TUGAS KALKULUS

Muyanto

09.0504.0038

 

3.i) Y= X3 – 6x2 + 9x + 6

=  3x2 -12x + 9

(3x – 3)            (x – 3)

 

X1 =    3x – 3 = 0

3x = 3

X =     = 1

X2 =     x – 3 = 0

X = 3

 

Y1 = (1)3 – 6 (1)2 + 9 (1) + 6

= 1 – 6 + 9 + 6

= 10

 

Y2 = (3)3 – 6 (3)2 + 9 (3) + 6

= 27 – 54 + 27 + 6

=  54 -54 + 6

= 6

 

= 2x -4       (X=1)                = 2x -4       (X=3)

= 2 (1) – 4                                  = 2 (3) – 4

= -2   < 0  (Max)                         = 2    > 0   (min)

 

Titik Belok  = 0

2x – 4 = 0

2x = 4

x =

= 2

 

Y = (2)3 – 6 (2)2 + 9 (2) + 6

= 8 – 24 + 18 + 6

= 8    ( 2 , 8 )

 

Eks I   = ( 1,10 )

Eks II  = ( 3,6 )

Titik Belok = ( 2 , 8 )

 

 

 

 

 

 

 

 

4i.)   y = (x – 2)2 (x – 7)

 

( x – 2)   ( x – 2)

(x2 – 4x + 4 )  (x – 7)

= ( x3 – 4x2 + 4x -7x2 + 28x -28)

Y = x3 – 11x2 + 32x – 28

 

 

= 3x2 – 22x + 32

3x2 – 22x + 32 = 0

( 3x – 16)  ( x – 2) = 0

3x – 16 = 0                  x – 2 = 0

3x = 16                        x = 2

X =                         X2 =  2

X1 = 5,3

 

Sifat

= 6x – 22    ( x = 5,3)

= 6 (5,3) – 22

= 31,8 – 22

= 9,8 > 0          (min)

 

=  6x – 22    ( x = 2)

= 6 (2) – 22

= 12 – 22

= -10  < 0        (max)

 

Y1 =  x3 – 11x2 + 32x – 28      ( x = 5,3)

= (5,3)3 – 11 (5,3)2 + 32 (5,3) – 28

= 148,9 – 308,9 + 169,6 – 28

= -18,4

Y2 = x3 – 11x2 + 32x -28        ( x = 2)

= (2)3 – 11(2)2 + 32(2) -28

= 8 – 44 + 64 – 28

= 0

=  6x – 22

6x – 22    = 0

6x = 22

X =                   = 3,67

Y = x3 – 11x2 + 32x -28          ( x = 3,67)

= (3,67)3 – (3,67)2 + 32 (3,67) – 28

=  49,43 – 148,15 + 117,44 – 28

= – 9,28

 

Eks I   = (5,3 ; 18,4)                (min)

Eks II  = (2,0)                        (max)

Eks III = ( 3,67 ; – 9,28)          ( belok)

 

 

 

.

 

 

 

 

 

 

 

4 ii) y = 4x3 + 3x2 – 18x – 9

 

= 2 x2 + 6x – 18

 

= 2x2 +  x – 3

( 2x + 3)   ( x – 1) = 0

2x + 3 = 0                    x – 1 = 0

2x = -3                         x = 1

X = – 1,5

 

Y1 = 4x3 + 3x2 – 18x – 9       ( x = – 1,5)

4(-1,5)3 + 3(-1,5)2 – 18(-1,5) – 9

= – 13,5 – 6,75 + 27 – 9

= – 2,25

Y2   = 4x3 + 3x2 – 18x – 9        ( x =   1)

= 4 + 3 – 18 – 9

=  -20

 

= 4x + 1                      (x = – 1,5)

=  4 (-1,5) + 1

= -5 < 0           (max)

 

 

= 4x + 1                      (x = 1)

= 4 (1) + 1

= 5 > 0             (min)

 

Titik belok

= 0

4x + 1 = 0

4x =  – 1

X =  –

X = – 0,25

 

Y = 4x3 + 3x2 – 18x – 9          ( x = – 0,25)

4 (- 0,25)3 + 3 (- 0,25)2 – 18 (- 0,25) – 9

= – 0,0625 + 0,1875 + 4,5 – 9

= – 4,375

 

Eks I   = (- 1,5 ; – 2,25)          (max)

Eks II   = ( 1 , – 20)                 (min)

Eks III  = ( – 0,25 ; -4,375)     (belok)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

6) y = 12 ln x + x2 – 10x

=  + 2x – 10

+ 2x – 10  = 0

= 0

2x2 – 10x + 12  = 0

X2 – 5x + 6 = 0

(x – 3)           (x – 2)

X1 = 3                         X2 = 2

 

Y1 =  12 ln x + x2 – 10x     (x = 3)

12 ln(3) + (3)2 – 10(3)

13,18 + 9 – 30

-7,82

 

Y2 =  12 ln x + x2 – 10x   (x = 2)

12 ln(2) + (2)2 – 10(2)

8,31 + 4 – 20

-7,68

= -12x-2 + 2        (x = 3)

= – 12 (3)-2 + 2

=

=

=                  = 0,67  > 0       (min)

 

= -12x-2 + 2                 (x = 2)

=  – 12 (2)-2 + 2

=        = -1  < 0          (max)

 

= -12x-2 + 2  = 0

-12x-2 = -2

x-2 =

x-2 =

x2 = 6

x          =                = 2,44              (belok)

 

Y1 =  12 ln x + x2 – 10x          (x = 2,44)

12 ln(2,44) + (2,44)2 – 10(2,44)

= 10,7 + 5,9 – 24,4

= -7,75

Eks I   = (3,-7,82)        (min)

Eks II  = (2,-7,68)       (max)

Eks III = (2,44; -7,55) (belok)

 

 

 

 

 

 

 

 

 

 

 

7) 4x2 + 8xy + 9y2 – 8x – 24y + 4 = 0

8x + 8y + 8x + 18y  – 8 – 24 = 0

 

(8x + 18y – 24) = -8x – 8y + 8

=

=    = 0

 

-4x – 4y + 4   : – 4

x + y -1 = 0

x +y = 1

 

u = -4x – 4y + 4                u’ = -4 – 4

v = 4x + 9y – 12               v’ = 4 + 9

 

=

 

=

 

 

 

 

 

8) y = 3e2x cos (2x – 3)

U = 3e2x u’ = 6e2x

V =  cos (2x – 3)            v’ =  – 2sin (2x – 3)

 

= 6e2x (cos (2x – 3)  ) + 3e2x -2sin (2x – 3)

= 6e2x (  cos (2x – 3)  ) – sin (2x – 3)

 

= 6e2x (  cos (2x – 3)  ) – sin (2x – 3) = 0

a . b = 0

a = 0       atau     b = 0

6e2x = 0

e2x = 0

ln e2x = ln 0

2x ln e  = ln 0

2x . 1 = ~

 

(cos (2x – 3)  ) – sin (2x – 3)

Cos (2x – 3) = sin (2x – 3)

Cos   45       = sin   45

(2x – 3) = 45o

2x = 45o + 3o = 48o

X = 24o

 

 

U =  6e2x u’ = 12e2x

 

V = (  cos (2x – 3)  ) – sin (2x – 3)         v’ =  -2 sin (2x – 3) – 2cos (2x – 3)

 

= 12e2x { cos (2x – 3)  ) – sin (2x – 3) } + 6e2x ( – 2sin (2x – 3)  – 2 cos(2x – 3)

= 12e2x { cos (2x – 3)  ) – sin (2x – 3) }  – 12e2x ( sin (2x – 3) +  cos(2x – 3)

= 12e2x { cos (2x – 3)  ) – sin (2x – 3) }  – sin (2x – 3)  –  cos(2x – 3)

=  12e2x ( -2 sin (2x – 3) )

 

– 24e2x sin (2x – 3)  = 0

 

– 24e2x = 0

e2x = 0

x     =  ~

 

sin (2x – 3)  = 0

sin (2x – 3)  =  sin 0o

(2x – 3)      = 0

2x               = 3

X              =

 

9) y3 = 6xy – x3 – 1

3y2 – 6y +6x + 3x2

=               =

=

 

~ oleh gusmuy pada November 2, 2009.

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