bounded by the cylinder x 2 + y 2 = 4, the plane z = 0, and the
8. F
paraboloid z = x 2 + y 2 .
⃗ = ⟨x 3 , y 3 , z 3 ⟩ and S surrounds the unit sphere.
9. F
⃗ =√ 1
10. F
2
2
x +y +z 2
⟨x, y , z ⟩ and S are the spherical shells that surround the region 1 ≤ x 2 + y 2 + z 2 ≤ 4.
⃗ = ⟨x, y , z ⟩ and surface S surrounds the space B, then prove that
11. If F
12. For vector field F (x, y , z) = ⟨P(x, y, z), Q(x, y , z), R(x, y, z)⟩, prove that
suming the favorable conditions of this chapter).
B
dV = 31
S
S
⃗ · N̂ d σ .
F
⃗ ×F
⃗ ) · N̂ d σ = 0 (as(∇
34
Vector Calculus
2.6.2
Answers
1. 0
1 1 1
2. −1 −1 −1 −2dzdydx = −16
3.
4.
5.
6.
7.
8.
9.
10.
111
0
0
0
111
0
0
0
(2x + 2y + 2z)dzdydx = 3
(x + y + z)dzdydx = 23
1 1−x 2 2−z
−1 0
0
123
0
0
0
(2x + 4xz + ex )dxdydz = 34 + 2e3
1 2π 2
0
0
0
(2r cos (θ ) + 2r sin (θ ) + 2z)r drd θ dz = 4π
2π 2 r 2
0
0
0
(r cos (θ ) − 1)r dzdrd θ = −8π
0
2π π 1
0
3ydydzdx = 184
35
0
(3ρ2 )ρ2 sin (ϕ) d ρd ϕd θ = 125π
2π π 2 2 2
( )ρ sin (ϕ)d ρd ϕd θ = 12π
0
0 1 ρ
11. Good luck.
12. Good luck.
Chapter 3
Infinite Sums
36
Infinite Sums
3.1
Sequences
3.1.1
Exercises
In Exercises 1 - 4, write the first four terms of the given sequence.
1. ak =
(−1)k
,k ≥1
2k + 1
3. a1 = 7 and an = 2 − an−1 , n ≥ 2
2. {n!}∞
n=0
4. a5 = 7, a6 = 3, a7 = −1, a8 = −5
In Exercises 5 - 16, determine an explicit formula for the given sequence.
4
5
11. {− 23 , 39 , − 27
, 81
,···}
5. {0.9, 0.09, 0.009, 0.0009, · · · }
6. {0, 1,
√ √
2,
3, 2, · · · }
12. {1, 0, −1, 0, 1, 0, −1, 0, · · · }
1
7. { 12 , 14 , 81 , 16
, · · · }, n ≥ 1
13. {0, 2, 0, 2, 0, 2, · · · }
1
8. { 21 , 14 , 81 , 16
, · · · }, n ≥ 0
14. {0, 1, 0, 1, 0, 1, · · · }
9. { 52 , 2, − 23 , − 72 , · · · }
1
1
, − 720
,···}
15. {1, − 12 , 24
4
8
10. {1, − 27 , 13
, − 19
,···}
1
1
16. {1, − 16 , 120
, − 5040
,···}
In Exercises 17 - 22, compute the limit of the given sequence graphically.
17. an = n2
19. an = arctan (n)
21. an = cos (nπ )
18. an = 1 − n
20. an = (−1)n
22. an = sin (nπ )
In Exercises 23 - 28, compute the limit of the given sequence.
23. an =
4 − 7n
3n + 1
r
26. an =
24. an =
2n
n+1
27. an =
25. an =
ln2 (n)
n
28. an =
3
1−
n
n
p
9n2 − n − 3n
(−1)n n2
n2 + 1
In Exercises 29 - 32, compute the limit of the given sequence with the sandwich rule.
29. an =
1 + cos (n)
n
31. an =
(−1)n n
n2 + 1
30. an =
tanh (n)
2n
32. an =
arctan (n)
n
3.1 Sequences
37
33. Compute the limit of the recursive sequence: a1 = 2 and an = 21 (an−1 + 6), n ≥ 2.
34. Order the following functions of n from least to greatest: nn ,
√
n, log7 (n), n!, 7, n7 , 7n .
In Exercises 35 - 42, determine the limit of the given sequence.
n
1
−√
35. an =
n1000
en
39. an =
36. an =
n!
100n
40. an =
ln100 (n)
n
37. an =
n!
nn
41. an =
n!
ln (n)
42. an =
3n+2
5n
38. an =
√
n
n2
2
38
Infinite Sums
3.1.2
Answers
1. a1 = − 13 , a2 = 15 , a3 = − 71 , a4 = 19
3. a1 = 7, a2 = −5, a3 = 7, a4 = −5
2. {1, 1, 2, 6, · · · }
4. a1 = 23, a2 = 19, a3 = 15, a4 = 11
5. an = 109n , n ≥ 1
√
9. an = 5−24n , n ≥ 0
13. an = 1 + (−1)n , n ≥ 1
n
n
1)
14. an = 1+(−
,n≥1
2
n
−1)
,n≥0
15. an = ((2n)!
6. { n − 3}∞
n=3
−2)
10. an = (1+6n
,n≥0
7. an = 21n , n ≥ 1
11. an = (−1)3n(n+1) , n ≥ 1
1
,n≥0
8. an = 2n+1
12. an = sin
nπ
2
n
n
(−1)
16. an = (2n+1)!
,n≥0
,n≥1
17. ∞
19. π2
21. DNE
18. -∞
20. DNE
22. 0
23. − 73
25. 0
27. − 16
26. e−3
28. DNE
24.
√
2
29. 0
31. 0
30. 0
32. 0
33. 6
34. 7, log7 (n),
35. 0
39. 0
36. ∞
40. 0
37. 0
41. ∞
38. 1
42. 0
√
n, n7 , 7n , n!, nn
3.2 Series
39
3.2
Series
3.2.1
Exercises
In Exercises 1 - 4, compute a formula for the partial sums of the given series to determine if it converges
or diverges. If the series converges, compute its sum.
1.
∞
X
1
n=1
2.
∞
X
n=1
2n−1
1
n(n + 1)
3.
∞
X
n
n=1
4.
∞
X
(−1)n
n=0
In Exercises 5 - 8, use the geometric test to prove whether the given series converges or diverges. If the
series converges, compute its sum.
5.
∞ X
√ n
2
7. 1 + c + c 2 + · · · , |c | < 1
n=0
6.
∞
X
1
n=−3
2n+3
8.
∞
X
1
n=0
πn
In Exercises 9 - 14, prove whether the given series converges or diverges. If the series converges, compute
its sum.
9.
∞ n
X
1
4
n=2
10.
∞ n+3
X
1
n=−3
4
40
10 20
11. 5 −
+
−
+ ···
3
9
27
12. 1 +
3 9 27
+ +
+ ···
2 4
8
n
∞ X
1
13.
−
+
3
n=1
14.
∞ X
1
n=0
πn
+e
n
3
2n−1
In Exercises 15 - 18, compute a formula for the partial sums of the given telescoping series to determine if
it converges or diverges. If the series converges, compute its sum.
15.
∞
X
n=1
16.
∞
X
n=1
1
n(n + 1)
1
4n2 − 1
17.
∞
X
[sin (n) − sin (n + 1)]
n=1
18.
∞
X
n=1
[e1/n − e1/(n+1) ]
40
Infinite Sums
In Exercises 19 - 20, use a series to convert the given repeating decimal into a fraction.
20. 6.235
19. 0.73
In Exercises 21 - 22, use the nth-term divergence test to prove the given series diverges.
21.
∞
X
n=1
n
3n + 1
22.
∞
X
sin (n)
n=1
23. Compute the value of c so that
∞
X
(1 + c)−n = 2.
n=2
24. Compute all the values of x so that
∞
X
[sin (x)]n converges.
n=0
25. Prove that 1 + r + r 2 + · · · + r n =
1 − r n+1
.
1−r
26. Prove the geometric test: The series
P∞
k =0 r
In Exercises 27 - 30, show examples of series
conditions.
27.
P
an and
P
bn converge, but
P
P P
k
1
= 1−
for |r | < 1 and diverges for |r | ≥ 1.
r
an and
P
bn where no terms are 0 and satisfy the given
(an bn ) does not converge to
an
bn
28.
P
an and
P
bn converge, but
29.
P
an and
P
bn diverge, but
P
30.
P
an and
P
bn diverge, but
P an diverges.
(an + bn ) converges.
bn
converges.
P
an ·
P
bn .
3.2 Series
3.2.2
41
Answers
1. Sn = 2 − 2n1−1 → 2
3. Sn = n(n+1)
→ ∞. Diverges.
2
1
2. Sn = 1 − n+1
→1
1)
4. Sn = 1+(−
and the limit DNE. Diverges.
2
5. Diverges
1
7. 1−
c
6. 2
π
8. π−
1
1
9. 12
n
12. Diverges
10. 43
13. 23
4
11. 3
14. Diverges
1
15. Sn = 1 − n+1
→1
1
→ 21
16. Sn = 12 − 4n+2
17. Sn = sin (1) − sin (n + 1) and the limit DNE. So the series diverges.
18. Sn = e − e1/(n+1) → e − 1
19. 73
99
20. 6173
990
n
→ 13 ̸= 0. So the series diverges by the nth term test.
21. 3n+1
22. limn→∞ sin (n) DNE, which is not zero. So the series diverges by the nth term test.
√
23. c =
3−1
2
25. Good luck.
24. All x except multiples of π2 .
26. Hint: Use the previous exercise.
27. an = bn = 21n
29. an = 2n , bn = −2n
28. an = bn = 21n
30. an = 2n , bn = 3n
42
Infinite Sums
3.3
The Comparison Tests
3.3.1
Exercises
In Exercises 1 - 4, use a p-series test to prove whether the given series converges or diverges.
1.
∞
X
1
√
n=1
3.
n
∞
X
1
n=1
n0.85
∞
X
2
√
2.
∞
X
1
√
4.
3
n=1
k=1
n n
k2
In Exercises 5 - 10, use the direct comparison test to prove whether the given series converges or diverges.
5.
∞
X
n=5
6.
∞
X
k =1
7.
∞
X
k =1
1
8.
n−4
∞
X
ln2 (n)
n
n=2
1
2
3k + k + 4
9.
∞
X
1
n=1
1
k
3 +k
10.
∞
X
n!
√
k=2
1
k2 − k − 1
In Exercises 11 - 18, use the limit comparison test to prove whether the given series converges or diverges.
11.
∞
X
n=1
12.
∞
X
n=1
13.
∞
X
k =1
1
n+4
15.
n
n=2
1
16.
2n − 1
∞
X
n=2
k
17.
2k 2 − k + 1
18.
5 + n5
1
n2 ln (n)
∞
X
ln (n)
√
n
n=2
∞
X
2n2 + 3n
√
14.
n=1
∞
X
ln (n)
∞
X
e
sin
k=1
19. Suppose the series
P
P
an and
(a) If an ≤ bn and
P
bn diverges, then
(b) If an ≥ bn and
P
(c) If an ≤ bn and
P
n
1
k
bn have positive terms. Prove or disprove the following statements.
P
an diverges.
bn converges, then
P
an converges.
bn converges, then
P
an diverges.
3.3 The Comparison Tests
43
20. Show an example of a pair of series
P
an and
P
bn with positive terms such that:
• abnn → 0
•
P
bn diverges
•
P
an converges
21. The decimal representation of any fraction can be expressed as:
0.d1 d2 d3 · · · =
d1
d2
d3
+
+
+ ··· ,
10 100 1000
where each di is any digit. Prove that this series always converges.
22. Prove that if
P
an converges, then so does
P
ln (1 + an ).
44
Infinite Sums
3.3.2
Answers
1. Diverges
3. Diverges
2. Converges
4. Diverges
5. Diverges. Compare to
P1
n
.
8. Diverges. Compare to
6. Converges. Compare to
P 1
7. Converges. Compare to
P 1
3k 2
11. Diverges. Limit compare to
3k
.
10. Diverges. Compare to
n
12. Converges. Limit compare to
.
2n
13. Diverges. Limit compare to
P1
14. Diverges. Limit compare to
P 1
19.
.
.
√ .
n
.
P 1
2n
P1
k
15. Diverges. Limit compare to
P 1
k
n
9. Converges. Compare to
.
P1
P1
.
.
P1
n
.
16. Converges. Limit compare to
P 1
.
17. Converges. Limit compare to
P 1
.
18. Diverges. Limit compare to
(a) False. For example, an = n12 and bn = n1 .
(b) False. For example, an = n1 and bn = n12 .
(c) False. For example, an = n13 and bn = n12 .
20. an = n12 and bn = n1 .
21. Hint: Compare to 109n .
22. Hint: Use the limit comparison test with L’Hopital’s rule.
n2
en
P1
k
.
3.4 The Ratio and Alternating Tests
45
3.4
The Ratio and Alternating Tests
3.4.1
Exercises
In Exercises 1 - 8, use the ratio test to prove whether the given series converges or diverges.
1.
∞
X
n=1
2.
∞
X
n=1
4.
5.
∞
X
∞
X
n!
n=1
∞
X
3n+2
n=1
3.
2n+1
n · 3 n −1
6.
ln (n)
10n
∞
X
n ln (n)
2n
n=1
n · 5n
(2n + 3) ln (n + 1)
7.
∞
X
n=1
2 −n
n e
8.
n=1
(2n − 1)!
[2 · 4 · · · (2n)] · 3n
∞
X
1 · 3 · · · (2n − 1)
n=1
[2 · 4 · · · (2n)] · 3n
In Exercises 9 - 10, use the alternating test to prove the given series converges.
9.
∞
X
(−1)n
n=2
10.
ln (n)
∞
X
n=1
1
(−1) ln 1 +
n
n
In Exercises 11 - 16, prove whether the given series converges conditionally, converges absolutely, or
diverges.
11.
∞
X
(−0.1)n
n=1
12.
∞
X
n
(−1)n
n=1
13.
∞
X
n=1
14.
(−1)n
∞
X
(−1)n
√
n=1
n
2n + 1
15.
n!
2n
16.
1+
n
∞
X
(−1)n
n3
n=1
∞
X
(−1)n
n=2
n ln (n)
17. Define the sequence:
(
an =
Prove whether
n
,
2n
1
,
2n
n is odd
n is even
P∞
n=1 an converges or diverges.
18. Define the sequence an recursively by a1 = 1 and an+1 = 1 + n1
converges or diverges.
n
an , n ≥ 1. Prove whether
P
an
46
Infinite Sums
P∞
n+1
19. For the series n=1 (−n1)p , compute all the values of p for which the series (1) converges absolutely,
(2) converges conditionally, and (3) diverges.
20. Let A =
P∞ (−1)n+1
n=1
n
. Rearrange the terms of this series by following these steps.
• Reorder all the terms into the following groups of three for k = 1, 2, 3, · · · :
··· +
1
2k − 1
−
1
1
−
+ ···
2(2k − 1)
4k
• Combine the first two fractions in each of these groups of three.
• Factor out a fraction from all terms so that the original series reappears.
What result is obtained?
3.4 The Ratio and Alternating Tests
3.4.2
47
Answers
1. ρ = 23 < 1; converges
5. ρ = ∞ > 1; diverges
2. ρ = 3 > 1; diverges
6. ρ = 12 < 1; converges
3. ρ = 5 > 1; diverges
7. ρ = ∞ > 1; diverges
4. ρ = e1 < 1; converges
8. ρ = 13 ; converges
9. Good luck.
10. Good luck.
11. Converges absolutely
14. Converges conditionally
12. Diverges
15. Converges absolutely
13. Diverges
16. Converges conditionally
17. Converges. Use the comparison test combined with the ratio test.
18. Diverges. Using the ratio test, ρ = e > 1
19. Converges absolutely for p > 1 by the p-series test. Converges conditionally for 0 < p ≤ 1 by the
alternating test. Diverges for p ≤ 0 by the nth-term divergence test.
20. 21 A
48
Infinite Sums
3.5
Power Series and Taylor Series
3.5.1
Exercises
In Exercises 1 - 8, compute the interval and radius of converges for the given series.
1.
∞
X
n
n=1
2.
∞
X
(2n)!
n=0
4.
∞
X
n=1
∞
X
nn (x − 1)n
n=1
xn
6.
∞
X
1
n=1
(−3)n
√
5.
(x + 2)
∞
X
n
n=1
3.
3n+1
n
n+1
x
n
7.
n
(x − 3)n
∞
X
(−1)n 32n
3n
n=1
1
(x − 1)n
n3 · 3n
8.
∞
X
[ln (x)]n
n=0
9. Compute the interval of convergence and the sum of the series
10. Compute the radius of convergence for the series
(x − 2)n
P∞ (n!)k
n=1 (kn)! x
n
P∞
1
2n
n=0 9n (x + 1) .
in terms of k .
In Exercises 11 - 20, compute the Taylor series for the given function at the given point. Write your answer
in sigma notation if there are infinitely many terms.
1
11. f (x) = 1−
at c = 0
x
16. f (x) = 2x at c = 1
12. f (x) = ex at c = 0
17. f (x) = x1 at c = 2
13. f (x) = sin (x) at c = 0
18. f (x) = sin (2x) at c = π
14. f (x) = cos (x) at c = 0
19. f (x) = 2x 3 + x 2 + 3x − 8 at c = 0
15. f (x) = ln (1 + x) at c = 0
20. f (x) = 2x 3 + x 2 + 3x − 8 at c = 1
In Exercises 21 - 24, you are given a Taylor series generated by f(x). Compute the given derivative value.
21.
∞
X
(−1)n
n=0
22.
n+3
∞
X
1
n=0
n
(x − 1) ; f ′′ (1)
n
(x + 2)n ; f ′′′ (−2)
n
23.
∞
X
n2 e(x − π )n ; f ′′′ (π )
n=1
24.
∞
X
nc
n=0
cn
(x − p)n ; f (k ) (p)
3.5 Power Series and Taylor Series
3.5.2
49
Answers
1. (−5, 1); R = 3
5. x = 1; R = 0
2. (−∞, ∞); R = ∞
6. [2, 4); R = 1
− 31 , 13 ; R = 13
7. ( 17
, 19
]; R = 19
9
9
3.
4. [−2, 4]; R = 3
9. (−4, 2); 8−2x9−x 2
11.
∞
X
x
n
2
−1
8. (e−1 , e); R = e 2e
10. R = k k
16.
n=0
12.
13.
17.
(2n + 1)!
(2n)!
x 2n
∞
X
(−1)n+1
n=1
x 2n+1
n
xn
18.
n!
∞
X
(−1)n
n=0
∞
X
(−1)n
n=0
15.
n!
xn
∞
X
(−1)n
n=0
14.
n=0
∞
X
1
n=0
∞
X
2[ln (2)]n
2n+1
(x − 1)n
(x − 2)n
∞
X
(−1)n 22n+1
n=0
(2n + 1)!
(x − π )2n+1
19. 2x 3 + x 2 + 3x − 8
20. −2 + 11(x − 1) + 7(x − 1)2 + 2(x − 1)3
21. f ′′ (1) = 52
23. f ′′′ (π ) = 54e
22. f ′′′ (−2) = 92
24. f (k) (p) = kc kk !
c
50
Infinite Sums
3.6
Application of Taylor Series
3.6.1
Exercises
In Exercises 1 - 6, use a Taylor series to compute the Taylor series for the given function.
1 2
x
2
1. f (x) = ex + e2x at c = 0
4. f (x) = x cos
1
2. f (x) = 1+x
at c = 0
5. f (x) = ex at c = 2
3. f (x) = sin (π x) at c = 0
1
6. f (x) = 2−
at c = 1
x
at c = 0
In Exercises 7 - 10, use a Taylor series to compute the given limit.
ex − e−x
x →0
x
9. lim
sin (x) − x + 16 x 3
10. lim
ex − 1 − x − 21 x 2
7. lim
x5
x →0
8. lim (x + 1) sin
x →∞
1
x +1
x3
x →0
In Exercises 11 - 14, use a Taylor series to compute the given derivative of the given function.
4
13. f (x) = x 5 ex ; f ′′′ (0)
4
14. f (x) = x 5 ex ; f (8) (0)
11. f (x) = 1x−x ; f ′′′ (0)
12. f (x) = 1x−x ; f (5) (0)
In Exercises 15 - 18, use a Taylor series to compute an explicit formula for f (x).
15. f (x) = x −
16. f (x) =
1 3 1 5 1 7
x + x − x + · · · , −1 ≤ x ≤ 1
3
5
7
∞
X
(−1)n
n!
n=0
x 2n , −∞ < x < ∞
17. f (x) = 1 + 2x + 3x 2 + 4x 3 + 5x 4 + · · · , −1 < x < 1
18. f (x) =
∞
X
(−1)n (4n + 2)
(2n + 1)!
n=0
x 4n+1 , −∞ < x < ∞
In Exercises 19 - 22, use a Taylor series to compute the given sum.
19.
∞
X
n=1
20.
∞
X
n=0
3n
21.
5n · n!
(−1)n
∞
X
n=0
(−1)n
π 2n+1
42n+1 (2n + 1)!
π 2n
62n (2n)!
22. 1 − ln (2) +
ln2 (2)
ln3 (2)
−
+ ···
2!
3!
3.6 Application of Taylor Series
51
In Exercises 23 - 24, use a Taylor series to estimate the value of the given definite integral so that the error
from the actual value is less than 0.001.
1
1
3
23.
x cos (x )dx
0
24.
2
e−x dx
0
25. Compute Exercise 10 again using L’Hopital’s rule and compare your work.
26. Use a Taylor series to prove that ei θ = cos (θ ) + i sin (θ ), where i =
√
−1.
52
Infinite Sums
3.6.2
1.
Answers
∞
X
1 + 2n
n!
n=0
2.
∞
X
x
n
4.
∞
X
(−1)n x n
5.
∞
X
e2
n=0
(−1)n
n=0
(−1)n
n=0
n=0
3.
∞
X
π 2n+1
(2n + 1)!
x 2n+1
6.
∞
X
n!
1
22n (2n)!
(x − 2)n
(x − 1)n
n=0
1
9. 120
7. 2
8. 1
10. 16
11. 0
13. 0
12. 120
14. 6720
15. f (x) = arctan (x)
17. f (x) = (1−1x)2
16. f (x) = e−x
2
19. e3/5 − 1
√
20.
3
2
23. 0.440
25. Good luck.
26. Good luck.
18. f (x) = 2x cos (x 2 )
√
21.
2
2
22. 12
24. 0.747
x 4n+1