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Файл:Lectures on differential calculus of functions of one variable. Textbook
.pdf
24. Application of differential calculus to the study of functions 211
3. Intersection points with coordinate axes. Since f(0) = −2,
the graph intersects the OY axis at a single point (0, −2). To find the intersection points with the OX axis, we solve the quadratic equation x2−3x−2 = 0:
x
1,2
=
3 ±
√
9 + 8
2
=
3 ±√17
2
.
Using the approximate value of 4.1 for√17, we get x1≈
3−4.1
2
= −0.55,
x2≈
3+4.1
2
= 3.55. Thus, the graph intersects the OX axis at points whose
approximate coordinates are (−0.6, 0), (3.6, 0).
4. Critical points. Find the first derivative of this function:
f0(x) =
x2− 3x − 2
x + 1
0
=
(2x − 3)(x + 1) − (x2− 3x − 2) ·1
(x + 1)
2
=
=
x2+ 2x − 1
(x + 1)
2
.
Now we can find the critical points of the function f , i. e., points at
which the derivative f0vanishes. To do this, solve the quadratic equation
x2+ 2x − 1 = 0:
x
∗
1,2
=
−2 ±√4 + 4
2
= −1 ±√2.
Using the approximate value of 1.4 for√2, we get x
∗
1
≈ −1 − 1.4 = −2.4,
x
∗
2
≈ −1 + 1.4 = 0.4.
5. Intervals of monotonicity and local extrema. Since in the
formula for the derivative f0(x) the denominator (x+1)2is non-negative, and
the numerator x2+ 2x −1 takes positive values for x ∈ (−∞, x
∗
1
) ∪(x
∗
2
, +∞)
and negative values for x ∈ (x
∗
1
, x
∗
2
), we obtain, by virtue of the first sufficient
condition for the existence of a local extremum, that the point x
∗
1
is a local
maximum point (the sign of the derivative changes from “+” to “−” in a
neighborhood of this point), and the point x
∗
2
is a local minimum point (sign
the derivative changes from “−” to “+” in a neighborhood of this point).
Let us also calculate the values of the function f at the points of local
extrema:
f(x
∗
1
) =
(−1 −√2)2− 3(−1 −√2) −2
(−1 −√2) + 1
=
=
1 + 2√2 + 2 + 3 + 3√2 −2
−√2
= −
4 + 5√2
√
2
=
= −2√2 −5 ≈ −2 · 1.4 −5 = −7.8,
212 M. E. Abramyan. Lectures on differential calculus
f(x
∗
2
) =
(−1 +√2)2− 3(−1 +√2) −2
(−1 +√2) + 1
=
=
1 −2√2 + 2 + 3 − 3√2 −2
√
2
=
4 −5√2
√
2
=
= 2√2 −5 ≈ 2 · 1.4 −5 = −2.2.
So the coordinates of the local maximum and local minimum of the func-
tion f are approximately equal to (−2.4, −7.8) and (0.4, −2.2).
Fig. 16. Graph of the function f (x) =
x2−3x−2
x+1
6. Intervals of convexity and inflection points. To find the
intervals of convexity and inflection points of the function f , we find its second
derivative:
24. Application of differential calculus to the study of functions 213
f00(x) =f0(x)
0
=
x2+ 2x − 1
(x + 1)
2
0
=
=
(2x + 2)(x + 1)2− (x2+ 2x − 1) · 2(x + 1)
(x + 1)
4
=
=
2(x + 1)2− 2(x2+ 2x − 1)
(x + 1)
3
=
4
(x + 1)
3
.
Thus, f00(x) < 0 for x ∈ (−∞, −1) and f00(x) > 0 for x ∈ (−1, +∞). By
virtue of the sufficient condition for convexity, we obtain that the function
f is convex upward on the interval (−∞, −1) and the function is convex
downward on the interval x ∈ (−1, +∞). The function f does not have
inflection points.
The graph of the function f is shown in Fig. 16. The figure also shows the
asymptotes x = −1, y = x−4 and points of the local extremum (−2.4, −7.8),
(0.4, −2.2).
References
1. Abramyan A. V. Continuous mathematics: theory and practice. Limit
of a sequence and limit of a function, continuous and differentiable functions. (In Russian.) – Rostov-on-Don, Taganrog: SFedU Press, 2018. –
253 p.
2. Abramyan A. V. Continuous mathematics: theory and practice. Limit
of a sequence and limit of a function, continuous and differentiable functions. – Rostov-on-Don, Taganrog: SFedU Press, 2020. – 253 p.
3. Bartle R. G., Sherbert D. R. Introduction to real analysis. – NY: John
Wiley & Sons, Inc., 2000. – 388 p.
4. Demidovich B. N. Collection of problems and exercises in mathematical
analysis. (In Russian.) – M.: Moscow University Press, 1997. – 624 p.
5. Fichtengolts G.M. Course of differential and integral calculus. Vol. 1.
(In Russian.) – M: Fizmatlit Publ., 2001. – 616 p.
6. Ilyin V. A., Poznyak E. G. Fundamentals of Mathematical Analysis.
Part 1. (In Russian.) – M .: Fizmatlit Publ., 2005. – 648 p.
7. Ilyin V. A., Sadovnichy V.A., Sendov B. Kh. Mathematical analysis.
Part 1. (In Russian.) – M.: Moscow University Press, 1985. – 662 p.
8. Konev V. V. Limits of sequences and functions. Textbook. –
Tomsk: TPU Press, 2009. – 100 p. – URL: http://portal.tpu.ru:
7777/SHARED/k/KONVAL/Textbooks/Tab1/Konev-Limits_of_Sequen
ces_and_Functions_Textbook.pdf (date of the access 31.01.2020).
9. Konev V.V. Higher mathematics. Part 2. Textbook. – Tomsk: TPU
Press, 2009. – 134 p. – URL: http://portal.tpu.ru:7777/SHARED/k/
KONVAL/Textbooks/Tab1/Konev-Higher_Mathematics_TextBook.pdf
(date of the access 31.01.2020).
10. Kozak A.V. Lectures on mathematical analysis. Semester 1. (In Rus-
sian.) – Rostov-on-Don: SFedU Press, 2015. – 106 p.
References 215
11. Kudryavtsev L.D. Course of mathematical analysis. Vol. 1. (In Rus-
sian.) – M.: Drofa Publ., 2003. – 704 p.
12. Nikolsky S.M. The course of mathematical analysis. Vol. 1. (In Rus-
sian.) – M.: Nauka Publ., 1983. – 464 p.
13. Pilidi V. S. Mathematical analysis. (In Russian.) – Rostov-on-Don:
Phoenix Publ., 2008. – 240 p.
14. Piskunov N. S. Differential and integral calculus. – M.: Mir Publ.,
1969. – 895 p.
15. Ter-Krikorov A. M., Shabunin M. I. The course of mathematical analy-
sis. (In Russian.) – M.: Nauka Publ., 1988. – 816 p.
16. Zorich V. A. Mathematical analysis. Part 1. (In Russian.) – M .: Nauka
Publ., 1997. – 568 p.
Index
Absolute value of a number, 17
Areacosine, 148
Areasine, 148
Areatangent, 148
Arithmetic mean and geometric mean, 51
Asymptote, 208
Axiom of continuity, 20
Bernoulli’s inequality, 49
Big-O, 123
Binomial formula, 174
Bolzano–Cauchy theorem on the limit
point, 56
Bolzano–Weierstrass theorem on a
convergent subsequence, 59
corollary, 61
Cantor’s theorem on uniform
continuity, 109
Cauchy criterion for sequence
convergence, 62
Cauchy criterion for the existence of the
limit of a function, 89
Cauchy sequence, 62
Cauchy’s mean value theorem, 170
Critical point (a point suspected for a local
extremum), 194
Derivative, 129
geometric sense, 154
of order n, 155
of the second order, 155
physical sense, 152
Differential, 131
arithmetic properties, 138
of the independent variable, 132
Discontinuity point, 112
of removable discontinuity, 112
of the first kind or of jump
discontinuity, 113
of the second kind or of essential
discontinuity, 113
Fermat’s theorem on interior local
extrema, 164
Function
basic notions, 18
bounded, 87
bounded in comparison with the other
function, 123
continuous at a point, 92
continuous on a set, 101
convex upwards (concave) and convex
downwards (convex), 197, 199
decreasing, 87
differentiable at a point, 128
differentiable on a set, 155
equivalent to the other
function, 83, 125
increasing, 87
infinite limit, 73
infinitely large of a higher order, 122
infinitesimal in comparison with
the other function, 121
infinitesimal of a higher order, 122
left-hand limit, 78
limit, 66
limit at the point at infinity, 72
lower-bounded or bounded from
below, 87
monotonic, 87
non-decreasing, 87
non-increasing, 87
one-sided limit, 78
right-hand limit, 78
strictly monotonic, 87
uniformly continuous on a set, 108
upper-bounded or bounded from
above, 87
Hyperbolic cosine, 146
Hyperbolic sine, 146
Hyperbolic tangent, 146
Increment of the argument and increment
of the function, 128
Index 217
Indeterminate forms, 42
Inflection point, 201
Integer part of a number, 17
Interior point of a set, 164
Isolated point of a set, 92
L’Hospital’s rule, 186
Lagrange’s theorem, 166
Leibniz rule for the differentiation of a
product, 159
Limit point of a set, 54
two-sided, 113
Little-o, 121
Local minimum, local maximum, local
extremum, 163
interior, 164
strict, 163
Maclaurin’s formula, 173
Mapping
basic notions, 18
Mathematical logic: operations
and laws, 15
Necessary condition, 35
Neighborhood, 27
of the points at infinity, 40, 72
punctured, 66
punctured symmetric, 66
right-hand and left-hand, 78
symmetric, 27, 40
Number of combinations, 158
Point suspected for inflection, 202
Points at infinity, 40
Principle of mathematical induction, 17, 49
Quantifiers, 16
Remainder term of Taylor’s formula, 175
in the Cauchy form, 178
in the Lagrange form, 178
in the Peano form, 179
Rolle’s theorem, 165
Secant line, 153
Sequence, 28
bounded, 32, 44
Cauchy, 62
convergent, 30
decreasing, 44
fundamental, 62
increasing, 44
infinite limit, 40
infinitely large, 41
infinitesimal or infinitely small, 34
limit, 29
lower-bounded or bounded from
below, 44
monotone, 44
non-decreasing, 44
non-increasing, 44
of contracting segments, 53
of nested segments, 53
strictly monotone, 44
upper-bounded or bounded from
above, 44
Set
basic notions, 15
bounded, 21
least upper bound (supremum)
and greatest lower bound
(infimum), 21, 22
lower-bounded or bounded from
below, 21
maximum and minimum element, 23
product of a set by a number, 24
sum of sets, 24
upper bound and lower bound, 21
upper-bounded or bounded from
above, 20
Subsequence, 58
Sufficient condition, 35
Summation signP, 137
Tangent line, 153
Taylor’s formula
for an arbitrary differentiable
function, 175
for the polynomial, 173
Theorem
Bolzano–Cauchy theorem on the limit
point, 56
218 M. E. Abramyan. Lectures on differential calculus
Bolzano–Weierstrass theorem on a
convergent subsequence, 59
Cantor’s theorem on uniform
continuity, 109
Cauchy criterion for sequence
convergence, 62
Cauchy criterion for the existence
of the limit of a function, 89
Cauchy’s mean value theorem, 170
convergence criterion for monotone
sequences, 45
corollary of the Bolzano–Weierstrass
theorem, 61
corollary of the intermediate value
theorem, 104
corollary of the theorem on points
of discontinuity of a monotone
function, 116
criterion for the continuity of a
monotone function, 117
criterion for the existence of a finite
limit in terms of infinitesimals, 35
criterion for the existence of a
non-vertical asymptote, 209
criterion for the existence of the limit
of a function in terms of one-sided
limits, 79
criterion for the existence of the limit
of a function in terms
of sequences, 68
Fermat’s theorem on interior local
extrema, 164
intermediate value theorem, 101
inverse function theorem, 118
L’Hospital’s rule, 186
Lagrange’s theorem, 166
Leibniz rule for the differentiation of a
product, 159
necessary condition for the existence
of a local extremum, 194
necessary condition for the existence
of an inflection point, 202
nested segments theorem, 53
on arithmetic properties of continuous
functions, 95
on arithmetic properties
of derivatives, 135
on arithmetic properties of the function
limit, 74
on arithmetic properties of the limit
of a sequence, 35
on differentiation of superposition, 138
on equivalence of functions, 125
on passing to the limit in inequalities
for functions (first theorem), 75
on passing to the limit in inequalities
for functions (second theorem), 75
on passing to the limit in inequalities
for sequences (first theorem), 37
on passing to the limit in inequalities
for sequences (second theorem), 38
on points of discontinuity of a
monotone function, 114
on properties of infinitesimals, 34
on subsequences of a convergent
sequence, 59
on Taylor’s formula with the remainder
term in the Peano form, 179
on the arithmetic property of infinitely
large sequences, 42
on the boundedness of a convergent
sequence, 33
on the continuity of a differentiable
function, 130
on the continuity of the superposition
of continuous functions, 98
on the convergence of the sequence
{(1 + 1/n)n}, 49
on the convergence of the sequence
{1/qn}, 46
on the convergence of the sequence
{
n
√
a}, 48
on the convergence of the sequence
{
n
√
n}, 48
on the convergence of the sequence
{n/qn}, 47
on the convergence of the sequence
{qn/n!}, 49
on the differentiation of inverse
function, 142
on the equivalence of the definitions
of the limit point, 55
Index 219
on the equivalence of the
differentiability and the existence
of a derivative, 129
on the equivalence of three definitions
of the limit of a function, 67
on the equivalence of two definitions
of the limit of a sequence, 29
on the exact boundaries of the product
of a set by a number (first
theorem), 25
on the exact boundaries of the product
of a set by a number (second
theorem), 26
on the existence of a maximum element
in an upper-bounded integer set, 23
on the existence of a minimum element
in a lower-bounded integer set, 23
on the existence of the greatest lower
bound, 22
on the existence of the least upper
bound, 21
on the greatest lower bound of the
sum of sets, 25
on the least upper bound of the sum of
sets, 24
on the limit of monotone
lower-bounded function, 89
on the limit of monotone
upper-bounded function, 88
on the limit of superposition
of functions, 76
on the location of a tangent in the
domain of convexity of a
function, 205
on the location of the tangent at an
inflection point, 207
on the relation between the equivalence
and O-notation, 126
on the simplest properties
of continuous functions, 94
on the uniqueness of exact
boundaries, 23
on the uniqueness of the limit of a
convergent sequence, 32
on the uniqueness of the limit of a
function, 67
on the uniqueness of the limit of an
infinitely large sequence, 41
on the use of equivalences in finding
limits, 126
Rolle’s theorem, 165
sufficient condition for convexity, 199
sufficient condition for the existence
of an inflection point (first
theorem), 203
sufficient condition for the existence
of an inflection point (second
theorem), 204
sufficient condition for the existence
of a local extremum (first
theorem), 195
sufficient condition for the existence
of a local extremum (second
theorem), 195
test of convergence for monotone
bounded sequences, 44
the first remarkable limit, 81
the second remarkable limit, 85
the superposition limit theorem in the
case when the external function
is continuous, 97
Weierstrass first theorem, 104
Weierstrass second theorem, 105
Weierstrass first theorem on functions
continuous on a segment, 104
Weierstrass second theorem on functions
continuous on a segment, 105
Educational edition
Abramyan Mikhail Eduardovich
Lectures on differential calculus
of functions of one variable
Computer typesetting by M. E. Abramyan
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