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g(t)=t4-t3+t2g(t)=t4−t3+t2
Paso 1
Considera la fórmula del cociente diferencial.
f(t+h)-fthf(t+h)−fth
Paso 2
Paso 2.1
Evalúa la función en x=t+hx=t+h.
Paso 2.1.1
Reemplaza la variable tt con t+ht+h en la expresión.
g(t+h)=(t+h)4-(t+h)3+(t+h)2g(t+h)=(t+h)4−(t+h)3+(t+h)2
Paso 2.1.2
Simplifica el resultado.
Paso 2.1.2.1
Simplifica cada término.
Paso 2.1.2.1.1
Usa el teorema del binomio.
g(t+h)=t4+4t3h+6t2h2+4th3+h4-(t+h)3+(t+h)2g(t+h)=t4+4t3h+6t2h2+4th3+h4−(t+h)3+(t+h)2
Paso 2.1.2.1.2
Usa el teorema del binomio.
g(t+h)=t4+4t3h+6t2h2+4th3+h4-(t3+3t2h+3th2+h3)+(t+h)2g(t+h)=t4+4t3h+6t2h2+4th3+h4−(t3+3t2h+3th2+h3)+(t+h)2
Paso 2.1.2.1.3
Aplica la propiedad distributiva.
g(t+h)=t4+4t3h+6t2h2+4th3+h4-t3-(3t2h)-(3th2)-h3+(t+h)2g(t+h)=t4+4t3h+6t2h2+4th3+h4−t3−(3t2h)−(3th2)−h3+(t+h)2
Paso 2.1.2.1.4
Simplifica.
Paso 2.1.2.1.4.1
Multiplica 33 por -1−1.
g(t+h)=t4+4t3h+6t2h2+4th3+h4-t3-3(t2h)-(3th2)-h3+(t+h)2g(t+h)=t4+4t3h+6t2h2+4th3+h4−t3−3(t2h)−(3th2)−h3+(t+h)2
Paso 2.1.2.1.4.2
Multiplica 33 por -1−1.
g(t+h)=t4+4t3h+6t2h2+4th3+h4-t3-3(t2h)-3(th2)-h3+(t+h)2g(t+h)=t4+4t3h+6t2h2+4th3+h4−t3−3(t2h)−3(th2)−h3+(t+h)2
g(t+h)=t4+4t3h+6t2h2+4th3+h4-t3-3(t2h)-3(th2)-h3+(t+h)2g(t+h)=t4+4t3h+6t2h2+4th3+h4−t3−3(t2h)−3(th2)−h3+(t+h)2
Paso 2.1.2.1.5
Elimina los paréntesis.
g(t+h)=t4+4t3h+6t2h2+4th3+h4-t3-3t2h-3th2-h3+(t+h)2g(t+h)=t4+4t3h+6t2h2+4th3+h4−t3−3t2h−3th2−h3+(t+h)2
Paso 2.1.2.1.6
Reescribe (t+h)2(t+h)2 como (t+h)(t+h)(t+h)(t+h).
g(t+h)=t4+4t3h+6t2h2+4th3+h4-t3-3t2h-3th2-h3+(t+h)(t+h)g(t+h)=t4+4t3h+6t2h2+4th3+h4−t3−3t2h−3th2−h3+(t+h)(t+h)
Paso 2.1.2.1.7
Expande (t+h)(t+h)(t+h)(t+h) con el método PEIU (primero, exterior, interior, ultimo).
Paso 2.1.2.1.7.1
Aplica la propiedad distributiva.
g(t+h)=t4+4t3h+6t2h2+4th3+h4-t3-3t2h-3th2-h3+t(t+h)+h(t+h)g(t+h)=t4+4t3h+6t2h2+4th3+h4−t3−3t2h−3th2−h3+t(t+h)+h(t+h)
Paso 2.1.2.1.7.2
Aplica la propiedad distributiva.
g(t+h)=t4+4t3h+6t2h2+4th3+h4-t3-3t2h-3th2-h3+t⋅t+th+h(t+h)g(t+h)=t4+4t3h+6t2h2+4th3+h4−t3−3t2h−3th2−h3+t⋅t+th+h(t+h)
Paso 2.1.2.1.7.3
Aplica la propiedad distributiva.
g(t+h)=t4+4t3h+6t2h2+4th3+h4-t3-3t2h-3th2-h3+t⋅t+th+ht+h⋅hg(t+h)=t4+4t3h+6t2h2+4th3+h4−t3−3t2h−3th2−h3+t⋅t+th+ht+h⋅h
g(t+h)=t4+4t3h+6t2h2+4th3+h4-t3-3t2h-3th2-h3+t⋅t+th+ht+h⋅hg(t+h)=t4+4t3h+6t2h2+4th3+h4−t3−3t2h−3th2−h3+t⋅t+th+ht+h⋅h
Paso 2.1.2.1.8
Simplifica y combina los términos similares.
Paso 2.1.2.1.8.1
Simplifica cada término.
Paso 2.1.2.1.8.1.1
Multiplica tt por tt.
g(t+h)=t4+4t3h+6t2h2+4th3+h4-t3-3t2h-3th2-h3+t2+th+ht+h⋅hg(t+h)=t4+4t3h+6t2h2+4th3+h4−t3−3t2h−3th2−h3+t2+th+ht+h⋅h
Paso 2.1.2.1.8.1.2
Multiplica hh por hh.
g(t+h)=t4+4t3h+6t2h2+4th3+h4-t3-3t2h-3th2-h3+t2+th+ht+h2g(t+h)=t4+4t3h+6t2h2+4th3+h4−t3−3t2h−3th2−h3+t2+th+ht+h2
g(t+h)=t4+4t3h+6t2h2+4th3+h4-t3-3t2h-3th2-h3+t2+th+ht+h2g(t+h)=t4+4t3h+6t2h2+4th3+h4−t3−3t2h−3th2−h3+t2+th+ht+h2
Paso 2.1.2.1.8.2
Suma thth y htht.
Paso 2.1.2.1.8.2.1
Reordena tt y hh.
g(t+h)=t4+4t3h+6t2h2+4th3+h4-t3-3t2h-3th2-h3+t2+ht+ht+h2g(t+h)=t4+4t3h+6t2h2+4th3+h4−t3−3t2h−3th2−h3+t2+ht+ht+h2
Paso 2.1.2.1.8.2.2
Suma htht y htht.
g(t+h)=t4+4t3h+6t2h2+4th3+h4-t3-3t2h-3th2-h3+t2+2ht+h2g(t+h)=t4+4t3h+6t2h2+4th3+h4−t3−3t2h−3th2−h3+t2+2ht+h2
g(t+h)=t4+4t3h+6t2h2+4th3+h4-t3-3t2h-3th2-h3+t2+2ht+h2g(t+h)=t4+4t3h+6t2h2+4th3+h4−t3−3t2h−3th2−h3+t2+2ht+h2
g(t+h)=t4+4t3h+6t2h2+4th3+h4-t3-3t2h-3th2-h3+t2+2ht+h2g(t+h)=t4+4t3h+6t2h2+4th3+h4−t3−3t2h−3th2−h3+t2+2ht+h2
g(t+h)=t4+4t3h+6t2h2+4th3+h4-t3-3t2h-3th2-h3+t2+2ht+h2g(t+h)=t4+4t3h+6t2h2+4th3+h4−t3−3t2h−3th2−h3+t2+2ht+h2
Paso 2.1.2.2
La respuesta final es t4+4t3h+6t2h2+4th3+h4-t3-3t2h-3th2-h3+t2+2ht+h2t4+4t3h+6t2h2+4th3+h4−t3−3t2h−3th2−h3+t2+2ht+h2.
t4+4t3h+6t2h2+4th3+h4-t3-3t2h-3th2-h3+t2+2ht+h2t4+4t3h+6t2h2+4th3+h4−t3−3t2h−3th2−h3+t2+2ht+h2
t4+4t3h+6t2h2+4th3+h4-t3-3t2h-3th2-h3+t2+2ht+h2t4+4t3h+6t2h2+4th3+h4−t3−3t2h−3th2−h3+t2+2ht+h2
t4+4t3h+6t2h2+4th3+h4-t3-3t2h-3th2-h3+t2+2ht+h2t4+4t3h+6t2h2+4th3+h4−t3−3t2h−3th2−h3+t2+2ht+h2
Paso 2.2
Reordena.
Paso 2.2.1
Mueve t3t3.
t4+4ht3+6t2h2+4th3+h4-t3-3t2h-3th2-h3+t2+2ht+h2t4+4ht3+6t2h2+4th3+h4−t3−3t2h−3th2−h3+t2+2ht+h2
Paso 2.2.2
Mueve t2t2.
t4+4ht3+6h2t2+4th3+h4-t3-3t2h-3th2-h3+t2+2ht+h2t4+4ht3+6h2t2+4th3+h4−t3−3t2h−3th2−h3+t2+2ht+h2
Paso 2.2.3
Mueve tt.
t4+4ht3+6h2t2+4h3t+h4-t3-3t2h-3th2-h3+t2+2ht+h2t4+4ht3+6h2t2+4h3t+h4−t3−3t2h−3th2−h3+t2+2ht+h2
Paso 2.2.4
Mueve t2t2.
t4+4ht3+6h2t2+4h3t+h4-t3-3ht2-3th2-h3+t2+2ht+h2t4+4ht3+6h2t2+4h3t+h4−t3−3ht2−3th2−h3+t2+2ht+h2
Paso 2.2.5
Mueve tt.
t4+4ht3+6h2t2+4h3t+h4-t3-3ht2-3h2t-h3+t2+2ht+h2t4+4ht3+6h2t2+4h3t+h4−t3−3ht2−3h2t−h3+t2+2ht+h2
Paso 2.2.6
Mueve t2t2.
t4+4ht3+6h2t2+4h3t+h4-t3-3ht2-3h2t-h3+2ht+h2+t2t4+4ht3+6h2t2+4h3t+h4−t3−3ht2−3h2t−h3+2ht+h2+t2
Paso 2.2.7
Mueve 2ht2ht.
t4+4ht3+6h2t2+4h3t+h4-t3-3ht2-3h2t-h3+h2+2ht+t2t4+4ht3+6h2t2+4h3t+h4−t3−3ht2−3h2t−h3+h2+2ht+t2
Paso 2.2.8
Mueve -t3−t3.
t4+4ht3+6h2t2+4h3t+h4-3ht2-3h2t-h3-t3+h2+2ht+t2t4+4ht3+6h2t2+4h3t+h4−3ht2−3h2t−h3−t3+h2+2ht+t2
Paso 2.2.9
Mueve -3ht2−3ht2.
t4+4ht3+6h2t2+4h3t+h4-3h2t-h3-3ht2-t3+h2+2ht+t2t4+4ht3+6h2t2+4h3t+h4−3h2t−h3−3ht2−t3+h2+2ht+t2
Paso 2.2.10
Mueve -3h2t−3h2t.
t4+4ht3+6h2t2+4h3t+h4-h3-3h2t-3ht2-t3+h2+2ht+t2t4+4ht3+6h2t2+4h3t+h4−h3−3h2t−3ht2−t3+h2+2ht+t2
Paso 2.2.11
Mueve t4t4.
4ht3+6h2t2+4h3t+h4+t4-h3-3h2t-3ht2-t3+h2+2ht+t24ht3+6h2t2+4h3t+h4+t4−h3−3h2t−3ht2−t3+h2+2ht+t2
Paso 2.2.12
Mueve 4ht34ht3.
6h2t2+4h3t+h4+4ht3+t4-h3-3h2t-3ht2-t3+h2+2ht+t26h2t2+4h3t+h4+4ht3+t4−h3−3h2t−3ht2−t3+h2+2ht+t2
Paso 2.2.13
Mueve 6h2t26h2t2.
4h3t+h4+6h2t2+4ht3+t4-h3-3h2t-3ht2-t3+h2+2ht+t24h3t+h4+6h2t2+4ht3+t4−h3−3h2t−3ht2−t3+h2+2ht+t2
Paso 2.2.14
Reordena 4h3t4h3t y h4h4.
h4+4h3t+6h2t2+4ht3+t4-h3-3h2t-3ht2-t3+h2+2ht+t2h4+4h3t+6h2t2+4ht3+t4−h3−3h2t−3ht2−t3+h2+2ht+t2
h4+4h3t+6h2t2+4ht3+t4-h3-3h2t-3ht2-t3+h2+2ht+t2h4+4h3t+6h2t2+4ht3+t4−h3−3h2t−3ht2−t3+h2+2ht+t2
Paso 2.3
Obtén los componentes de la definición.
g(t+h)=h4+4h3t+6h2t2+4ht3+t4-h3-3h2t-3ht2-t3+h2+2ht+t2g(t+h)=h4+4h3t+6h2t2+4ht3+t4−h3−3h2t−3ht2−t3+h2+2ht+t2
g(t)=t4-t3+t2g(t)=t4−t3+t2
g(t+h)=h4+4h3t+6h2t2+4ht3+t4-h3-3h2t-3ht2-t3+h2+2ht+t2g(t+h)=h4+4h3t+6h2t2+4ht3+t4−h3−3h2t−3ht2−t3+h2+2ht+t2
g(t)=t4-t3+t2g(t)=t4−t3+t2
Paso 3
Inserta los componentes.
g(t+h)-gth=h4+4h3t+6h2t2+4ht3+t4-h3-3h2t-3ht2-t3+h2+2ht+t2-(t4-t3+t2)hg(t+h)−gth=h4+4h3t+6h2t2+4ht3+t4−h3−3h2t−3ht2−t3+h2+2ht+t2−(t4−t3+t2)h
Paso 4
Paso 4.1
Simplifica el numerador.
Paso 4.1.1
Aplica la propiedad distributiva.
h4+4h3t+6h2t2+4ht3+t4-h3-3h2t-3ht2-t3+h2+2ht+t2-t4--t3-t2hh4+4h3t+6h2t2+4ht3+t4−h3−3h2t−3ht2−t3+h2+2ht+t2−t4−−t3−t2h
Paso 4.1.2
Multiplica --t3−−t3.
Paso 4.1.2.1
Multiplica -1−1 por -1−1.
h4+4h3t+6h2t2+4ht3+t4-h3-3h2t-3ht2-t3+h2+2ht+t2-t4+1t3-t2hh4+4h3t+6h2t2+4ht3+t4−h3−3h2t−3ht2−t3+h2+2ht+t2−t4+1t3−t2h
Paso 4.1.2.2
Multiplica t3t3 por 11.
h4+4h3t+6h2t2+4ht3+t4-h3-3h2t-3ht2-t3+h2+2ht+t2-t4+t3-t2hh4+4h3t+6h2t2+4ht3+t4−h3−3h2t−3ht2−t3+h2+2ht+t2−t4+t3−t2h
h4+4h3t+6h2t2+4ht3+t4-h3-3h2t-3ht2-t3+h2+2ht+t2-t4+t3-t2hh4+4h3t+6h2t2+4ht3+t4−h3−3h2t−3ht2−t3+h2+2ht+t2−t4+t3−t2h
Paso 4.1.3
Resta t4t4 de t4t4.
h4+4h3t+6h2t2+4ht3-h3-3h2t-3ht2-t3+h2+2ht+t2+0+t3-t2hh4+4h3t+6h2t2+4ht3−h3−3h2t−3ht2−t3+h2+2ht+t2+0+t3−t2h
Paso 4.1.4
Suma h4h4 y 00.
h4+4h3t+6h2t2+4ht3-h3-3h2t-3ht2-t3+h2+2ht+t2+t3-t2hh4+4h3t+6h2t2+4ht3−h3−3h2t−3ht2−t3+h2+2ht+t2+t3−t2h
Paso 4.1.5
Suma -t3−t3 y t3t3.
h4+4h3t+6h2t2+4ht3-h3-3h2t-3ht2+h2+2ht+t2+0-t2hh4+4h3t+6h2t2+4ht3−h3−3h2t−3ht2+h2+2ht+t2+0−t2h
Paso 4.1.6
Suma h4h4 y 00.
h4+4h3t+6h2t2+4ht3-h3-3h2t-3ht2+h2+2ht+t2-t2hh4+4h3t+6h2t2+4ht3−h3−3h2t−3ht2+h2+2ht+t2−t2h
Paso 4.1.7
Resta t2t2 de t2t2.
h4+4h3t+6h2t2+4ht3-h3-3h2t-3ht2+h2+2ht+0hh4+4h3t+6h2t2+4ht3−h3−3h2t−3ht2+h2+2ht+0h
Paso 4.1.8
Suma h4+4h3t+6h2t2+4ht3-h3-3h2t-3ht2+h2+2hth4+4h3t+6h2t2+4ht3−h3−3h2t−3ht2+h2+2ht y 00.
h4+4h3t+6h2t2+4ht3-h3-3h2t-3ht2+h2+2hthh4+4h3t+6h2t2+4ht3−h3−3h2t−3ht2+h2+2hth
Paso 4.1.9
Factoriza hh de h4+4h3t+6h2t2+4ht3-h3-3h2t-3ht2+h2+2hth4+4h3t+6h2t2+4ht3−h3−3h2t−3ht2+h2+2ht.
Paso 4.1.9.1
Factoriza hh de h4h4.
h⋅h3+4h3t+6h2t2+4ht3-h3-3h2t-3ht2+h2+2hthh⋅h3+4h3t+6h2t2+4ht3−h3−3h2t−3ht2+h2+2hth
Paso 4.1.9.2
Factoriza hh de 4h3t4h3t.
h(h3)+h(4h2t)+6h2t2+4ht3-h3-3h2t-3ht2+h2+2hthh(h3)+h(4h2t)+6h2t2+4ht3−h3−3h2t−3ht2+h2+2hth
Paso 4.1.9.3
Factoriza hh de 6h2t26h2t2.
h(h3)+h(4h2t)+h(6ht2)+4ht3-h3-3h2t-3ht2+h2+2hthh(h3)+h(4h2t)+h(6ht2)+4ht3−h3−3h2t−3ht2+h2+2hth
Paso 4.1.9.4
Factoriza hh de 4ht34ht3.
h(h3)+h(4h2t)+h(6ht2)+h(4t3)-h3-3h2t-3ht2+h2+2hthh(h3)+h(4h2t)+h(6ht2)+h(4t3)−h3−3h2t−3ht2+h2+2hth
Paso 4.1.9.5
Factoriza hh de -h3−h3.
h(h3)+h(4h2t)+h(6ht2)+h(4t3)+h(-h2)-3h2t-3ht2+h2+2hthh(h3)+h(4h2t)+h(6ht2)+h(4t3)+h(−h2)−3h2t−3ht2+h2+2hth
Paso 4.1.9.6
Factoriza hh de -3h2t−3h2t.
h(h3)+h(4h2t)+h(6ht2)+h(4t3)+h(-h2)+h(-3ht)-3ht2+h2+2hthh(h3)+h(4h2t)+h(6ht2)+h(4t3)+h(−h2)+h(−3ht)−3ht2+h2+2hth
Paso 4.1.9.7
Factoriza hh de -3ht2−3ht2.
h(h3)+h(4h2t)+h(6ht2)+h(4t3)+h(-h2)+h(-3ht)+h(-3t2)+h2+2hthh(h3)+h(4h2t)+h(6ht2)+h(4t3)+h(−h2)+h(−3ht)+h(−3t2)+h2+2hth
Paso 4.1.9.8
Factoriza hh de h2h2.
h(h3)+h(4h2t)+h(6ht2)+h(4t3)+h(-h2)+h(-3ht)+h(-3t2)+h(h)+2hthh(h3)+h(4h2t)+h(6ht2)+h(4t3)+h(−h2)+h(−3ht)+h(−3t2)+h(h)+2hth
Paso 4.1.9.9
Factoriza hh de 2ht.
h(h3)+h(4h2t)+h(6ht2)+h(4t3)+h(-h2)+h(-3ht)+h(-3t2)+h(h)+h(2t)h
Paso 4.1.9.10
Factoriza h de h(h3)+h(4h2t).
h(h3+4h2t)+h(6ht2)+h(4t3)+h(-h2)+h(-3ht)+h(-3t2)+h(h)+h(2t)h
Paso 4.1.9.11
Factoriza h de h(h3+4h2t)+h(6ht2).
h(h3+4h2t+6ht2)+h(4t3)+h(-h2)+h(-3ht)+h(-3t2)+h(h)+h(2t)h
Paso 4.1.9.12
Factoriza h de h(h3+4h2t+6ht2)+h(4t3).
h(h3+4h2t+6ht2+4t3)+h(-h2)+h(-3ht)+h(-3t2)+h(h)+h(2t)h
Paso 4.1.9.13
Factoriza h de h(h3+4h2t+6ht2+4t3)+h(-h2).
h(h3+4h2t+6ht2+4t3-h2)+h(-3ht)+h(-3t2)+h(h)+h(2t)h
Paso 4.1.9.14
Factoriza h de h(h3+4h2t+6ht2+4t3-h2)+h(-3ht).
h(h3+4h2t+6ht2+4t3-h2-3ht)+h(-3t2)+h(h)+h(2t)h
Paso 4.1.9.15
Factoriza h de h(h3+4h2t+6ht2+4t3-h2-3ht)+h(-3t2).
h(h3+4h2t+6ht2+4t3-h2-3ht-3t2)+h(h)+h(2t)h
Paso 4.1.9.16
Factoriza h de h(h3+4h2t+6ht2+4t3-h2-3ht-3t2)+h(h).
h(h3+4h2t+6ht2+4t3-h2-3ht-3t2+h)+h(2t)h
Paso 4.1.9.17
Factoriza h de h(h3+4h2t+6ht2+4t3-h2-3ht-3t2+h)+h(2t).
h(h3+4h2t+6ht2+4t3-h2-3ht-3t2+h+2t)h
h(h3+4h2t+6ht2+4t3-h2-3ht-3t2+h+2t)h
h(h3+4h2t+6ht2+4t3-h2-3ht-3t2+h+2t)h
Paso 4.2
Reduce la expresión mediante la cancelación de los factores comunes.
Paso 4.2.1
Cancela el factor común de h.
Paso 4.2.1.1
Cancela el factor común.
h(h3+4h2t+6ht2+4t3-h2-3ht-3t2+h+2t)h
Paso 4.2.1.2
Divide h3+4h2t+6ht2+4t3-h2-3ht-3t2+h+2t por 1.
h3+4h2t+6ht2+4t3-h2-3ht-3t2+h+2t
h3+4h2t+6ht2+4t3-h2-3ht-3t2+h+2t
Paso 4.2.2
Simplifica la expresión.
Paso 4.2.2.1
Mueve h2.
h3+4th2+6ht2+4t3-h2-3ht-3t2+h+2t
Paso 4.2.2.2
Mueve h.
h3+4th2+6t2h+4t3-h2-3ht-3t2+h+2t
Paso 4.2.2.3
Mueve h.
h3+4th2+6t2h+4t3-h2-3th-3t2+h+2t
Paso 4.2.2.4
Mueve h.
h3+4th2+6t2h+4t3-h2-3th-3t2+2t+h
Paso 4.2.2.5
Mueve -h2.
h3+4th2+6t2h+4t3-3th-3t2-h2+2t+h
Paso 4.2.2.6
Mueve -3th.
h3+4th2+6t2h+4t3-3t2-3th-h2+2t+h
Paso 4.2.2.7
Mueve h3.
4th2+6t2h+4t3+h3-3t2-3th-h2+2t+h
Paso 4.2.2.8
Mueve 4th2.
6t2h+4t3+4th2+h3-3t2-3th-h2+2t+h
Paso 4.2.2.9
Reordena 6t2h y 4t3.
4t3+6t2h+4th2+h3-3t2-3th-h2+2t+h
4t3+6t2h+4th2+h3-3t2-3th-h2+2t+h
4t3+6t2h+4th2+h3-3t2-3th-h2+2t+h
4t3+6t2h+4th2+h3-3t2-3th-h2+2t+h
Paso 5