Mathematical analysis 1
(2016/2017)
Compiled by: dr Marian Gewert, doc. dr Zbigniew Skoczylas
This list of exercises∗ covers the entire course and is divided into 14 parts corresponding to individual lectures. Solve one or two items from every exercise in class, except for exercises marked as (S)
or (*). The exercises marked with the letter are simple and should be solved on your own, and the
exercises marked with the asterisk (*) are more difficult and directed at the mroe ambitious students.
List 1
1. Which of the following are logical statements? For those that are, give their logic value (true or
false):
(a) “Wrocław is the capital of Poland”; (b) “2016 is divisible by 8”;
(c) “a2 + b2 = c2 ”;
(e)
“25
(d) “A triangle with sides 3, 4, 5 is a right triangle”;
­ 32”;
(f) “∆ = b2 − 4ac”.
2. Give the negations of the following esntences:
(a) “I’m eating breakfast and I’m listening to the radio”;
(b) “A square is not a pentagon”;
(c) “he capital of Poland is Gniezno or Warsaw”;
(d) “If the function f is increasing, then the function −f is decreasing”’;
(e) “A number is divisible by 6 if and only if it is divisible both by 2 and by 3”.
3. Decide, whether the following statements are true or false:
(a) “It is not true that the function f (x) = x2 is increasing on R”;
(b) “(−1)4 4 = −1 or 2008 is an even number”;
(c) “The function g(x) = sin(x) is periodic and the function f (x) = 3x + 3−x is an odd function”;
(d) “If Peter is Tom’s father, then Tom is older than Peter”;
(e) “13579 is divisible by 9 if and only if the sum 1 + 3 + 5 + 7 + 9 is divisible by 9”.
4. Rewrite the following statements, using only quantifiers, logical connectives and relation symbols
=, ̸=, <, ¬:
(a) The function f is not increasing on the interval [a, b];
(b) The interval [p, q) contains the point x;
{
(c) The system of equations
x2 + y 2 = 4,
has no solutions;
x + y = 10
(d) The equation x7 + 3x5 + 1 = 0 has only one real solution;
(e) 2017 is a prime number.
∗
The exercises come from the books: Analiza matematyczna 1 (Definicje, twierdzenia, wzory; Przykłady i zadania; Kolokwia i egzaminy), Wstp do analizy i algebry and Algebra i analiza. Egzaminy na
ocen celującą
1
5. Decide whether the following formulas using quantifiers are true:
(a)
∨
xx = 27;
(b)
x∈R
(d)
∨
∧
x2 + 4x + 1 > 0;
(c)
x∈R
∧
xy = 0;
(e)
y∈R x∈R
∧
∧
∨
x2 + y 3 = 0;
x∈R y∈R
∧
(y ¬ x) ∨ (y > x);
(f)
x∈R y∈R
∨
∨
sin x + cos y = 0.
x∈R y∈R
6. For every pair of sets A, B ⊂ R determine A ∪ B, A ∩ B, A \ B, B \ A, Ac , B c :
(a) A = (0, 5), B = [0, 7]; (b) A = (−∞, 3), B = [−1, ∞); (c) A = {1, 2}, B = {1, 2, 3, 4}.
For which pairs A, B we have A ⊂ B?
7.(S) Write the following quadratic functions in factored form (if it exists) and draw their graphs:
1
(a) f (x) = −x2 + x;
(b) f (x) = 2x2 + 1;
(c) f (x) = x2 + x + ;
4
3
9
(d) f (x) = x2 + 2x − 3;
(e) f (x) = −2x2 − 2x + ;
(f) f (x) = −x2 − 3x − .
2
4
List 2
8. Determine and draw the natural domains of the following functions:
√
√
√
4−x
3−x
x
4
(a) f (x) = 2
; (b) f (x) = 2
; (c) f (x) = 16 − x ; (d) f (x) = √
.
x − 2x − 3
x +2
x+1
9. Directly using the definition, prove that the following functions are monotone on the indicated sets:
(a) f (x) = 3 − 2x, R;
(b) f (x) = x2 , (−∞, 0].
10. Give the formulas for the compositions f ◦f , f ◦g, g ◦f , g ◦g and determine their natural domains:
1
(a) f (x) = x − 1, g(x) = 3x + 2;
(b) f (x) = , g(x) = x2 ;
x
√
√
(c) f (x) = x, g(x) = x4 ;
(d) f (x) = |x|, g(x) = x + 1.
11. Prove that the following functions are injective on the indicateed sets:
1
(a) f (x) = x3 , R;
(b) f (x) = , (0, ∞).
x
12.(S) Using the properties of logarithms, calculate:
(a) log6 3 + log6 12;
(b) log3 18 − log3 2;
(d) 3 log2 3 · log3 4;
(e) 3 log4
(c) 9 log6
√
1
3 − log4 3 + 3 log4 2 − log4 6;
2
13. Draw the graphs of the following functions:
√
(a) y = (x + 1)4 ;
(b) y = x − 2;
(c) y =
(d) y = 2
x+1
( )x−2
;
(g) y = 5 + log2 x;
(e) y =
1
3
;
(h) y = |log 100x|;
(f)
log2 54 − log2 6
.
log2 27 − log2 9
1
;
(x + 3)2
(f) y = 4|x| ;
(i) y = log 1
14. Find the inverse functions to the following:
√
x+1
(a) f (x) =
;
(b) f (x) = 3 − 3 x + 2;
x−1
(d) f (x) = log(x + 2); (e) f (x) = x2 (x ¬ 0);
3
|x|
.
9
(c) f (x) = 2x−1 ;
(f*) f (x) = x2 sgn (x).
2
√
3
36;
List 3
y
15. The figure shows the graph y = f (x). Draw the following graphs:
(a) y = f (x) + 3;
(b) y = f (x + 1);
(c) y = −f (x);
(d) y = f (−x);
(e) y = f (x)/2;
(f) y = f (3x);
(g) y = |f (x)|;
(h) y = f (|x|).
1
−1
−1
16.(S) Based on the graph y = sin x, draw the following graphs:
x
(b) y = sin ;
2
1
(e) y = sin x − 1;
2
(a) y = sin 3x;
(d) y = 1 + sin x;
(
y = f (x)
1
x
2
)
π
(c) y = sin x +
;
4
)
(
π
.
(f) y = sin 2 x −
6
17. Draw the following graphs:
(a) y = |cos x|;
sin x ;
(b) y = sin x − 2 (c) y = |tg x| ctg x.
18. Prove the following trigonometric identities:
1 + tg α
= tg α;
1 + ctg α
x
2 tg
2 ;
(d) sin x =
2 x
1 + tg
2
(a)
1
(b) sin4 α + cos4 α = 1 − sin2 2α;
2
x
1 − tg2
2;
(e) cos x =
2 x
1 + tg
2
(c) tg α + ctg α =
2
;
sin 2α
(f) cos4 α − sin4 α = cos 2α.
For what angles α are these identities true?
19. Express the following functions using sine and cosine of multiples of α:
(a) sin2 α;
(b) cos2 α;
(c) sin4 α;
(d) cos4 α.
20.(S) Find the values of the following expressions:
( √
)
√
arc
sin
−
3/2
2
1
(a) arc sin
+ arc cos ; (b) arc ctg 1 · arc tg 1; (c)
;
2
2
arc sin 1
21. Find the domains of the functions:
(
)
1
2
(a) f (x) = arc sin(2x + 1);
(b) f (x) = arc cos x +
;
2
1
(c) f (x) = arc tg
;
(d) f (x) = arc ctg 2x .
x+1
(d) arc tg
√
3 − arc ctg
List 4
22. Determine if the following sequences are bounded from below and bounded from above:
√
√
2 + cos n
(a) an =
;
(b) an = n 2n − 1; (c) an = 1 − n;
3 − 2 sin n
√
√
1
1
1
(d) an = n + 8 − n + 3; (e*) an = 1
+ 2
+ ... + n
.
4 +1 4 +2
4 +n
23. Determine whether the following sequences are monotone from some point onwards:
n
n!
2n + 1
;
(b) an = 2
;
(c) an = n ;
(a) an =
n+2
n +1
10
n
√
1
4
(d) an = 2
;
(e) an = n
;
(f) an = n2 + 1 − n.
n − 6n + 10
2 + 3n
3
√
3.
24. Using the definition of the proper or improper limit of a sequence, prove the following equalities:
1
3−n
(a) lim
= −1;
(b) lim 2 = 0;
(c) lim 2n = ∞.
n→∞ n + 4
n→∞ n
n→∞
25. Using the theorems about the sequence limit arithmetics, calculate the following limits:
3n − 1
;
n→∞ n + 4
( 2
)30
n +2
(d) lim
;
n→∞ (n3 + 1)20
n+1
;
n→∞ 2n2 + 1
1 + 3 + . . . + (2n − 1)
(e) lim
;
n→∞
2 + 4 + . . . + 2n
n2 + 1 n! + 1
(g) lim
;
n→∞ (2n + 1)(n + 1)!
(h) lim
(a) lim
(
(b) lim
)
(√
n→∞
n2 + 4n + 1 −
√
n3 + 2n2 + 1
;
n→∞
n − 3n3
5n+1 − 4n
(f) lim n
;
n→∞ 5 − 4n+2
√
2n n + 1
√
(i) lim
.
n→∞
n3 + 1
(c) lim
)
n2 + 2n ;
26. Using the squeeze theorem, calculate the following limits:
√
2n + (−1)n
⌊nπ⌋
(a) lim
;
(b) lim
;
(c) lim n 3 + sin n;
n→∞
n→∞ n
n→∞
3n + 2
√
n
(d) lim
n→∞
√
1
3
2
+ 2 + 3;
n n
n
n
(e) lim
n→∞
3n + 2n
;
5n + 4n
(
lim
n→∞
27. Calculate the following limits involving the number e:
(
(a) lim
1+
n→∞
(
(e) lim
n→∞
1
n
)3n−2
3n + 1
3n − 1
(
;
(b) lim
n→∞
)n
(
;
(f) lim
n→∞
5n + 2
5n + 1
3n
3n + 1
)15n
;
)
1
1
1
.
+ 2
+ ... + 2
2
n +1 n +2
n +n
(
(c) lim
n→∞
)n
(
;
(g) lim
n→∞
n+4
n+3
)5−2n
1 + ln n
ln n
(
;
(c) lim
n→∞
)ln n2
;
2n + 1
5n
)n (
5n + 1
2n
(n + 1)n − (n + 2)n
.
n→∞ (n + 2)n − (n + 3)n
(h) lim
28. Using the theorem about improper limits of sequences, calculate the following limits:
n2 + 1
;
n→∞
n
(
(b) lim
(a) lim
(d) lim
(√
n→∞
n→∞
)
)
n4 − 3n3 − 2n2 − 1 ;
1 − (n + 1)!
(e) lim
;
n→∞
n! + 2
n2 + 1 − n ;
(c) lim (1 + 2n − 3n );
n→∞
(f) lim
n→∞
arc tg n
.
arc ctg n
List 5
29. Using Heine’s definition of limits, calculate the following:
1
2
(a) lim (x − 2)5 = 1;
(b) lim
= 0;
(c) lim
= ∞.
x→∞ x
x→3
x→ 2+ x − 2
30. Using the theorems about the function limit arithmetic, calculate the
√
x2 − 1
x2 − 1
x+ x
√
(a) lim 2
;
(b) lim 2
; (c) lim
;
x→1 x − x + 1
x→−1 x + 4x + 3
x
x→0+
√
(√
)
x2 − 5x + 4
x−2−2
(e) lim
; (f) lim
;
(g) lim
x2 + 1 + x ;
x→∞ x(x − 5)
x→−∞
x→6
x−6
√
2
2
tg x + 1
sin x
x2 + x + 2
(i) lim
;
(j)
lim
;
(k)
lim
;
2
x→∞
x→0 1 − cos x
x+1
x→ π2 − tg x + 5
following:
x3 − 8
;
x→2 x4 − 16
(d) lim
2x + 1
;
x→∞ 3x + 2
(
)
1
3
(l) lim
−
.
x→1 1 − x
1 − x3
(h) lim
31. Determine whether the following limits exist, by calculating the one-sided limits:
(a) lim x sgn x;
x→0
1
x
(b) lim 2 ;
x→0
x2 − 4
;
x→2 |x − 2|
(c) lim
1
(d) lim x arc tg .
x→0
x
32. Use the squeeze theorem to prove the following equalities:
√
1
1
2+sin x
(a) lim x cos 2 = 0; (b) lim x2 arc tg = 0; (c) lim
= 0.
+
x→∞
x→0
x
x
x2
x→0
4
)n
;
33. Using the limits of the basic indeterminate forms, calculate the following:
x
sin2
sin(x − 4)
arc sin 2x
2;
(a) lim
(b) lim √
;
(c) lim
;
2
x→4
x→0 arc tg x
x→0 x
x−2
e3x − 1
;
x→0 sin 2x
xπ − xe
(i) lim
;
x→1 x − 1
√
√
1
3
1+x− 61−x
x
ctg x
(j) lim (1 + 2x) ;
(k) lim [1 + tg(2x)]
;
(l) lim
;
x→0
x→0
x→0
x
34. Find the vertical and oblique asymptotes of the following functions:
2x3
x−3
x3 + x2
;
(b) f (x) =
;
(c) f (x) = √
(a) f (x) = 2
;
2
x −4
(x + 1)
x2 − 9
√
1 + x2
2x2 + sin x
3x
;
(f)
f
(x)
=
(d) f (x) =
;
(e) f (x) = x
;
x
3 − 2x
x
√
cos x
(x + 1) x − 2
(g) f (x) = x
;
(h) f (x) = x − arc tg x;
(i) f (x) =
.
e −1
x−1
1
(d) lim x2 arc tg ;
x→∞
x
√
ln (1 + 3 x)
(g) lim
;
x→0
x
cos 5x
;
x→ 2 cos 3x
( 2
)
ln x − 3
(h) lim
;
x→−2
x+2
(e) limπ
(f) lim
List 6
35. Choose the parameters a, b ∈ R to make the following functions continuous on R:

 2
a

−1
for x < 0,



 +1 for x < −1,
 ax +1 for x < −1,
for −1 ¬ x ¬ 0,
(a) f(x) = a+b sin x for 0 ¬ x ¬ π/2, (b) f(x) = x
(c) f(x) = 2x

 b − 2x for x ­ −1;


 3
1
x +bx for x > 0.
for x > π/2;
Draw the graph of the function in (a).
36. Find the discontinuities of the following functions and determine their type:

x+2




1

 x2 + x + 2 for x ̸= 1, 2
arc tg for x ̸= 0,
(b) f (x) =
(a) f (x) = 0
x
for x = 1,
0


for x = 0;

1
for x = 2;




1
for x ̸= 0,
2
(c) f (x) = ln (x ) − ln (x2 + 1)

0
for x = 0;
(d) f (x) =

1 − cos
0
1
for x ̸= 0,
x
for x = 0.
37. Prove that the following equations have unique solutions in the indicated intervals:
[
]
[
]
5π
π
(a) x3 + 6x − 2 = 0, [0, 1];
(b) x sin x = 7, 2π,
;
(c) 2 − 2x = sin x, 0,
;
2
2
[
]
1
(d) x100 + x − 1 = 0,
,1 ;
(e) 3x + x = 3, [0, 1];
(f) x2x = 1, [0, 1].
2
Find the solution of (a) to the nearest 0.125.
38. Directly using the definition, calculate the derivatives of the following functions:
1
(a) f (x) = x4 (x ∈ R);
(b) f (x) = 3 (x ̸= 0);
x
√
(c) f (x) = x (x > 0);
(d) f (x) = cos x (x ∈ R).
List 7
39. Calculate the one-sided derivatives to determine, whether the following functions are differentiable
at the indicated points:
5
{
}
(a) f (x) = x2 − x , x0 = 1; (b) f (x) = sin x · sgn (x), x0 = 0; (c) f (x) = min x2 , 1 , x0 = −1.
Draw the graphs of these functions.
40. Directly using the definitions, decide whether the following functions have improper derivatives
at x0 = 0:
√
√
(a) f (x) = 3 − 5 x;
(b) f (x) = | sin x|.
41. Assuming that f and g have proper derivatives on some interval, calculate the derivatives of the
following functions:
( )
( )
1
f x2
(a) y = xf
;
(b) y =
;
(c) y = e−x f (ex );
x
x
√
(d) y = f (x) cos g(x);
(g) y = ln
f (x)
;
g(x)
(e) y =
f 2 (x) − g 2 (x);
(h) y = tg
f (x)
;
g(x)
(f) y = arc tg [f (x)g(x)];
( )
(i) y = f (x)g
1
.
x
42. Using the rules of differentiation, calculate the derivatives of the following functions:
x2 + 1
ex+1
(a) y =
;
(b) y = 3 cos x + tg x;
(c) y =
;
sin x
(x − 1
)
√ ) √
(
1
(d) y = x3 + 2 ex ;
(e) y = 1 + 4 x tg x;
(f) y = ex arc tg x;
x
(
√
)
(g) y = ln sin2 x + 1 ;
(h) y =
2
(j) y =
2sin x
;
3cos2 x
3
(
arc sin (x2 );
2
)3
(k) y = ex + 1 ;
x
(i) y = ee ;
(l) y = (sin x)x (0 < x < π).
(
43.* Using the theorem about the derivative of the inverse function, calculate f −1
(a) f (x) = x + ln x, y0 = e + 1;
√
√
√
(c) f (x) = 3 x + 5 x + 7 x, y0 = 3;
)′
(y0 ), if:
(b) f (x) = cos x − 3x, y0 = 1;
(d) f (x) = x3 + 3x , y0 = 4.
44.(S) Determine the equations of the tangents to the graphs of the following functions at the indicated points:
(
( ))
(
)
π
π
x
,f
;
(a) f (x) = arc sin , (1, f (1)); (b) f (x) = ln x2 + e , (0, f (0)); (c) f (x) = etg x ,
2
4
4
(√ ))
(√
√
1
2x
(d) f (x) = 2x + 1, (3, f (3)); (e) f (x) =
,
2, f
2 ; (f) f (x) = e1+ x , (x0 , 1) .
2
1+x
45. (a) Find the tangent to the graph of f (x) = x4 − 2x + 5, which is parallel to the line y = 2x + 3.
√
π
(b) Find the tangent to the graph of f (x) = x, which forms the angle with the positive x-axis.
4
(c) Find the tangent to the graph of f (x) = x ln x, which is perpendicular to the line 2x + 6y − 1 = 0.
1
(d) Find the tangent to the graph of f (x) = x arc tg at the point of intersection between the graph
x
and the line πx = 4y.
(e) Find the tangent to the graph of f (x) = sin 2x − cos 3x at the point of intersection between the
graph and the y-axis.
46. Using L’Hospital’s rule, calculate the following limits:
6
π
ln sin x
2 ;
(b) lim
x→1
ln x
ln cos x
(e) lim
;
x→0 ln cos 3x
x
(h) lim (π − x) tg ;
−
2 )
x→π
(
1
1
(k) lim
+
;
x→1 ln x
1−x
ln (2x + 1)
(a) lim
;
x→∞
x
x10 − 10x + 9
(d) lim 5
;
x→1 x − 5x + 4
(g) lim x ln x;
x→0+
(
)
1
(j) lim
− ctg x ;
x→0− x
(
)x
2
(m) lim
arc tg x ;
x→∞ π
x − arc tg x
;
x→0
x2
(c) lim
(f) lim x arc ctg x;
x→∞ (
)
1
1
(i) lim
− 2 ;
x
x→0+ 1 − cos x
(l) lim (− ln x)x ;
x→0+
(n) lim (1 + x)ln x ;
(o)
x→0+
lim (tg x)cos x .
−
x→( π2 )
List 8
47. Find the intervals of monotonicity for the following functions:
1
x4 x3
−
− x2 ;
(c) f (x) = 4x + ;
(a) f (x) = x3 − 30x2 + 225x;
(b) f (x) =
4
3
x
√
3
2
x
x −1
(d) f (x) =
;
(e) f (x) =
;
(f) f (x) = xe−3x ;
3 − x2
x
x
1
(g) f (x) = x ln2 x;
(h) f (x) =
;
(i) f (x) =
.
ln x
x ln x
48. Find the local extrema of the following functions:
1
2x
(a) f (x) = x3 − 4x2 ;
(b) f (x) = x + ;
(c) f (x) = ;
x
x
x+1
2
−x
(d) f (x) = (x + 1)e ;
(e) f (x) = 2
;
(f) f (x) = x − 5x − 6 ;
x +1
(g) f (x) = x ln x;
(h) f (x) =
√
(
3x − x3 ;
49. Find the smallest and greatest values of the following functions on the following intervals:
x+1
, [−4, 2];
(a) f (x) = 2x3 − 15x2 + 36x, [1, 5];
(b) f (x) = 2
x +1
(c) f (x) = (x − 3)2 e|x| , [−1, 4];
[
]
3
(e) f (x) = 2 sin x + sin 2x, 0, π ;
2
(d) f (x) = 1 − 9 − x2 , [−5, 1];
(f) f (x) =
1
1
+
, [−1, 2].
1 + |x| 2 + x
50.(S) Calculate f ′ , f ′′ for the following functions:
2
;
x
(a) f (x) = 4x7 − 5x3 + 2x;
(b) f (x) = x3 −
(d) f (x) = arc tg x;
(e) f (x) = sin3 x + cos3 x;
(c) f (x) =
ex
;
x
(f) f (x) = x3 ln x.
51. Find the intervals of convexity and inflection points for the following functions:
x3
(a) f (x) = x(x − 1)(x − 3);
(b) f (x) = xe−x ;
(c) f (x) = 2
;
x + 12
(
)
2
1
;
(f) f (x) = x − x3 − 4 ln |x|;
(d) f (x) = ln 1 + x2 ;
(e) f (x) =
2
1−x
3
1
ln
x
(g) f (x) = sin x + sin 2x;
(h) f (x) = earc tg x ;
(i) f (x) = √ .
8
x
List 9
52. Investigate the following functions and draw their graphs:
7
)
(i) f (x) = 2 arc tg x − ln 1 + x2 .
√
x
(c) f (x) =
;
x−1
x
(f) f (x) =
;
ln x
(i) f (x) = x2 ln x.
x3
(b) f (x) =
;
x−1
(a) f (x) = (x − 1) (x + 2);
2
√
4
4
− 2;
x x
2x
(g) f (x) = xe ;
(d) f (x) = 3 −
(e) f (x) = x 1 − x2 ;
(h*) f (x) = sin x + sin 3x;
53. An oil platform is anchored at sea, 10 kilometers away from the shore which forms a straight line.
The oil from the platform will be transported by pipeline to the refinery, which lies at the shore, 16
kilometers from the point closest to the platform. The cost of constructing 1 kilometer of pipeline on
sea floor is 200,000 euros, and on land – 100,000 euro. To what point on the shore should the pipeline
lead so that the cost is the smallest?
b
Platforma
wiertnicza
10 km
b
x
b
b
Rafineria
16 km
54. A container has the shape of a cuboid, with volume 22.50 m3 and a square floor. 1 m2 of the metal
needed to make the floor and the lid costs 20 zł, and for the sides – 30 zł. What dimensions should
the container have to make its cost the smallest?
55. Let α denote the angle at the vertex of the isosceles triangle with a given area. What value of α
gives the greatest radius r of the circle inscribed in the triangle?
α
r
56. Choose the dimensions a, b of a rectangular area of size S, naturally bounded on one side by the
straight bank of the river, so that the least amount of fence is needed to surround the area?
rzeka
S
a
b
57. A rectangle has been inscribed in the parabola y = 4 − x2 , as shown on the figure. Find the
dimensions of this rectangle for which it has maximum area.
y
y = 4 − x2
x
8
List 10
58. Write the Taylor expansions with Lagrange remainder for the following functions f , points x0 and
n:
1
(a) f (x) = x3 , x0 = −1, n = 4;
(b) f (x) = 2 , x0 = 1, n = 2;
x
(c) f (x) = sin 2x, x0 = π, n = 3; (d) f (x) = e−x , x0 = 0, n = 5.
59. Write the Maclaurin expansions with the n-th Lagrange remainder for the following functions:
x
x
(a) f (x) = sin ;
(b) f (x) = cosh x;
(c) f (x) = cos x;
(d) f (x) = x .
3
e
60. Estimate the accuracy of the following approximations on the indicated intervals:
π
(a) tg x ≈ x, |x| ¬ ;
(b) cos2 x ≈ 1 − x2 , |x| ¬ 0.1;
12
√
x x2
x2
x3
(c) 1 + x ≈ 1 + − , |x| ¬ 0.25;
(d) ln(1 − x) ≈ −x −
− , |x| < 0.1.
2
8
2
3
61. Use the Maclaurin formula to calculate:
√
1
3
(a) with accuracy 10−3 ;
(b) 0.997 with accuracy 10−3 ;
e
(c) ln 1.1 with accuracy 10−4 ;
(d) sin 0.1 with accuracy 10−5 .
List 11
62. Using the uniform partition in the definition of Riemann integral, calculate:
∫1
(a)
∫3
(2x − 1) dx;
x2 dx.
(b)
−2
2
Hint: Use the following identities
(a)
1 + 2 + ... + n =
n(n + 1)
,
2
(b)
1 2 + 22 + . . . + n 2 =
n(n + 1)(2n + 1)
.
6
63. Use the Newton-Leibniz theorem to calculate the following integrals:
∫2 (
(a)
√
1
x+ √
x
)
∫1
dx;
(b)
1
(c)
0
∫2
(
∫2 (
)
x 1 + x3 dx;
(d)
∫9
x−1
dx;
x+1
(e)
−1
1
2
1
−
+
x3 x2 x4
dx
;
x2 + 1
0
π/3
∫
)
dx;
tg2 x dx.
(f)
1
0
64. Use the definition of the definite integral and the fact that continuous functions are integrable to
prove the following equalities:
[
(
1
1
π
2π
nπ
13 + 23 + . . . + n3
= ; (b) lim
cos
+ cos
+ . . . + cos
(a) lim
n→∞ n
n→∞
n4
4
2n
2n
2n
[
]
(
)
(
)
√
√
√
1
2 √
√
(c) lim
1 + n + 2 + n + ... + n + n =
2 2−1 .
n→∞ n n
3
)]
=
2
;
π
65. Calculate the following indefinite integrals:
∫ (
x3 +
(a)
∫
(d)
√
4
−3 x
x
cos 2x dx
;
cos x − sin x
)
∫
dx;
(1 − x) dx
√ ;
1+ x
∫ 3 √
3
x + x2 − 1
√
(e)
dx;
x
(b)
9
∫
(c)
∫
(f)
x4 dx
;
x2 + 1
2x − 5x
dx.
10x
66.* Find the function f if:
( )
1
(a) f ′ x2 = (x > 0);
x
(b) f ′ (3x ) = 3−2x (x ∈ R);
67. Calculate the areas bounded by the curves:
1
(a) y = 2x − x2 , x + y = 0; (b) y = x2 , y = x2 , y = 3x;
2
4
(d) y = 1, y = 2
;
(e) y = 2x , y = 2, x = 0;
x +1
(g) y = πx2 , x = πy 2 ;
(c) f ′ (sin x) = sin x (x ∈ R).
1
, y = x, y = 4;
x2
(c) y =
(f) y = x + sin x, y = x, (0 ¬ x ¬ 2π);
(i) y 2 = −x, y = x − 6, y = −1, y = 4.
(h) yx4 = 1, y = 1, y = 16;
List 12
68. Using integration by parts, calculate the following indefinite integrals:
∫
(a)
xe
−3x
∫
dx;
(b)
∫
∫
x2 sin x dx;
(e)
(f)
∫
∫
(i)
e
2x
sin x dx;
(j)
∫
(x + 1) e dx;
(c)
arc cos x dx
√
;
x+1
(g)
sin x sin 3x dx;
(k)
cos4 x dx;
(o)
(d)
ln(x + 1) dx;
(h)
x dx
;
cos2 x
∫
∫
arc cos x dx;
∫
sin 3x cos x dx;
∫
(n)
∫
√
√
x arc tg x dx;
∫
∫
sin2 x dx;
(m)
∫
2 x
(
(l)
cos x cos 5x dx;
∫
)
ln 1 + x2 dx;
(p*)
x sin xex dx.
69. Using integration by parts, calculate the following definite integrals:
∫1
(a)
xe
−x
∫1
dx;
(b)
x e
−1
π
4
∫
(d)
x sin 2x dx;
∫e
2 2x
dx;
(c)
0
√
∫π
∫1
(e)
x(1 + cos x) dx;
0
(f)
0
ln x
dx;
x2
e
arc sin x dx.
0
70. Using appropriate substitutions, calculate the following indefinite integrals:
√
∫
∫ √
∫
∫
(
)
cos x
1 + 4x
cos x dx
√
√
(a)
dx; (b)
dx; (c)
;
(d) x sin x2 + 4 dx;
x
x
1 + sin x
∫
(e)
∫
(i)
∫
dx
;
ch x
(f)
ln x
dx;
x
(j)
∫
10
∫
(5−3x) dx;
(g)
ex dx
;
e2x + 1
(k)
2
x
∫
√
5
5x3 +1 dx;
5 sin x dx
;
3−2 cos x
∫
dx
√ ;
2+ x
(h)
∫
(l)
2
x3 ex dx.
71. Using the indicated substitutions, calculate the definite integrals:
∫π
(c)
∫3
sin xecos x dx, cos x = t;
(a)
(b)
0
1
∫1 √
∫e
x 1 + x dx,
√
1 + x = t;
(d)
0
∫
0
ln x dx, ln x = t;
1
1
4
(e)
x dx
√
, 1 + x = t;
x+1
dx
√
, x = t2 ;
x(1 − x)
(f)
∫3 √
1
2
9 − x2 dx, x = 3 sin t;
0
∫ln 3
(g)
0
10
ex dx
, ex = t.
1 + e2x
72.(S) Calculate the following integrals of partial fractions:
∫
(a)
dx
;
(x − 3)7
∫
(b)
∫
dx
;
x+5
(c)
∫
5 dx
;
(2 − 7x)3
8 dx
.
9x + 20
(d)
List 13
73. Calculate the following integrals of partial fractions:
∫
(a)
dx
;
2
x + 4x + 29
∫
(6x + 3) dx
;
x2 + x + 4
(b)
∫
∫
(4x + 2) dx
;
2
x − 10x + 29
(c)
(d)
(x − 1) dx
.
9x2 + 6x + 2
74. Calculate the following integrals of rational functions:
∫
(a)
∫
(e)
∫
(i)
∫
(x + 2) dx
;
x(x − 2)
(b)
dx
;
2
(x + 1) (x2 + 4)
(f)
(5 − 4x) dx
;
x2 − 4x + 20
(j)
∫
∫
∫
x2 dx
;
x+1
(c)
(4x + 1) dx
;
2x2 + x + 1
(g)
x2 dx
;
x2 + 2x + 5
(k)
∫
∫
∫
dx
;
(x − 1)x2
(d)
2 dx
;
2
x + 6x + 18
(h)
dx
;
x (x2 + 4)
(l)
∫
∫
x4 dx
;
x2 − 9
dx
;
x (x2 − 4)
x dx
.
x4 − 1
75. Calculate the following integrals of trigonometric functions:
∫
∫
3
(a)
sin x dx;
(b)
∫
3
sin x cos x dx;
6
sin x cos x dx;
(e)
cos4 x dx;
(c)
∫
3
(d)
∫
4
∫
2
cos x cos 2x dx;
(f*)
sin2 2x sin2 x dx.
76. Calculate the following integrals of trigonometric functions:
∫
(a)
∫
(d)
∫
(g)
dx
;
sin x + tg x
∫
(b)
∫
sin2 x dx
;
1 + cos x
(e)
dx
;
cos x
(h)
∫
∫
1 + tg x
dx;
cos x
(c)
dx
;
1 − tg x
(f)
dx
;
sin x + cos x
(i)
∫
∫
dx
;
1 + 2 cos2 x
sin5 x dx
;
cos3 x
dx
.
3 sin x + 4 cos x + 5
List 14
77. Calculate the areas bounded by the curves:
√
√
(a) x + y = 1, x = 0, y = 0;
(b) 4y = x2 , y =
(d) y = tg x, y = ctg x (0 < x < π/2);
8
;
+4
√
(e) y = 9 − x2 , y = 1, y = 2;
x2
78. Calculate the lengths of the following curves:
ex + 1
, 2 ¬ x ¬ 3;
(b) y = x2 , 0 ¬ x ¬ 1;
(a) y = ln x
e −1
1
x5
1
1
ln 2 ¬ x ¬ ln 3; (g) y =
+ 3 , 1 ¬ x ¬ 2;
(e) y = ex ,
2
2
10 6x
(c) y = ln x, x = e, y = −1;
(f) y = 2x , y = 4x , y = 16.
√
(c) y = 2 x3 , 0 ¬ x ¬ 11;
(h) y = 1 − ln cos x, 0 ¬ x ¬
π
.
4
79. Calculate the volumes of the solids obtained by rotating the figure T around the indicated axis:
√
2
(a) T : 0 ¬ x ¬ 2, 0 ¬ y ¬ 2x − x2 , Ox; (b) T : 0 ¬ x ¬ 5, 0 ¬ y ¬ √
, Oy;
2
x +4
√
π
(c) T : 0 ¬ x ¬ , 0 ¬ y ¬ tg x, Ox;
(d) T : 0 ¬ x ¬ 1, x2 ¬ y ¬ x, Oy;
4
11
1
, Oy;
x
(e) T : 0 ¬ x ¬ 1, 0 ¬ y ¬ x3 , Oy;
(f) T : 1 ¬ x ¬ 3, 0 ¬ y ¬
4
¬ y ¬ 5−x, Ox;
x
(i) T : 0 ¬ x ¬ π, 0 ¬ y ¬ sin x, y = 2;
π
(h) T : 0 ¬ x ¬ , 0 ¬ y ¬ sin x+cos x, Ox;
2
(j) T : 0 ¬ x ¬ 1, 0 ¬ y ¬ x − x2 , x = 2.
(g) T : 1 ¬ x ¬ 4,
80. Calculate the areas of the surfaces obtained by rotating the graph of the functions f around the
indicated axis:
√
π
(a) f (x) = cos x, 0 ¬ x ¬ , Ox;
(b) f (x) = 4 + x, −4 ¬ x ¬ 2, Ox;
√2
(c) f (x) = ln x, 1 ¬ x ¬ 3, Oy;
(d) f (x) = |x − 1| + 1, 0 ¬ x ¬ 2, Oy;
(
)
√
√
x
(e) f (x) = 4 − x2 , −1 ¬ x ¬ 1, Ox; (f) f (x) = x 1−
, 1 ¬ x ¬ 3, Ox;
3
√
x−1
x2
(g) f (x) =
, 1 ¬ x ¬ 10, Oy;
(h) f (x) =
, 0 ¬ x ¬ 3, Oy.
9
2
81. (a) The force stretching the spring is directly proportional to the increase in length (the proportionality coefficient is k). Calculate the work needed to stretch a spring of length l to length L.
(b) A water tank has cylindrical shape with horizontal axis, diameter D = 2 m, and length L = 6 m.
Calculate the work needed to empty the completely full tank. The aperture for emptying the tank is
in its upper part. The density of water is γ = 1000 kg/m3 .
82. (a) A material point starts moving in a straight line with initial speed v0 = 10 m/s and acceleration
a0 = 2 m/s2 . After t1 = 10 s it switches to decelerating at a1 = −1 m/s2 . Find the position of the point
at t2 = 20 s from the beginning.
(b) Two elementary particles at distance d = 36 begin moving towards each other at speeds v1 (t) =
10t + t3 , v2 (t) = 6t, where t ­ 0. After what time will the particles collide?
12