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Higher_Mathematics_Part_1
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2) Points of intersection with the coordinate axis: if x = 0 , then y = 0 ; if y = 0 , then x = 0 or x = 2 . So, the curve passes through the points (0; 0) and (2; 0).
3)Function is neither even nor odd.
4)The points of discontinuity and vertical asymptotes do not exist.
5) Let’s find the derivative y′ = |
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3x 2 − 4x |
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x(x −4 / 3) |
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3 3 (x3 − 2x 2 )2 |
3 x4 (x −2)2 |
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Critical points are |
x = 0 ; 4/3; 2. These points break the numerical axis into |
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four intervals (−∞;0) , |
(0; 4 / 3) , (4 / 3; 2) , (2; ∞) . On each of these intervals the |
derivative y′ has constant sign, so: if x (−∞;0) , then y′ > 0 and the function increases; if x (0; 4) , then y′ < 0 and the function decreases; if x ( 4 ;2) (2; ∞) ,
3 |
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then y′ > 0 and the function |
increases. During transition through the point |
x = 0 derivative changes the sign from plus to minus, so, x = 0 is the point of
maximum, and ymax = y(0) = 0 . |
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x = 4 / 3 derivative |
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During transition through the point |
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from minus to plus, so, at this point it is minimum: |
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ymin = y(4 3) = 3 (4 3)2 (4 3 − 2) |
= − 3 32 27 = − 23 4 |
>> – 1.1. |
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During transition through the point |
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sign, so, this point is not the point of extremum. |
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6) Let’s find the second derivative |
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x(x − 4 3) |
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x − 4 3 |
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y′′ = |
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x(x − 2) |
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3 x(x − 2)2 − (x − 4 3) (x − 2)2 + 2x(x − 2) |
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33 x2 (x − 2)4 |
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9x(x − 2)2 − (3x − 4)(x − 2)(3x − 2) = − 8 |
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(x − 2) |
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(x − 2) |
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9 3 |
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2) |
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Second derivative doesn’t exist if |
x = 0 or |
x = 2 . So, points x = 0; 2 |
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the critical points of the second type. Let’s consider the intervals (−∞; 0) , (0; 2) and (2; ∞) . On the intervals (−∞; 0) and (0; 2) y′′ > 0 and the curve is concave; if x (2; ∞) , then y′′ < 0 and the curve is convex. The inflection point has the coordinates (2; 0).
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7) Let’s find the inclined asymptote:
k = lim |
f (x) |
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x3 − 3x2 |
= 1, |
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= lim |
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x→∞ |
x→∞ |
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b = lim ( f (x) − kx) = lim (3 |
x3 − 2x 2 |
− x) = |
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x→∞ |
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x→∞ |
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= lim |
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x3 − 2x2 − x3 |
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2 |
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x→∞ 3 (x3 − 2x2 )2 + x 3 x3 − 2x2 + x2 |
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In such a way, straight line y = x −2 / 3 is inclined asymptote of our curve.
Other asymptotes don’t exist.
8) Taking into account the spent investigations, we can draw the graph of our function (Fig.3.35).
у
4/3
О |
1 |
2 |
х |
•
Fig. 3.35
Micromodule 22
CLASS AND HOME ASSIGNMENTS
Let’s find the intervals of increasing and decreasing of the functions:
1. |
y = 6−3x2 −x3 . |
2. |
y = x4 −2x2 . |
3. y = x ln x . |
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4. |
y = x2 e−x . |
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y = |
x2 +2x |
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x −1 |
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Investigate the function on extremum: |
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6. |
y = x3 −9x2 +15x −10 . |
7. |
y = x5 −5x4 +5x3 +5 . |
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8. |
y = (x −1)2 (x −2)2 . |
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9. |
y = x(x −1)2 (x +1)3 . |
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10. |
y = |
1−x + x2 |
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11. |
y = 3x + |
1 |
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12. y = |
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1+ x |
−x2 |
x3 |
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+ x2 |
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Let’s find maximum and minimum values of the functions:
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13. y = xe−x . |
14. y = x −ln(1+ x) . |
15. y = |
ln x |
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x |
Investigate the behavior of the function in neighbourhood of the giving points with the help of the derivatives of higher order:
16.y = 6ex −x3 −3x2 −6x −5 , x0 = 0 .
17.y = x sin x −x2 , x0 = 0 .
Let’s find the intervals of the concavity and convexity, and the inflection points of the curves.
18. |
y = x2 |
− 2x + 1 . |
19. |
y = x3 |
− 1 . |
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y = x3 |
− 3x2 + 9x + 6 . |
21. |
y = x2 |
ln x . |
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22. |
y = e− x2 |
. 23. y = xex . |
24. y = ln x + 2x2 . |
Let’s find the asymptotes of the curves:
25. y = |
6x4 +3x3 |
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1 |
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27. y = x e2 / x +1. |
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5x3 +1 |
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x2 −3x +2 |
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Investigate the functions and sketch the graphs:
28. |
y = |
x3 +4 |
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29. |
y =(2x+3)e−2/(x+1) . 30. |
y = 3 (x +3)x2 . |
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x2 |
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3x4 +1 |
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31. |
y = arctg(sin x) . |
32. |
y = |
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33. |
y = 3ln |
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−1 . |
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ex+1 |
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34. |
y = |
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35. |
y = |
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4x2 |
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36. |
y = |
x3 |
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x +1 |
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x2 − |
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ln x |
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37. |
y = 3 1−x2 . 38. |
y = x2 + |
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39. y =16x(x −1)3 . 40. y = |
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Answers |
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1. |
(−∞;−2) і (0; ∞) – |
decreases; |
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(−2;0) – increases. 2. |
(−∞;−1) and |
(0;1) – |
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decreases; |
(−1;0) and (1; ∞) – increases. 3. |
(0;1/ e) – decreases; |
(1/ e; ∞) – increases. |
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4. (−∞;0) and (2; ∞) – decreases; (0;2) – increases. 5. |
(−∞;1− |
3) and (1+ |
3; ∞) – |
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increases; |
(1− 3;1) and |
(1;1+ |
3) – |
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decreases. 6. |
x = 2 – |
maximum, |
x = 3 – |
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minimum. 7. |
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x = 1– maximum, |
x = 3 – minimum. 8. |
x = 1;2 – minimum, |
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x = 3 / 2 – |
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maximum. |
9. x = −4 / 3;1– minimum, x = 1/ 2 – maximum. 10. x = 1/ 2 – minimum. |
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x = −1 – maximum, x =1– minimum. 12. x = 1 – maximum. 13. ymax = y(1) = 1/ e . |
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14. |
ymin = y(0) = 0 . |
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ymin |
= y(1) = 0 , ymax = y(e2 ) = 4 / e2 .16. |
Minimum. |
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17. |
Maximum. 20. |
(−∞;1) – |
convex, |
(1; ∞) – |
concave, |
(1;13) is inflection point. |
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21. |
(0; e−3 / 2 ) – convex, (e−3 / 2 ; ∞) – concave, (e−3 / 2 ;−3 /(2e3 )) is inflection point. |
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22. |
(−∞;−1/ |
2) (1/ |
2; ∞) – concave, (−1/ |
2;1/ |
2) – convex, (±1/ |
2;e−1/ 2 ) |
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are inflection points. 24. |
(0;1/ 2) – convex, |
(1/ 2; ∞) – concave, (1/ 2;1/ 2 − ln 2) – point |
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of |
bend. 25. |
6x − 5y + 3 = 0 – inclined asymptote, |
x = −1/ 3 5 |
is a vertical asymptote. |
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26. |
x = 1;2 |
are vertical asymptotes, y = 0 |
is a |
horizontal |
asymptote. |
27. |
y = x + 3 . |
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28. |
y = x , |
x = 0 are |
asymptotes; (−∞;0) (2; ∞) – |
increases, |
(0;2) – |
decreases; |
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xmin = 2 , |
ymin = 3 ; (−∞;0) (0; ∞) – concave. 29. Designated everywhere, except |
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x = −1; extremum |
doesn’t |
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function increases; |
(−2;−e2 ) – inflection point; |
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x = −1– asymptote. 32. Odd; (−∞;−1) (1; ∞) – increases, |
(−1;0) (0;1) – decreases; |
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ymax = −4 |
if x = −1, |
ymin |
= 4 |
if x = 1 ; y = 3x , |
x = 0 – asymptotes; |
(−∞;0) – |
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convex, (0; ∞) – concave. 34. Designated |
everywhere, except |
x = −1 ; |
ymin = e |
if |
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x = 0 ; |
x = −1– |
asymptote; |
(−∞;−1) – |
concave down, |
(−1; ∞) – |
concave |
up. |
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35. |
Even; |
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ymax = 0 |
if x = 0 ; |
y = 4 – |
asymptote; |
(±1; 1) |
is inflection point; |
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(−∞;−1) (1; ∞) – |
convex, |
(−1;1) – |
concave. |
39. |
ymin |
= −27 /16 |
if |
x = 1/ 4 ; |
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(−∞;1/ 2) (1; ∞) – concave, (1/ 2;1) – convex; |
(1/ 2;−1) , |
(1;0) are inflection points; |
asymptotes don’t exist.
Micromodule 22
SELF-TEST ASSIGNMENTS
22.1. Find the intervals of increasing and decreasing of the functions:
22.1.1. y = x2 − ln x2 . |
22.1.2. y = |
x − 1(x − 2) . |
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22.1.3. y = (x − 1)2 (x + 1)3 . |
22.1.4. y = arcsin(1+ x) . |
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22.1.5. y = |
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22.1.6. y = |
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22.1.7. y = |
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22.1.8. y = ln x − arctgx . |
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22.1.9. y = |
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22.1.10. y = x4 ln x . |
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22.1.11. y = ln(x2 + 1) − x .
22.1.13. y = 9− x − 3− x .
22.1.15. |
y = |
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22.1.17. y = x2 ln x . 22.1.19. y = x2 e− x2 .
22.1.21.y = x2 + 3x − 4 .
x− 5
22.1.23. y = |
2x − x2 . |
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22.1.25. y = |
x2 − 3x + 2 |
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22.1.27.y = x2 − 7x + 6 .
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22.1.29. y = e− x − e−2x .
22.1.12. y = x3 . ex
22.1.14. y = x ln3 x .
22.1.16. y = (x − 5)3 (x + 4)2 .
22.1.18. y = x ln2 x .
22.1.20. |
y = 2x − 4x . |
22.1.22. |
y = (x − 4)3 (x + 5)2 . |
22.1.24. |
y = x ln x . |
22.1.26. y = ln |
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22.1.30. y = x + |
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22.2. Find the intervals of the concavity and convexity, and the inflection points of the curves:
22.2.1. |
y = x2 |
x + 1 . |
22.2.2. y = |
x2 + 1 . |
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22.2.3. |
y = (x − 4)4 (x + 7)3 . |
22.2.4. y = xx . |
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22.2.5. |
y = ln(1+ x2 ) . |
22.2.6. y = x + sin x . |
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22.2.7. |
y = 3x2 − x3 . |
22.2.8. y = xe− x . |
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22.2.9. y = x + x5 / 3 . |
22.2.10. y = 3x2 − 4x |
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22.2.11. y = ln(x4 + 1) . |
22.2.12. y = ln x + ln2 x . |
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22.2.13. y = e− x4 . |
22.2.14. y = x4 + 8x3 + 18x2 + 8 . |
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22.2.15. y = (x − 1)2 x . |
22.2.16. y = x3 − 3x2 + 6x + 7 . |
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22.2.17. y = |
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22.2.18. y = |
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22.2.19. y = |
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22.2.20. y = |
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22.2.21. y =
22.2.23. y =
22.2.25. y =
22.2.27. y =
22.2.29. y =
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22.2.22. y = |
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22.2.24. y = x |
x (4 − x)−1/ 2 . |
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22.2.26. y = |
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22.2.28. y = x4 + 6x3 + 12x2 . |
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22.2.30. y = 3 x(x + 1)2 . |
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22.3. Investigate the functions and sketch the graphs.
22.3.1. |
y = x2 + |
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22.3.2. y = |
x2 − 4x + 3 |
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22.3.3. y = |
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x − 2 |
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22.3.5. y = |
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22.3.6. |
y = |
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x − 3 |
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266
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LITERATURE
1.Бахвалов Н.С., Жидков Н.П., Кобельков Г.М. Численные методы. —
М.: Наука, 1987. — 600 с.
2.Валєєв К.Г., Джалладова І.Л. Вища математика: Навчальний посіб-
ник: У 2-х ч.— К.: КНЕУ, 2001.— Ч.2. — 451 с.
3.ДубовикВ.П., ЮрикІ.І. Вищаматематика. — К.: А.С.К., 2001. — 648 с.
4.Заварыкин В.М., Житомирский В.Г., Лапчик М.П. Численные мето-
ды. — М.: Просвещение, 1991. — 172 с.
5.Краснов М.Л., Киселев А.И., Макаренко Г.И., Шикин Е.В., Заляпин В. И.,
Соболев С.К. Вся высшая математика: Учебник, Т.4. — М.: Эдиториал УРСС, 2001. — 352 с.
6.Овчинников П.П., Яремчук Ф.П., Михайленко В.М. Вища математика:
Підручник. У 2 ч. — Ч. 1: Лінійна і векторна алгебра. Аналітична геометрія. Вступ до математичного аналізу. Диференціальне і інтегральне числення. —
К.: Техніка, 2000. — 592 с.
7.Овчинников П.П. Вища математика: Підручник. У 2 ч. — Ч. 2: Диференціальні рівняння. Операційне числення. Ряди та їх застосування. Стійкість за Ляпуновим. Рівняння математичної фізики. Оптимізація і керування. Теорія ймовірностей. Числові методи. — К.: Техніка, 2000. — 792 с.
8.Пак В.В., Носенко Ю.Л. Вища математика: Підручник.— Д.: «Видавництво Сталкер», 2003. — 496 с.
9.Пискунов Н.С. Дифференциальное и интегральное исчисления. — М.:
Наука, 1985. — Т. 2. — 456 с.
10.Письменный Д.Т. Конспект лекций по высшей математике. 2 часть. — 2-е изд., испр.— М.: Айрис-пресс, 2003. — 256 с.
11.ФильчаковП.Ф. Численные и графические методы прикладной математики. — К.: Наукова думка, 1970. — 792 с.
12.Денисюк В. П., Репета В. К. Вища математика: Навч. посіб.: У 4 ч. — Ч. 1. — К.: Вид-воНац. авіа. ун-ту«Нау-Друк», 2009. — 296 с.
267
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CONTENTS
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
. . . . . 3 |
Module 1. ELEMENTS OF LINEAR AND VECTOR ALGEBRA . |
. . . . . 4 |
Micromodule 1. Determinants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
. . . . . 6 |
Micromodule 2. Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
. . . . 22 |
Micromodule 3. Systems of linear algebraic equations . . . . . . . . . . . . . |
. . . . 42 |
Micromodule 4. Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
. . . . 58 |
Micromodule 5. Dot product of two vectors . . . . . . . . . . . . . . . . . . . . |
. . . . 69 |
Micromodule 6. Cross and triple products . . . . . . . . . . . . . . . . . . . . . . |
. . . . 77 |
Module 2. ELEMENTS OF ANALYTICAL GEOMETRY . . . . . . . |
. . . . 85 |
Micromodule 7. Straight line on a plane . . . . . . . . . . . . . . . . . . . . . . . |
. . . . 87 |
Micromodule 8. Plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
. . . 100 |
Micromodule 9. Straight line in space. Mutual position of straight line |
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and plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
. . . 109 |
Micromodule 10. Curves of the second order . . . . . . . . . . . . . . . . . . . |
. . . 121 |
Micromodule 11. Surfaces of the second order . . . . . . . . . . . . . . . . . . . |
. . 129 |
Module 3. INTRODUCTION TO MATHEMATICAL ANALYSIS. |
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DERIVATIVES AND DIFFERENTIALS OF A FUNCTION |
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OF ONE VARIABLE. APPLICATIONS OF DERIVATIVES 138 |
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Micromodule 12. Sequence. The limit of a numerical sequence. Theorems |
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about limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
. . 141 |
Micromodule 13. The concept of a function. Classification of functions. |
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Limit of function. Theorems about limits . . . . . . . . . . |
. . 152 |
Micromodule 14. Honorable limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
. . 172 |
Micromodule 15. Comparison of infinitesimals. Equivalent infinitesimals. |
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Their application in calculation of limits . . . . . . . . . . |
. . 181 |
Micromodule 16. Continuity of a function . . . . . . . . . . . . . . . . . . . . . . |
. . 188 |
Micromodule 17. Derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
. . 196 |
Micromodule 18. Derivative and its calculation (continued) . . . . . . . . . . |
. . 211 |
Micromodule 19. Differential of function. Tangent . . . . . . . . . . . . . . . . |
. . 219 |
Micromodule 20. Higher order derivatives and differentials . . . . . . . . . . |
. . 229 |
Micromodule 21. Basic theorems of differential calculus . . . . . . . . . . . . |
. . 239 |
Micromodule 22. The usage of derivative in investigation of a function |
. . . 249 |
Literature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
. . 267 |
268
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Навчальне видання
ДЕНИСЮК Володимир Петрович ГРИШИНА Людмила Іллівна КАРУПУ Олена Вальтерівна ОЛЕШКО Тетяна Анатоліївна ПАХНЕНКО Валерія Валеріївна РЕПЕТА Віктор Кузьмич
ВИЩА МАТЕМАТИКА
У чотирьох частинах
Частина 1
Навчальний посібник
(Англійською мовою)
В авторській редакції
Художник обкладинки Т. Зябліцева
Верстка О. Іваненко
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Тираж 300 пр. Замовлення №
Видавництво Національного авіаційного університету «НАУ-друк» 03680, Київ-58, просп. Космонавта Комарова, 1
Свідоцтво про внесення до Державного реєстру ДК № 977 від 05.07.2002
Тел. (044) 406-78-28. Тел./факс: (044) 406-71-33 E-mail: publish@nau.edu.ua
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