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Hanoi University of Science and Technology
School of Applied Mathematics and Informatics
CALCULUS I EXERCISE
COURSE ID: MI 1016
1.1-1.3. Functions. Essential functions
Exercise 1. Determine the domain of the following functions.
a) y =
x √ 4 x^2 − 1
b) y = arcsin
x
x + 2
c) y = ln
1 − x
1 + x
d) y =
arctan x.
Exercise 2. Find the range of the following functions.
a) y = ln(1 − 2 sin x).
b) y = arctan (2ex).
c) y =
arccos x.
d) y =
x^2 − 1
x^2 + 1
Exercise 3. As dry air moves upward, it expands and cools. The ground temperature is
◦ C and the temperature at a height of 1 km is 20 ◦ C.
a) Express the temperature T (in ◦C ) as a function of the height h (in kilometers),
assuming that a linear model is appropriate.
b) Draw the graph of the function.
c) What is the temperature at a height of 4 km?
Exercise 4. The figure shown here shows a rectangle inscribed in an isosceles right
triangle whose hypotenuse is 2 units long.
a) Express the y-coordinate of P in terms of x.
b) Express the area of the rectangle in terms of x.
Exercise 5. Determine whether f is even, odd, or neither.
a) f (x) =
ex^ − e−x
2
=: sinh x.
b) f (x) =
2 x^ − x^2
2 x^ + x^2
c) f (x) = ln
x +
x^2 + 1
d) f (x) = ln
1 − x
1 + x
e) f (x) = sin x + cos 2x.
f) f (x) = sin x + sin 2x.
Exercise 6. Find the functions f ◦ g, g ◦ f, f ◦ f , and g ◦ g and their domains.
a) f (x) =
x + 1
, g(x) = x − 1.
b) f (x) =
2 x + 3, g(x) = x 2
c) f (x) = sin x, g(x) =
x + 1.
d) f (x) = 1 − x^2 , g(x) =
1 − x
1 + x
Exercise 7. Find the inverse functions of the following functions.
a) y = arcsin 2x.
b) y =
e^2 x^ − e−^2 x
2
c) y =
1 − 3 x
1 + 3x
d) y = ln
ex^ − 1
ex^ + 1
g) lim x→∞
sin
x
x
x
.
h) lim x→ 0
ln(1 + 3 tan x)
ex^ − cos x
i) lim x→ 0 +
x^5 √ sin x ln (1 − 3 x^2 )
j) lim x→ 0
cos(sin x) − 1
sin(cos x − 1)
Exercise 14. If lim x→ 1
f (x) − 8
x − 1
= 9, find lim x→ 1
f (x).
3.1 - 3.6. Continuity of functions
Exercise 15. For what value of a is f (x) =
x 2
2 ax^3 + 1, if x ≥ 1
continuous at every
x?
Exercise 16. Show that f is continuous on (−∞, ∞).
a) f (x) =
sin x if x <
π
4
cos x if x ≥
π
4
. b) f (x) =
x^2 if x < 1
√ x if x ≥ 1
Exercise 17. Locate the discontinuity of the function and illustrate by graphing
a) y =
1 + e^1 /x^
. b)^ y^ = ln (tan
(^2) x). c) y =
sin x
2 x^ − 1
Exercise 18. Find the numbers at which f is discontinuous. At which of these numbers
is f continuous from the right, from the left, or neither?
a) f (x) =
2 x^ − 1
x
if x < 0
2 x + c if x ≥ 0
. b)^ f^ (x) =
sin 2 (πx)
ln (1 + 2x^2 )
if x < 1
√ x if x ≥ 1
Exercise 19. Prove that there is a root of the given equation in the specified interval.
a) x^6 − 3 x + 1 = 0, (0, 1). b) x^3 =
3 x + 1, (1, 2).
Exercise 20. A train starts at 8AM from Hanoi to Haiphong, arriving at 11AM. The
next day it starts at 8AM from Haiphong to Hanoi, arriving at 11AM. Is there a point on
the route the train will cross at exactly the same time of day on both days?
Exercise 21. Let f be a continuous function on a close interval [0, 1] and f (0) = 1, f (1) =
- Prove that there is a number c ∈ (0, 1) at which f (c) = c.
4.1 - 4.6. Derivatives
Exercise 22. Find the derivative of the following functions:
a) y = (x^2 + 1) 3
x^2 + 2.
b) y = sin(tan x).
c) y =
p x +
x.
d) y = ln
x +
x^2 + 5
e) y = sin n x cos nx.
f) y =
x
x
.
Exercise 23. For what values of a and b will
f (x) =
ax if x < 2
ax 2
be differentiable for all values of x? Discuss the geometry of the resulting graph of f.
Exercise 24. Let f (x) =
x 2 if x ≤ 2
mx + b if x > 2
. Find the values of m and b that make
f differentiable everywhere.
Exercise 25. Let r(x) = f (g(h(x))), where h(1) = 2, g(2) = 3, h ′ (1) = 4, g ′ (2) = 5, and
f ′(3) = 6. Find r′(1).
Exercise 26. If F (x) = f (3f (4f (x))), where f (0) = 0 and f ′(0) = 2, find F ′(0).
Exercise 27. Is the derivative of
h(x) =
x 2 sin(1/x) if x ̸= 0
0 if x = 0
continuous at x = 0? How about the derivative of k(x) = xh(x)? Give reasons for your
answer.
Exercise 28. Find y′(x) if y is defined implicitly as a function of x by the equation
a) arctan(2x + y) = y 3 .
b) x 3
c) cos(x − y) = xe y .
Exercise 29. Find the equation of the tangent line of the curve 2 x^3 + 4y^2 = 6 at the
point (1, 1).
Exercise 38. Find a cubic function y = ax^3 + bx^2 + cx + d whose graph has horizontal
tangents at the points (− 2 , 6) and (2, 0).
Exercise 39. Find all the point on the curve y = 2x^3 − 3 x^2 − 12 x + 20 where the tangent
of the curve at such a point is
a) perpendicular to the line y = 1 −
x
24
, b) parallel to the line^ y^ =^
2 − 12 x.
Exercise 40. Show that the tangents to the curve y =
π sin x
x
at x = π and x = −π
intersect at right angles.
Exercise 41. Given f (x) = ln
x + 1
x + 2
, find df (x), d^10 f (x).
Exercise 42. Given f (x) = (x + 2) ln x, find d 2 f (1), d 20 f (1).
Exercise 43. Find the n th-degree Taylor polynomials centered at x = 0 of f (x). Deter-
mine the remainder.
a) f (x) = x cos x, n = 5.
b) f (x) =
x √ 1 + x^2
, n = 5.
c) f (x) =
2 + 2x, n = 3.
d) f (x) = e^2 x^ + 1, n = 4.
5.6 - 5.9. Applications of Derivatives and Differentials
Exercise 44. Evaluate the following limits
a) lim x→ 0
arcsin x − x
x^3
b) lim x→ 0
x 5 − ln (1 + x 5 )
sin 10 x
c) lim x→ 0
x ln |x|.
d) lim x→+∞
x[π − 2 arctan(3x)].
e) lim x→ 1
x
x − 1
ln x
f) lim x→ 0
x
e^2 x^ − 1
g) lim x→−∞
(x 2
x (^).
h) lim x→ 0
[ln(e + 2x)]
sin x (^).
i) lim x→ 0
[2x + e^3 x]
sin x (^).
j) lim x→ 0 +
[arcsin 2x] tan x .
k) lim x→ 0
sin x ln(x + 1) − x 2
x^3
l) lim x→ 0 +^
x sin x .
Exercise 45. Show that
a) sin(arccos x) = cos(arcsin x) =
1 − x^2 for all x ∈ [− 1 , 1].
b)
arctan
2 x
1 − x^2
= arctan x for all x ∈ (− 1 , 1).
c) arcsin(tanh x) = arctan(sinh x).
Exercise 46. Prove that
a) | arcsin x − arcsin y| ≥ |x − y| for all x, y ∈ [− 1 , 1].
b)
y − x
1 + y^2
< arctan y − arctan x <
y − x
1 + x^2
for all 0 < x < y.
c)
x
8
x
ex^ − 1
for all x > 0.
d)
x
x + 1
≤ ln(x + 1) ≤ x for all x > − 1.
Exercise 47. Show that the equation 3 x + 2 cos x + 5 = 0 has exactly one real root.
Exercise 48. Show that the equation a cos x + b cos 2x + c cos 3x = 0 has at least one
root on (0, π).
Exercise 49. Suppose that f (x) is a continuous function on a close interval [a, b] and
differentiable on (a, b), and f (a) = f (b) = 0. Show that there exists a number c ∈ (a, b)
such that f ′(c) = 2021f (c).
Exercise 50. For what values of a and b is the following equation true?
lim x→ 0
sin 2x
x^3
b
x^2
Exercise 51. Determine the local extreme values
a) y = x^2 /^3 (x + 2).
b) y = x 2 / 3 (x 2 − 4).
c) y = x
4 − x^2.
d) y = x^2 ln x.
e) y = (x^2 − 3) ex.
f) y = 3 arctan x − ln (x^2 + 1).
g) y = ln(x + 3) + arccot x.
Exercise 52. Find the absolute maximum and minimum values of f (x) = x^2 +
x
over
[1, 10].
Exercise 60. Determine the asymptotes of the curves.
a) x = t 3 − 3 π, y = t 3 − 6 arctan t. b) x =
t^2
t − 1
, y =
t
t^2 − 1
Exercise 61. If f ′^ is continuous, f (2) = 0, and f ′(2) = 7, evaluate
lim x→ 0
f (2 + 3x) − f (2 + 5x)
x
6.1 - 6.5. Indefinite Integrals 1
Exercise 62. Find the function f.
a f ′(x) = 1 + x, f (0) = 1.
b f ′(x) = 5x^4 − 3 x^2 + 4, f (−1) = 2.
c f ′′ (x) = − 2 − 12 x 2 , f (0) = 4, f ′ (0) = 12.
d f ′′(x) = 20x^3 + 12x^2 , f (0) = 0, f ′(0) = 1.
Exercise 63. The graph of a function f is shown. Which one (a,b or c) is the anti-
derivative of f? Give your explanation.
a) b)
Exercise 64. What constant acceleration is required to increase the speed of a car from
30 km/h to 50 km/h in 5 s?
Exercise 65. Find a function f (x) such that f ′(x) = x^3 and the line x + y = 0 is tangent
to the graph of f (x).
Exercise 66. Evaluate the following integrals
a)
R
x sin(x 2 )dx.
b)
R (^) x + 1
x^2 + 2x + 2
dx.
c)
R 1
x ln 2 x
dx.
d)
R (^) x √ x + 1
dx.
e)
R
x sin xdx.
f)
R
x 2 e x dx.
g)
R
tan 2xdx.
h)
R
ex^ sin xdx.
6.6 - 6.8. Indefinite Integrals 2
Exercise 67. Evaluate the following integrals
a)
R (^) x^3 + 1
x^2 + 4
dx.
b)
R
tan 4 xdx.
c)
R (^1) − 2 x √ 2 + x^2
dx.
d)
R (^) x
(x^2 + 1) (x + 2)
dx.
e)
R (^) sin 2x √ sin^4 x + 1
dx.
f)
R (^) dx
3 sin x − 4 cos x
g)
R (^) dx
1 +
x^2 + 4x + 5
h)
R (^) x + 1 √ x^2 − 2 x − 1
dx.
Exercise 68. Evaluate the following integrals
a)
R
(x + 1) arctan xdx.
b)
R
(x + 2) ln xdx.
c)
R
arcsin 2 xdx.
d)
R (^) arctan x
x^2
dx.
e)
R (^) x
(x^2 + 2x + 2)
2 dx.
f)
R (^) e^2 x
1 + ex^
dx.
g)
R
r x
x − 1
dx.
h)
R (^) x^2 + 2
x^3 − 1
dx.
i)
R (^) x^2 + 1
x^4 + 1
dx.
j)
R (^) sin^2 x
cos^3 x
dx.
k)
R 1
x^2
x^2 + 1
dx.
7.6 - 7.8. Definite Integrals 2
Exercise 75. Find the area of the region enclosed by the parabolas x = 2y − y^2 , x =
y^2 − 4 y.
Exercise 76. Find the area of the region enclosed by the curve y^2 = x^2 − x^4.
Exercise 77. Find the area of the region enclosed by y =
x
, y = x and y =
x, x > 0.
Exercise 78. Find the number b such that the line y = b divides the region bounded by
the curves y = x 2 and y = 4 into two regions with equal area.
Exercise 79. Find the volume of the solid obtained by rotating the region bounded by
the given curves about the specified line.
a) y = 2x − x^2 , y = 0; about the x-axis.
b) y = ln x, y = 1, y = 2, x = 0; about
the y-axis.
c) x = y^2 , x = 1; about x = 1.
d) y = x 2 , x = y 2 ; about y = − 1.
Exercise 80. Find the volume of the solid generated by revolving the region bounded on
the left by the parabola x = y^2 + 1 and on the right by the line x = 5 about
a) the x-axis.
b) the y-axis.
c) the line x = 5.
Exercise 81. Find the length of the curves
a) y =
x^2
8
− ln x, 4 ≤ x ≤ 8.
b) x = y^2 /^3 , 1 ≤ y ≤ 8.
c) x = 5 cos t − cos 5t, y = 5 sin t − sin 5t, 0 ≤ t ≤ π/ 2.
Exercise 82. Find the area of the surface generated by revolving the curve
a) y =
x^2 + 2, 0 ≤ x ≤
2 , about the x-axis.
b) y =
x^2 −
ln x, 1 ≤ x ≤ 2 , about the y-axis.
7.9 - 7.10. Definite Integrals 3
Exercise 83. Determine whether each integral is convergent or divergent. Evaluate those
that are convergent.
a)
R ∞
0
x
(x^2 + 2)
2 dx.
b)
R ∞
1
x + 2
x^2 + 3x
dx.
c)
R ∞
−∞
x
x^2 + 1
dx.
d)
R 0
−∞ xe
−xdx.
e)
R 1
0
ln x √ x
dx.
f)
R ∞
0
ex
e^2 x^ + 3
dx.
g)
R ∞
0
x arctan x
(1 + x^2 )
2 dx.
h)
R ∞
1
x + 1 √ x^4 − x
dx.
i)
R 1
0
p x(1 − x)
dx.
Exercise 84. Determine whether the improper integral is convergent or divergent.
a)
R ∞
e
x(ln x)p^
dx.
b)
R ∞
1
dx √ x + x^3
c)
R ∞
1
sin x
x^2 + x + 1
dx.
d)
R 1
0
xdx √ 1 − x^4
e)
R 1
0
dx
x − sin x
f)
R ∞
0
x^3 + 1 − x
dx.
g)
R ∞
0
sin x
x
dx.
h)
R ∞
0
cos x − cos 3x
x^2 ln(1 +
x)
dx.
Exercise 85. Find the value of the constant C for which the integral
R ∞
0
x
x^2 + 1
C
3 x + 1
dx
converges. Evaluate the integral for this value of C.
Exercise 86. Suppose f is continuous on [0, ∞) and lim x→∞
f (x) = 1. Is it possible that
R (^) ∞
0 f (x)dx is convergent?
8.1 - 8.3. Functions of Several Variables 1
Exercise 87. Find and sketch the domain of the function.
a) f (x, y) =
1 − x^2 −
p 4 − y^2.
b) f (x, y) = arcsin (x 2
c) f (x, y) =
p y − x^2
x^2 − 1
d) f (x, y) =
x − y ln(x + y).
Exercise 88. Find the domain and range of the function f (x, y) =
p 4 − x^2 − y^2.
Exercise 95. Show that the function u = sin x cosh y + cos x sinh y is a solution of
Laplace’s equation
∂^2 u
∂x^2
∂^2 u
∂y^2
Exercise 96. Let f (x, y) =
x^3 y − xy^3
x^2 + y^2
, if (x, y) ̸= (0, 0),
0 , if (x, y) = (0, 0)
a) Find fx(x, y) and fy(x, y).
b) Show that fxy(0, 0) = − 1 and fyx(0, 0) = 1.
Exercise 97. Find the linear approximation of the function f (x, y) =
p 20 − x^2 − 7 y^3 at
(2, 1) and use it to approximate f (1. 98 , 1 .05).
Exercise 98. Find the differential of the function.
a) z = x^2 ln (x + y^2 ).
b) z = arctan
y
x
c) z = xyexz^.
d) z = xy + sinh(xy).
8.7 - 8.9. Functions of Several Variables 3
Exercise 99. Find the d^2 f (x, y).
a) f (x, y) = x^2 y + y^2 x, (x, y) = (1, 1).
b) f (x, y) = sin(xy)ex, (x, y) = (0, 1).
c) f (x, y, z) = x^2 + y^3 + z^4 , (x, y) = (− 1 , 0 , 1).
d) f (x, y, z) = ln(1 + xyz), (x, y, z) = (0, 0 , 0).
Exercise 100. Use the Chain Rule to find dz/dt.
a) z =
p 1 + x^4 + y^2 , x = ln t, y = sin t.
b) z = cos(x + 2y), x = 3t 2 , y = 1/t.
c) z = xy + yz + xz, x = sin t, y = cos t, z = tan t.
d) z =
x + y
y − z
, x = t^2 , y = t^3 , z = t^4.
Exercise 101. Use the Chain Rule to find
∂z
∂r
∂z
∂s
a) z = cos(1 + xy), x = r^2 , y = s^2.
b) z = x 2 y 3 , x = r cos s, y = r sin s.
c) z = ex^ sin y, x = rs, y = r + s.
d) z = tan
x
y
, x = r^2 + s^2 , y = 2rs.
Exercise 102. Use the Chain Rule to show that if z = f (x, y) and x = r cos θ, y = r sin θ
then
∂ 2 z
∂x^2
2 z
∂y^2
2 z
∂r^2
r^2
2 z
∂θ^2
r
∂z
∂r
Exercise 103. Find the gradient of f.
a) f (x, y) = ex^ sin y.
b) f (x, y) = arctan(xy).
c) f (x, y) = y^2 exy.
d) f (x, y, z) = xe y
Exercise 104. Find the directional derivative of f in the given direction v.
a) f (x, y) = ey^ cos x, v = (1, 1).
b) f (x, y) =
x^2
x^2 + y^2
, v = (− 1 , 1).
c) f (x, y) = sin(x^2 + y^2 ), v = (0, 2).
d) f (x, y) =
p x^2 + y^2 , v = (1, −1).
Exercise 105. Use implicit differentiation to find
∂z
∂x
and
∂z
∂y
a) xy = ln (y + z^2 ).
b) x − z = arctan(yz).
c) sin(xyz) = x + 2y + 3z^3.
d) x^3 + y^2 + z^3 + 6xyz = 1.
e) 2 x 2 y + 4y 2
f) xyz = arcsin(x + y + z).
