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© h.hofstede (h.hofstede@hogeland.nl) |
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| 1. | a. | f(x) = 4/(x +
3) F(x) = 4ln|x + 3| |
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| b. | f(x) = x + 2/x F(x) = 1/2x2 + 2ln|x| |
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| c. | f(x) = 1/(4x) F(x) = ln|4x| • 1/4 |
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| d. | f(x) = 1/(3 - x)
F(x) = ln|3 - x| • -1 = -ln|3 - x| |
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| e. | f(x) = 2/(3x
+ 1) F(x) = 2ln|3x + 1| • 1/3 = 2/3ln|3x + 1| |
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| f. | f(x) = 1/(x2)
= x-2 F(x) = -x-1 = -1/x |
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| 2. | a. |
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| F(x) = x + 8ln|3x - 2| • 1/3 = x + 8/3ln|3x - 2| | |||
| b. |
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| F(x) = ln|x + 3| maar voor x = 2 bestaat de functie niet. | |||
| c. |
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| F(x) = 2x - 13ln|x + 4| | |||
| 3. | snijpunten: 4x2 = 7,5x + 1 4x2 - 7,5x - 1 = 0 ABC-formule: x = (7,5 ±√(56,25 + 16))/8 = 2 of -1/8 4x2 = 1/2x 8x3 = 1 x3 = 1/8 x = 1/2 1/2x = 7,5x + 1 1 = 15x2 + 2x 15x2 + 2x - 1 = 0 ABC-formule: x = (-2 ±√(4 + 60))/30 = 1/5 of -1/3 |
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| Verdeel het gebied in twee stukken: I en II (zie de figuur) | |||
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| = (0,9375 + 0,5 - 0,5 • ln1) - (0,15 + 0,2 - 0,5 • ln0,4) = 1,4375 - 0,35 + 0,5ln0,4 = 1,0875 - 0,5ln0,4 | |||
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| = (15 + 2
- 102/3)
- (0,9375 + 0,5 - 1/6)
= 61/3
- 61/48
= 81/16 Samen is dat 6,15 - 0,5ln0,4 |
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| 4. | f(x) = 3,5 x2 + x + 1 = 3,5x x2 - 2,5x + 1 = 0 (x - 2)(x - 0,5) = 0 x = 2 ∨ x = 0,5 f(x) = x + 1 + 1/x |
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| = (5 - 2 - ln2) - (1,25 - 1/8 - ln(1/2)) = 17/8 - 2ln2 | |||
| 5. | f(x) = 1 + 2/x + 2/x2 | ||
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| = (-1 + 0 + 2) - (-4 + 2ln4 + 1/2) = 41/2 - 2ln4 | |||
| 6. | Het is wél dezelfde
op een constante na: 1/2ln(2x) = 1/2(ln2 + lnx) = 1/2lnx + 1/2ln2 |
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| 7. |
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| = 2lne - 2ln1 = 2 | |||
| 8. |
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| (6x
- 2)(2 + 1) = 8x2 12x2 + 6x - 4x - 2 = 8x2 4x2 + 2x - 2 = 0 ABC-formule: x = (-2 ±√(4 + 32))/8 = 1/2 of -1 |
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| = (6ln4 +
1/2
- 4ln9) - (6ln0,5 + 4
- 4ln2) = 12ln2 + 1/2 - 8ln3 + 6ln2 - 4 + 4ln2 = 22ln2 - 31/2 - 8ln3 = 2,96 |
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