Arkusz-Pochodna funkcji
1. Wyznaczyć pochodną funkcji f, gdzi:
1 3 3 2
(1) f(x) = x5 + x4 - 7x2 - 13x + 18 (2) f(x) = 4x2 - +
5 4 x x2
" "
"
3 3
1 9
"
(3) f(x) = 2x + 2 x + (4) f(x) = 3 x4 + 6 x2 +
3
x2 x
2x-3
(5) f(x) = x2 cos x (6) f(x) =
3x-2
x3
(7) f(x) = (8) f(x) = (x2 + 3 sin x)2
x2
"+3
(9) f(x) = x2 - x + 11 (10) f(x) = 1 + sin 5x
Ą
(11) f(x) = 9 sin 7x + (12) f(x) = 5 sin3 x
24
x+2
(13) f(x) = ln(3x - 2) (14) f(x) = ln
x-1
2
(15) f(x) = e-3x+5 (16) f(x) = e2x -4x+5
2. Wyznaczyć przedziały monotoniczności funkcji f.
x
(1) f(x) = x4e-x (2) f(x) = (3) f(x) = ln(x2 + 12)
1 - ln x
x2-x+4
ln x 4 - ln x
x-1
(4) f(x) = (5) f(x) = e (6) f(x) =
x2 x
3. Wyznaczyć ekstrema lokalne funkcji f i zbadać jej monotoniczność.
2
(1) f(x) = 2x3 - 3x2 - 36x - 8 (2) f(x) = x4 - 4x - 2 (3) f(x) =(1-x)
2x
x - 1 x 4x
(4) f(x) = (5) f(x) = (6) f(x) =
2
x xx - 1 x2 + 4
"+ 2
4
(7) f(x) = x3 (8) f(x) = (9) f(x) = x - ln 1 + x2
ln xx
ln x e ex
(10) f(x) = " (11) f(x) = (12) f(x) =
x 1 - x x + 1
2
(13) f(x) = ex -1 (14) f(x) = x2e-x (15) f(x) = xe1-2x
"
x2+x+1
x2-6x
x+1
(16) f(x) = (-x2 + 2)ex-2 (17) f(x) = e (18) f(x) = e
"
(19) f(x) = ln(1 + 4x2) (20) f(x) = 1 - 4x2
4. Wyznaczyć przedziały wypukłości, wklęsłości i punkty przegięcia funkcji f, jeśli:.
(1) f(x) = (x2 + 1)e-x (2) f(x) = ln(x2 + 16)
2 ex
(3) f(x) = xe- x
(4) f(x) =
x + 2
1 - ln x
(5) f(x) = x3 ln x (6) f(x) =
x
x
(7) f(x) = ex-1
5. Zbadać przebieg zmienności i naszkicować wykres funkcji f:
ln x
(1) f(x) = 6x2 - 8x3 + 3x4 (2) f(x) =
x
1
(3) f(x) = e-x (4) f(x) = x2 ln x
x
x
(5) f(x) =
1-ln x
Magdalena Górajska, CMF
1
Odpowiedzi
1.
3 4
(1) f (x) = x4 + 3x3 - 14x - 13 (2) f (x) = 8x + -
x2 x3
"
1 2 4 3
3
" "
(3) f (x) = 2 + - (4) f (x) = 4 x + - "
3 3
x x3 x x x
5
(5) f (x) = 2x cos x - x2 sin x (6) f (x) =
(3x-2)2
x4+9x2
(7) f (x) = (8) f (x) = 2(x2 + 3 sin x)(2x + 3 cos x)
(x2+3)2
2x-1
"
(9) f (x) = (10) f (x) = 5 cos 5x
2 x2-x+11
Ą
(11) f (x) = 63 cos 7x + (12) f (x) = 15 sin2 x cos x
24
3 -3
(13) f (x) = (14) f (x) =
3x-2 (x+2)(x-1)
2
(15) f (x) = -3e-3x+5 (16) f (x) = (4x - 4)e2x -4x+5
2. (1) f jest rosnąca na przedziale (-2, 2), f jest malejącą na przedziałach (-", -2) i (2, +").
(2) f jest rosnąca na przedziałach (0, e) i (e, e2), f jest malejącą na przedziale (e2, +").
(3) f jest rosnąca na przedziale (0, +"), f jest malejącą na przedziale (-", 0).
" "
(4) f jest rosnąca na przedziale (0, e), f jest malejącą na przedziale ( e, ").
(5) f jest rosnąca na przedziałach (-", -1) i (3, +"), f jest malejącą na przedziale (-1, 3).
(6) f jest rosnąca na przedziale (e5, +"), f jest malejącą na przedziale (0, e5).
3. (1) x = -2 -local maximum point, f(-2) = 36, x = 3 local minimum point f(3) = -89 , f is
increasing on (-", -2) and (3, +"), f is decreasing on (-2, 3).
(2) x = 1- local minimum point, f(1) = -5, f is decreasing on (-", 1)and is increasing on (1, +").
(3) x = -1- -local maximum point, f(-1) = -2, x = 1 -local minimum point, f(1) = 0, f is increasing
on (-", -1) and on (1, +"), f is decreasing on (-1, 0) and (0, 1).
(4) there is no extremum, f is increasing on whole domain.
(5) there is no extremum, f is increasing on each interval of domain.
(6) x = -2-local minimum point, f(-2) = -1, x = 2-local maximum point, f(2) = 1. f is increasing
on (-2, 2) and is decreasing on (-", -2), (2, +").
(7) there is no extremum, f is increasing on each interval of domain.
(8) x = e-local minimum point, f(e) = e, increasing on (e, +"), decreasing on (0, 1) (1, e).
(9) there is no extremum, f is increasing on each interval of domain.
2
(10) x = e2 local maximum point, f(e2) = , increasing on (0, e2) and decreasing on (e2, +").
e
(11) x = 2 - local maximum point, f(2) = -e2, increasing on (-", 1) and (1, 2), decreasing on (2, +").
Magdalena Górajska, CMF
2
(12) x = 0 - local minimum point, f(0) = 1, increasing on (0, +") and decreasing on (-", -1) and
(-1, 0).
1
(13) x = 0- local minimum point, f(0) = , increasing on (0, +"), decreasing on (-", 0).
e
4
(14) x = 0 -local minimum point, f(0) = 0, x = 2 - local maximum point, f(2) = , increasing on (0, 2),
e2
decreasing on (-", 0), (2, +").
1 1 1
(15) x = - local maximum point, f(1) = , increasing on (-", ), decreasing on (1, +").
2 2 2 2 2
"
" " " "
3-3
(16) x = 3-1 -local maximum point, f( 3-1) = (2 3-2)e , x = - 3-1 - local minimum point,
"
" " " " "
f(- 3 - 1) = (-2 3 - 2)e- 3-3. increasing on (- 3 - 1, 3 - 1), decreasing on (-", - 3 - 1),
"
( 3 - 1, +").
(17) there is no extremum, decreasing on x " (-", 0), increasing on (6, +").
(18) x = -2 - local maximum point, f(-2) = e-3, x = 0-local minimum point, f(0) = e. is increasing
on (-", -2), (0, +"), decreasing on (-2, -1), (-1, 0).
(19) x = 0 -local minimum point, f(0) = 0, decreasing on x " (-", 0), increasing on (0, +").
1
(20) x = 0 local maximum point, f(0) = 1, increasing on (-1, 0), decreasing on (0, ).
2 2
4. (1) P (1, 2e-1), P (3, 10e-3)- inflection points, convex on (-", 1), (3, "), concave on (1, 3).
(2) P (-4, ln 32), P (4, ln 32)- inflection points, convex on (-4, 4), concave on (-", -4), (4, ").
(3) there is no point of inflection, convex on (0, "), concave on (-", 0).
(4) there is no point of inflection, convex on (-2, "), concave on (-", -2).
5 5 5 5
(5) P (e- 6 -5e- 2
, )- inflection points, convex on (e- 6
, "), concave on (0, e- 6
).
6
5 5 5 5
(6) P (e2 , -3e- 2
)- inflection points, convex on (0, e2 ), concave on (e2 , ").
2
1
(7) P (1, e)- inflection point, convex on (1, 1), (1, "), concave on (-", ).
2 2 2
5. (1) D = R, f (x) = 12x(x - 1)2, f (x) = 12(3x - 1)(x - 1).
1
x -"... 0 ... ... 1 ... + "
3
f(x) 0 -13 1
54
min
convex convex concave convex
1-ln x -3+2 ln x
(2) D = (0, "), f (x) = , f (x) = .
x2 x3
3
x 0 ... e ... e2 ... + "
1 3
f(x) |
3
e
2
2e
concave concave convex
max
-1-x x2+2x+2
(3) D = R \ {0}, f (x) = e-x, f (x) = e-x.
x2 x3
x -"... -1 ... 0 ... + "
f(x) -e |
max
concave concave convex
Magdalena Górajska, CMF
3
(4) D = (0, "), f (x) = x(2 ln x + 1), f (x) = 2 ln x + 3.
3 1
x 0 ... e- 2
... e- 2
... + "
f(x) | -3e-3 -1e-1
2 2
concave convex convex
min
2-ln x 3-ln x
(5) D = (0, e) *" (e, "), f (x) = , f (x) = .
(1-ln x)2 x(1-ln x)3
x 0 ... e ... e2 ... e3 ... + "
f(x) | | -e2 -1e3
2
max
convax concave concave convex
Magdalena Górajska, CMF
4