Calculo. Granville-Smith-Longley. Soluciones Problemas Pag 32

Calculo Diferencial e Integral Granville / Smith / Longley UTEHA Impreso en Espaa! "#$% &A'TE III &ro(lemas &aginas )* Soluciones por +enito Camela ,ergara ,ergara

Calcular la -eriva-a -e ca-a una -e las siguientes funciones usan-o la regla general! "!

y . *  )0 Se sustituye en la función 101 por 10 2

01 y se calcula el nuevo valor de la función y 2

y + ∆y = 2 - 3 (x + ∆X) . Se resta el valor dado de la función del nuevo valor y se obtiene ∆y. Y + ∆y = 2 - 3 (x + ∆X) Y + ∆y = 2  3X -3 ∆X

y + ∆y - y ∆y = - 3∆x.

=

2 - 3x 3x - 3∆x  2 + 3X

Se divide ∆y para ∆y

.

∆x.

- 3∆x ∆x ∆x =

Se calcula el l!"ite de este cociente cuando cuando #l l!"ite as! $allado es la derivada buscada. ∆y

=

4 lim

- 3∆x . ∆x .

0

-y .  ) -0

3

56 . )

0

3 .

y!

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*!

y . m0 2 (. y + ∆y = " (x + ∆x) + b. Y + ∆y = "X + " ∆x + b

y + ∆y - y = "x "x + + "∆x + b  "X - b ∆y = "∆X ∆y = " ∆x ∆x ∆x ∆y = " ∆x lim 0

3

-y . m ! -0 )!

5 7 . m

y . a0*

y + ∆y = a ( X + ∆x)2. y + ∆y = a ( X2 +2X ∆X + ∆x2 ) y + ∆y = aX2 +2aX ∆X + a∆x2 y + ∆y - y = aX2 +2aX ∆X + a∆x2 - aX2 2 ∆y = 2aX ∆X + a∆x a.∆x2 ∆y = 2ax. ∆x + a. ∆x ∆x ∆x ∆y = 2ax + a.∆x . ∆x !

lim 0

3

-y . *a0 2 a 839 -0 -y . *a0 . -0 :!

5 7 . * a0

s . *t  t*!

s + ∆s = 2(t + ∆t) - (t + ∆t)2. s + ∆s 2t + 2∆t - (t2 +2t ∆t + ∆t 2 )

!









∆s t lim 0

3

-s -t

2 - 2t - ∆t

=

.

56 . * ; *t

*  *t  3

y . c0) y + ∆y = c ( x + ∆x)3. y + ∆y = c ( x3 + 3x2. ∆x + 3x.∆x2 + ∆x3 ) y + ∆y = c x3 + 3cx2. ∆x + 3cx.∆x2 + c.∆x3 y + ∆y - y = c x3 + 3cx2. ∆x + 3cx.∆x2 + c.∆x3 - c x3 2 2 3 ∆y = 3cx .∆x + 3cx.∆x + c ∆x 2 3c x. ∆x2 + c∆x3 ∆y = 3cx . ∆x + 3cx. ∆x ∆x ∆x ∆x . ∆y

3cx2 + 3cx.∆x + c∆x2.

=

∆x lim 0

3

-y . )c0* 2 )c08 3 9 2 c 8 3 9 * -0

5 7 . )c0*

=!

y . )0  0)! y + ∆y = 3 (x + ∆x) - (x + ∆x)3. y + ∆y = 3 x + 3∆x - (x3 +3X2 ∆x + 3X ∆x2 + ∆x3 ) y + ∆y = 3 x + 3∆x - x3 -3X2 ∆x - 3X ∆x2 - ∆x3 x + 3∆x - x3 -3X2 ∆x - 3X ∆x2 - ∆x3  3X + X3. y + ∆y - y = 3 x + 2 2 3 ∆y = 3∆x - 3X ∆x - 3X ∆x - ∆x 2 3x .(∆x)2 - (∆x)3 . ∆y = 3.∆x - 3x . ∆x - 3x.( ∆x ∆x ∆x ∆x ∆x 2 2 ∆y = 3 - 3x - 3x (%) - ( ∆x) ∆x lim 0

3

-y )  )0*

56 . ) ; )0*









u + ∆u = & ( v2 + 2v∆v + ∆v2 ) + 2 ( v3 + 3v2∆v + 3v ∆v2 + ∆v3 ) u + ∆u = & v2 + 'v∆v +&∆v2 + 2v3 + v2∆v + v ∆v2 + 2∆v3 u 2 ∆u  u = u = & v2 + 'v∆v +&∆v2 + 2v3 + v2∆v + v∆v2 + 2∆v3  &v2  2v3

u . 'v∆v +&∆v2 + v2∆v + v∆v2 + 2∆v3 2 2 2 3 ∆u = 'v. ∆v + &. ∆v + v . ∆v + v. ∆v + 2. ∆v . ∆v ∆v ∆v ∆v ∆v ∆v ∆u

'v + &. ∆v + v2 + v. ∆v + 2. ∆v2

=

∆v ∆u

'v + &(%) + v2 + v(%) + 2(% ) 2

=

∆v v

lim

3

-u . %v 2 3 2 =v* 2 3 2 3 -v -u . %v 2 =v* . U6 . %v 2 =v* -v %!

y . 0:!

y + ∆y = (x + ∆x)&. y + ∆y = x& + &X3∆x + X2 ∆x2 + &X∆x3 + ∆x& y + ∆y - y = x& + &X3∆x + X2 ∆x2 + &X∆x3 + ∆x& - x& 3 2 2 3 & ∆y = &x . ∆x + x .∆x + &x.∆x + ∆x . ∆y

3 &x .

=

∆x

∆y

∆x + x

∆x

2

(∆x)2 + &x (∆x)3 + (∆x)& ∆x ∆x ∆x

&x3 + x2. ∆x + &x (∆x)2 + (∆x)3

=

∆x ∆y ∆x lim 0

&x3 + x2(%) + &x(%)2 + ( % )3

=

3

-y . :0) . -0 #!

e.

5 7 . :0)

* . 2 "

e + ∆e

=

2 (θ + ∆θ) + 

.

.









∆e

=

∆e

=

2θ + 2 - 2θ - 2. ∆θ - 2 = *(θ + ∆θ) +  ( θ + ) - 2.∆θ *(θ + ∆θ) + (θ + )( )(∆θ)

∆θ ∆e

-2 . *(θ + %) + ( θ + )

=

∆θ

lim 0

3

-e -

.

de -

* 8 2 "9 8 2 "9 .

Y,

"3!

=

2. ∆θ . *(θ + ∆θ) + (θ + ) -2 = *(θ + ∆θ) + (θ + )

.

.

* . * 8 2 "9  * 8 2 "9*

.

) . 0 2 * y + ∆y = 3 . 2 (x + ∆x) + 2 y + ∆y - y = 3 3 . 2 2 (x + ∆x) + 2 x + 2 y.

*

3 (x 2 + 2) - 3 *(x + ∆x)2 + 2 *(x + ∆x)2 + 2 (x2 + 2)

∆y

=

∆y

=

∆y

=

∆y

=

3x2 +  -3 *x 2 + 2x. ∆x +( ∆x)2 +2 *(x + ∆x)2 + 2 (x2 + 2) 3x2 +  - 3x2 - x.∆x - 3(∆x)2 -  *(x + ∆x)2 + 2 (x2 + 2) ∆x

(- x -3. ∆x) 2 2 ∆x *(x + ∆x) + 2 (x + 2) ∆x

=

=

- x. ∆x - 3(∆x)2 . *(x+∆x)2 + 2 (x2 + 2)

- x - 3. ∆x *(x + ∆x)2 + 2 (x2 + 2)

=









80* 2 *9 80 * 2 *9

-0

""!

80* 2 *9*

s . t 2 : t

s + ∆s

=

(t + ∆t) + & t + ∆t

s + ∆s - s = t + ∆t + & - t + & t + ∆t t ∆s

=

t (t + ∆t + &) - (t + &) (t + (t + ∆t) t

∆s

=

∆s

=

∆s

t2 + t. ∆t + &t - t 2 - &t - t. ∆t - &. ∆t (t + ∆t) t

∆t

- & ( ∆t ) . (t + ∆t) t ( ∆t )

∆s

=

t t

.

S >. : / t*

y.

-& . (t + %)t

3

-s .  : -t t!t

"*!

=

t2 + t.∆t + &t & t -(t - (t 2 + &t + t. ∆t + &. ∆t) = (t + ∆t) t

=

li"

∆t)

" ! "  *0

: . t*

=

- &. ∆t . (t + ∆t) t











∆y

=

∆y

=

∆y

=

( - 2x) - * -2(x + ∆x) * - 2(x+∆x)( - 2x)  - 2x -  + 2x + 2∆x * - 2(x+∆x)( - 2x)

0

3

-y . * -0 8"  *09 8"  *09 -y . -0 ")!

2 . * - 2(x + ∆x)( - 2x)

2 . * - 2 (x + %)( - 2x)

=

0

li"

=

2∆x . * - 2 (x + ∆x)( - 2x)

=

2 ∆x = ∆x * - 2(x + ∆x)( - 2x)

∆x ∆y

 - 2x -( - 2x - 2∆x) * - 2(x+∆x)( - 2x)

=

.

* . * 8"  *09

* 8"  *09*

e.

!

2 *

e + ∆e =

+ ∆θ (θ + ∆θ) + 2 θ

e + ∆e - e =

e .8

2 *9 8

+ ∆θ  (θ + ∆θ) + 2 θ

2

?8 2 ∆e

.

9  ?8 2

.

θ +

9 2 *@ .

9 2 *@8 2 *9

2 ∆θ

θ

*

2 2 *

2

!

?8 2

2*



*



!

9 2 *@ 8 2 *9

 *

!









":!

s . At 2 + Ct 2 D

s + ∆s = 0(t + ∆t) + 1 (t + ∆t) + 

.

s + ∆s - s = 0(t + ∆t) + 1 - 0t + 1 *(t + ∆t) +  t +  ∆s

=

*0(t + ∆t) + 1 (t + ) - *(t + ∆t) +  (0t + 1) *(t + ∆t) +  (t + )

∆s

=

∆s

=

*0.t + 0.∆t + 1(.t 1( .t + ) - *.t + .∆t + (0.t + 1) *(t + ∆t) +  (t + ) 0t 2 + 0t + 0 . t . ∆t + 0∆t + 1t 1 t + 1 1 *(t + ∆t) +  (t + )

.

- 0t 2 - 1 1tt - 0t 0 t∆t - 1 ∆t - 0dt 0d t - 1 . *(t + ∆t) +  (t + ) ∆s

∆s

=

=

∆t ∆s

t

=

0.. ∆t - 1.. ∆t *(t + ∆t) +  (t + ) ∆t

(0. - 1.) *(t + ∆t) +  (t + )

∆t

(0. - 1.) *(t + ∆t) +  (t + )

=

=

(0. - 1.) *(t + %) +  (t + )

(0. - 1.) . (∆t) *(t + ∆t) +  (t + ) =

(0. - 1.) (t + )(t + )

.

.









∆y

=

∆y

=

∆y

=

∆y

=

∆y

=

*(x + ∆x)3 +  x - (x3 + ) (x + ∆x) (x + ∆x) x

3 3 ∆x) - x - ∆x *x 3 + 3x 2∆x + 3x( 3x (∆x)2 + (∆x)  + 4(x) 4( x) - x& - x ( (x + ∆x) x

2 2 3 3 x& + 3x3∆x + 3x ( (x ) + x - x & - x ( ∆x) + (∆x) (x) ∆x) - x -∆x (x + ∆x) x

3 2 2 3 2 2 3 2x . 3 x ( ( x) - 3x 3 x ( ∆x + 3x ∆x) + (∆x) (x) ∆x) + (∆x) (x) - ∆x (x + ∆x) x

3

* 2x ∆x *2

2 2 2 + 3x2 ∆x + ( ∆x) (x) ( x) - 3x 3 x ( ∆x) + (∆x) (x) -  (x + ∆x) x

ividiendo a a"bos "ie"bros para ∆x5 tene"os / ∆y

3

=

* 2x ∆x *2

=

∆x

∆x ∆y ∆x ∆y

0 lim 0

2 2 2 + 3x2 ∆x + ( ∆x) (x) ( x) - 3x 3 x ( ∆x) + (∆x) (x) -  (x + ∆x) (x) (∆x)

2 2 2 *2x 3 + 3x2. ∆x + ( ∆x) .x . x - 3x . ∆x + ( ∆x) .x -  (x + ∆x)(x)( x)(x)( ∆x)

2 2 2 2 *2x 3 + 3x ( % ) + (%) (% ) (x) ( x) - 3x 3 x (%) ( %) + (%) ( %) (x) -  (x+%)x

=

3

y = 2x3 + % + % - % + % -  = 2x3 -  = 2x3 -  x.x x2 x2 x2 ’

-y . *0  " -0 0*









∆y

- ∆x (2x + ∆x) *(x + ∆x)2 + a2 (x2 + a2)

=

ividiendo a a"bos "ie"bros para

∆x5

∆y

=

=

-

∆y

(2x + ∆x) *(x + ∆x) + a2(x2 + a2). ∆x - (2x + %) *(x + %)2 + a2 (x2 + a2)

=

0 0

∆x 2

∆x

lim

-y . -0

y.

tene"os /

- (2x + ∆x) . 2 2 2 2 *(x + ∆x) + a  (x + a )

.

3

-y .  *0 * -0 80 2 a*9 80* 2 a*9

"$!

.

.

 *0 80 2 a* 9*

!

*

 *0 80 2 a* 9* *

0 . 0 2 " *

y + ∆y =

x + ∆x . (x + ∆x)2 + 

y + ∆y - y =

∆y

x + ∆x x . 2 2 *(x + ∆x) +  (x + )

(x + ∆x) (x ( x2 + ) - x *(x + ∆x)2 +  .

=









*(x + %)2 + 

0

li"

0

-y . -0

"%!

(x2 + )

3

880*  "9 80* 2 "9 80 * 2 "9

.

"  0* 80* 2 "9*

0* . :  0* y + ∆y = (x + ∆x)2 . & - (x + ∆x)2 y.

y + ∆y - y =

∆y

=

∆y

=

(x + ∆x)2 x2 . *& - (x + ∆x)2 (& - x2)

2 2 (x + ∆x)2 (& - x ) - *& - (x + ∆x)  x2 . *& - (x + ∆x)2 (& - x2)

2 2 2 *x2 + 2x. ∆x + ( ∆x) (&  (& - x ) - *&-( *& -(x x 2 +2x. ∆x + ( ∆x) ( x 2) *& - (x + ∆x)2 (& - x2)

* * * y . :0* 2 %0! %0 ! 0 2 :8 09*  0:  *0 )! 0  0 !8 09* ?: ? : 0 **0! *0 ! 08 09 @80 9 * * ?:  80 2 09 @ 8:  0 9

: ) : y . :0* 2 %0! 0 2 :8 09 * 0  *0 ! 0  0*!8 09*  :0 * 2 0 2 *0)! 0 2 0*! 8 09* ?:  80 2 09*@8:  0 *9

∆y

=

'x.∆x + &. (∆x)2 2

= 2

∆x

('x + &. ∆x) 2

. 2











y + ∆y = 3 (x + ∆x)2 - & (x + ∆x) - 6 y + ∆y - y = 3 (x + ∆x)2 - & (x + ∆x) - 6 - (3x 2 - &x -6)

3 *x2 + 2x. ∆x + (∆x)2 - & (x + ∆x) - 6 - (3x 2 - &x -6)

∆y

=

∆y

=

∆y

=

3x2 + x. ∆x + 3.(∆x)2 - &x - &. ∆x - 6 - 3x2 + &x + 6

x. ∆x + 3 (∆x)2 - &.(∆x) = (∆x) *x + 3 (∆x) - &

ividiendo para ∆y 0

lim

0

∆x

/

(∆x)* x) *x x + 3 (∆x) - & = ∆x *x * x + 3 ( ∆x) - & = x + 3(%) - & ∆x ∆x

=

3

-y . =0  : . *8)0  *9 -0

*3!

.

s . at* 2 (t 2 c .

s + ∆s a (t + ∆t)2 + b (t + ∆t) + c .









*"!

u . *v)  )v* u + ∆u = 2 (v + ∆v)3 - 3 (v + ∆v)2 u + ∆u - u = 2(v + ∆v)3 - 3 (v + ∆v)2 - (2v3 - 3v2) u . *?v) 2 )v*! v 2 )v!8 v9 * 2 8 v)9@  )?v * 2 *v! v 2 8 v9*@  *v) 2 )v* u . *v) 2 =v*! v 2 =v 8 v9* 2 * 8 v9)  )v*  =v! v  )8 v9*  *v) 2 )v* ! ∆u

v2. ∆v + v ( ∆v)2 + 2 (∆v)3 - v. ∆v - 3(∆v)2

=

8actori7ando y dividiendo para ∆u

∆v

=

∆v ∆u = v lim v 3 ∆u

=

∆v

/

*v 2 + v. ∆v + 2. (∆v)2 - v - 3. ∆v ∆v .

v2 + v. ∆v + 2 (∆v)2 - v - 3. ∆v

v2 + v (%) + 2 (%) 2 - v - 3 (%) .

∆v

-u =v*  =v









-0

*)!

e . 8a  ( 9*

e + ∆e = *a - b (θ + ∆θ )2 e + ∆e - e = *a - b (θ + ∆θ )2 - (a - b θ)2 ∆e

=

∆e

=

∆e

∆e

=

(a - bθ - b.∆θ)2 - (a - b θ)2

*a + (-bθ) + (- b.∆θ)2 - (a - b θ)2

a2 + (-bθ)2 + (- b.∆θ)2 + 2a.(-bθ) +2a.(- b.∆θ) +2.(-bθ).(- b. ∆θ) - *a2-2a.bθ + (bθ)2

=

a2 + (bθ)2 +(b.∆θ)2- 2a(bθ) -2a(b.∆θ) +2(bθ)(b.∆θ) - a2 + 2a.b θ - (bθ)2

∆e

=

∆e

=

(b. ∆θ)2 -2a(b. ∆θ) + 2(bθ)(b. ∆θ)

b2(∆θ)2 - 2a(b. ∆θ) + 2(bθ)(b.∆θ)

8actorando y dividiendo para ∆θ. ∆

∆θ

2 b .( 2 a.b + 2b 2 θ4 ∆θ) - 2a.b









∆y

=

∆x

(-6 + &x + 2 ∆x) ∆x

∆x ∆y 0

li"

.

- 6 + &x + 2( % )

=

3

-y .  < 2 :0 . :0  < -0 *
y . 8A0 2 +9 8C0 2 D9

y + ∆y = *0 (x + ∆x) + 1 * (x + ∆x) +  y + ∆y - y y  y 2

=

*0 (x + ∆x) + 1 * (x +

∆x)

+  - (0x + 1) (x + ).

y . 8A0 2 A! 0 2 + 9 8C0 2 C! 0 2 D 9  8A0 2 +9 8C0 2 D9!

y . AC0 * 2 AC0! 0 2 AD0 2 AC0! 0 2 AC8 09* 2 AD8 09 2 +C0 2 +C! 0 2 +D  AC0 *  AD0  +C0  +D ! ∆y

2 0x. ∆x + 0(∆x)2 + 0(∆x) + 1. ∆x

=









)a8( t9* 2 )8(t98( t9* 2 =a8(t9 8( t9  a )  )a*8(t9 8(t9  )a8(t9 *  8(t9

)

s . 8( t9)2 )a*8( t9 2 )8(t9 * 8( t9 2 )a 8( t9* 2 ) 8(t9 8( t9* 2 =a 8(t9 8( t9

8actorando 5 dividiendo y si"plificando para ∆t . ∆s ∆t

=

∆t

2 3 2 3 2 (b ∆t)2 + 3a b + 3b 3 b t + 3ab 3a b2∆t + 3b t. a b t4 ∆t + ab ∆t .

∆s = *b ∆t li" ∆t→%

(%)2 + 3a2 b + 3b3t2 + 3ab2.(%) + 3b3t.(%) + ab2t

ds = % + 3a2 b + 3b3t2 + % + % + ab2t = 3a2 b + 3b3t2 + ab2t. dt ds = 3a2 b + ab2t + 3b3t2 = 3b ( a2 + 2abt + b2t2 ) = dt -s . )( ?a 2 8(t9@*B -t











2 ∆x - bx.( a. ∆x - bx . bx .( ∆x)2 . 8actorando y dividiendo para ∆x/ *a + b (x + ∆x)2 *a + bx2

∆y

=

∆y

=

∆x ∆y 0

(a - bx b x2 - bx.∆x) . 2 2 *a + b (x + ∆x)  *a + bx  ∆x . 2 2 a - bx - bx.∆x a - bx - bx ( % ) . = = *a + b (x + ∆x)2 *a + bx2 a + b *x + ( % )24*a + bx2 ∆x

li" ∆x→%

∆y 0

=

a - bx2 - % . 2 2 *a + bx  *a + bx 

∆x→%

-y

a  (0*









-y .  *a0 .  *a! 0 0 : ) -0 0 0 !0 *#!

y.

0* . a 2 (0*

y + ∆y =

(x + ∆x)2 . a + b (x + ∆x)2

y + ∆y - y =

∆y

 *a 0)

.

(x + ∆x)2 x2 . *a + b (x + ∆x)2 (a + bx2)

2 2 2 (x + ∆x)2 (a + bx ) - a + b (x + ∆x) 4( x ) *a + b (x + ∆x)2  ( a + bx2 )

=

Calculo. Granville-Smith-Longley. Soluciones Problemas Pag 32

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