Precálculo Ejemplos

Hallar el ratio de cambio medio d(t)=(100t)/(2t+15)
d(t)=100t2t+15d(t)=100t2t+15
Paso 1
Considera la fórmula del cociente diferencial.
f(t+h)-fthf(t+h)fth
Paso 2
Obtén los componentes de la definición.
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Paso 2.1
Evalúa la función en x=t+hx=t+h.
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Paso 2.1.1
Reemplaza la variable tt con t+ht+h en la expresión.
d(t+h)=100(t+h)2(t+h)+15d(t+h)=100(t+h)2(t+h)+15
Paso 2.1.2
Simplifica el resultado.
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Paso 2.1.2.1
Aplica la propiedad distributiva.
d(t+h)=100(t+h)2t+2h+15d(t+h)=100(t+h)2t+2h+15
Paso 2.1.2.2
La respuesta final es 100(t+h)2t+2h+15100(t+h)2t+2h+15.
100(t+h)2t+2h+15100(t+h)2t+2h+15
100(t+h)2t+2h+15100(t+h)2t+2h+15
100(t+h)2t+2h+15100(t+h)2t+2h+15
Paso 2.2
Obtén los componentes de la definición.
d(t+h)=100(t+h)2t+2h+15d(t+h)=100(t+h)2t+2h+15
d(t)=100t2t+15d(t)=100t2t+15
d(t+h)=100(t+h)2t+2h+15d(t+h)=100(t+h)2t+2h+15
d(t)=100t2t+15d(t)=100t2t+15
Paso 3
Inserta los componentes.
d(t+h)-dth=100(t+h)2t+2h+15-(100t2t+15)hd(t+h)dth=100(t+h)2t+2h+15(100t2t+15)h
Paso 4
Simplifica.
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Paso 4.1
Simplifica el numerador.
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Paso 4.1.1
Para escribir 100(t+h)2t+2h+15100(t+h)2t+2h+15 como una fracción con un denominador común, multiplica por 2t+152t+152t+152t+15.
100(t+h)2t+2h+152t+152t+15-100t2t+15h100(t+h)2t+2h+152t+152t+15100t2t+15h
Paso 4.1.2
Para escribir -100t2t+15100t2t+15 como una fracción con un denominador común, multiplica por 2t+2h+152t+2h+152t+2h+152t+2h+15.
100(t+h)2t+2h+152t+152t+15-100t2t+152t+2h+152t+2h+15h100(t+h)2t+2h+152t+152t+15100t2t+152t+2h+152t+2h+15h
Paso 4.1.3
Escribe cada expresión con un denominador común de (2t+2h+15)(2t+15)(2t+2h+15)(2t+15), mediante la multiplicación de cada uno por un factor adecuado de 11.
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Paso 4.1.3.1
Multiplica 100(t+h)2t+2h+15100(t+h)2t+2h+15 por 2t+152t+152t+152t+15.
100(t+h)(2t+15)(2t+2h+15)(2t+15)-100t2t+152t+2h+152t+2h+15h100(t+h)(2t+15)(2t+2h+15)(2t+15)100t2t+152t+2h+152t+2h+15h
Paso 4.1.3.2
Multiplica 100t2t+15100t2t+15 por 2t+2h+152t+2h+152t+2h+152t+2h+15.
100(t+h)(2t+15)(2t+2h+15)(2t+15)-100t(2t+2h+15)(2t+15)(2t+2h+15)h100(t+h)(2t+15)(2t+2h+15)(2t+15)100t(2t+2h+15)(2t+15)(2t+2h+15)h
Paso 4.1.3.3
Reordena los factores de (2t+2h+15)(2t+15)(2t+2h+15)(2t+15).
100(t+h)(2t+15)(2t+15)(2t+2h+15)-100t(2t+2h+15)(2t+15)(2t+2h+15)h100(t+h)(2t+15)(2t+15)(2t+2h+15)100t(2t+2h+15)(2t+15)(2t+2h+15)h
100(t+h)(2t+15)(2t+15)(2t+2h+15)-100t(2t+2h+15)(2t+15)(2t+2h+15)h100(t+h)(2t+15)(2t+15)(2t+2h+15)100t(2t+2h+15)(2t+15)(2t+2h+15)h
Paso 4.1.4
Combina los numeradores sobre el denominador común.
100(t+h)(2t+15)-100t(2t+2h+15)(2t+15)(2t+2h+15)h100(t+h)(2t+15)100t(2t+2h+15)(2t+15)(2t+2h+15)h
Paso 4.1.5
Reescribe 100(t+h)(2t+15)-100t(2t+2h+15)(2t+15)(2t+2h+15)100(t+h)(2t+15)100t(2t+2h+15)(2t+15)(2t+2h+15) en forma factorizada.
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Paso 4.1.5.1
Factoriza 100100 de 100(t+h)(2t+15)-100t(2t+2h+15)100(t+h)(2t+15)100t(2t+2h+15).
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Paso 4.1.5.1.1
Factoriza 100100 de 100(t+h)(2t+15)100(t+h)(2t+15).
100((t+h)(2t+15))-100t(2t+2h+15)(2t+15)(2t+2h+15)h100((t+h)(2t+15))100t(2t+2h+15)(2t+15)(2t+2h+15)h
Paso 4.1.5.1.2
Factoriza 100100 de -100t(2t+2h+15)100t(2t+2h+15).
100((t+h)(2t+15))+100(-t(2t+2h+15))(2t+15)(2t+2h+15)h100((t+h)(2t+15))+100(t(2t+2h+15))(2t+15)(2t+2h+15)h
Paso 4.1.5.1.3
Factoriza 100100 de 100((t+h)(2t+15))+100(-t(2t+2h+15))100((t+h)(2t+15))+100(t(2t+2h+15)).
100((t+h)(2t+15)-t(2t+2h+15))(2t+15)(2t+2h+15)h100((t+h)(2t+15)t(2t+2h+15))(2t+15)(2t+2h+15)h
100((t+h)(2t+15)-t(2t+2h+15))(2t+15)(2t+2h+15)h100((t+h)(2t+15)t(2t+2h+15))(2t+15)(2t+2h+15)h
Paso 4.1.5.2
Expande (t+h)(2t+15)(t+h)(2t+15) con el método PEIU (primero, exterior, interior, ultimo).
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Paso 4.1.5.2.1
Aplica la propiedad distributiva.
100(t(2t+15)+h(2t+15)-t(2t+2h+15))(2t+15)(2t+2h+15)h100(t(2t+15)+h(2t+15)t(2t+2h+15))(2t+15)(2t+2h+15)h
Paso 4.1.5.2.2
Aplica la propiedad distributiva.
100(t(2t)+t15+h(2t+15)-t(2t+2h+15))(2t+15)(2t+2h+15)h100(t(2t)+t15+h(2t+15)t(2t+2h+15))(2t+15)(2t+2h+15)h
Paso 4.1.5.2.3
Aplica la propiedad distributiva.
100(t(2t)+t15+h(2t)+h15-t(2t+2h+15))(2t+15)(2t+2h+15)h100(t(2t)+t15+h(2t)+h15t(2t+2h+15))(2t+15)(2t+2h+15)h
100(t(2t)+t15+h(2t)+h15-t(2t+2h+15))(2t+15)(2t+2h+15)h100(t(2t)+t15+h(2t)+h15t(2t+2h+15))(2t+15)(2t+2h+15)h
Paso 4.1.5.3
Simplifica cada término.
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Paso 4.1.5.3.1
Reescribe con la propiedad conmutativa de la multiplicación.
100(2tt+t15+h(2t)+h15-t(2t+2h+15))(2t+15)(2t+2h+15)h100(2tt+t15+h(2t)+h15t(2t+2h+15))(2t+15)(2t+2h+15)h
Paso 4.1.5.3.2
Multiplica tt por tt sumando los exponentes.
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Paso 4.1.5.3.2.1
Mueve tt.
100(2(tt)+t15+h(2t)+h15-t(2t+2h+15))(2t+15)(2t+2h+15)h100(2(tt)+t15+h(2t)+h15t(2t+2h+15))(2t+15)(2t+2h+15)h
Paso 4.1.5.3.2.2
Multiplica tt por tt.
100(2t2+t15+h(2t)+h15-t(2t+2h+15))(2t+15)(2t+2h+15)h100(2t2+t15+h(2t)+h15t(2t+2h+15))(2t+15)(2t+2h+15)h
100(2t2+t15+h(2t)+h15-t(2t+2h+15))(2t+15)(2t+2h+15)h100(2t2+t15+h(2t)+h15t(2t+2h+15))(2t+15)(2t+2h+15)h
Paso 4.1.5.3.3
Mueve 1515 a la izquierda de tt.
100(2t2+15t+h(2t)+h15-t(2t+2h+15))(2t+15)(2t+2h+15)h100(2t2+15t+h(2t)+h15t(2t+2h+15))(2t+15)(2t+2h+15)h
Paso 4.1.5.3.4
Reescribe con la propiedad conmutativa de la multiplicación.
100(2t2+15t+2ht+h15-t(2t+2h+15))(2t+15)(2t+2h+15)h100(2t2+15t+2ht+h15t(2t+2h+15))(2t+15)(2t+2h+15)h
Paso 4.1.5.3.5
Mueve 1515 a la izquierda de hh.
100(2t2+15t+2ht+15h-t(2t+2h+15))(2t+15)(2t+2h+15)h100(2t2+15t+2ht+15ht(2t+2h+15))(2t+15)(2t+2h+15)h
100(2t2+15t+2ht+15h-t(2t+2h+15))(2t+15)(2t+2h+15)h100(2t2+15t+2ht+15ht(2t+2h+15))(2t+15)(2t+2h+15)h
Paso 4.1.5.4
Aplica la propiedad distributiva.
100(2t2+15t+2ht+15h-t(2t)-t(2h)-t15)(2t+15)(2t+2h+15)h100(2t2+15t+2ht+15ht(2t)t(2h)t15)(2t+15)(2t+2h+15)h
Paso 4.1.5.5
Simplifica.
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Paso 4.1.5.5.1
Reescribe con la propiedad conmutativa de la multiplicación.
100(2t2+15t+2ht+15h-12tt-t(2h)-t15)(2t+15)(2t+2h+15)h100(2t2+15t+2ht+15h12ttt(2h)t15)(2t+15)(2t+2h+15)h
Paso 4.1.5.5.2
Reescribe con la propiedad conmutativa de la multiplicación.
100(2t2+15t+2ht+15h-12tt-12th-t15)(2t+15)(2t+2h+15)h100(2t2+15t+2ht+15h12tt12tht15)(2t+15)(2t+2h+15)h
Paso 4.1.5.5.3
Multiplica 1515 por -11.
100(2t2+15t+2ht+15h-12tt-12th-15t)(2t+15)(2t+2h+15)h100(2t2+15t+2ht+15h12tt12th15t)(2t+15)(2t+2h+15)h
100(2t2+15t+2ht+15h-12tt-12th-15t)(2t+15)(2t+2h+15)h100(2t2+15t+2ht+15h12tt12th15t)(2t+15)(2t+2h+15)h
Paso 4.1.5.6
Simplifica cada término.
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Paso 4.1.5.6.1
Multiplica tt por tt sumando los exponentes.
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Paso 4.1.5.6.1.1
Mueve tt.
100(2t2+15t+2ht+15h-12(tt)-12th-15t)(2t+15)(2t+2h+15)h100(2t2+15t+2ht+15h12(tt)12th15t)(2t+15)(2t+2h+15)h
Paso 4.1.5.6.1.2
Multiplica tt por tt.
100(2t2+15t+2ht+15h-12t2-12th-15t)(2t+15)(2t+2h+15)h100(2t2+15t+2ht+15h12t212th15t)(2t+15)(2t+2h+15)h
100(2t2+15t+2ht+15h-12t2-12th-15t)(2t+15)(2t+2h+15)h100(2t2+15t+2ht+15h12t212th15t)(2t+15)(2t+2h+15)h
Paso 4.1.5.6.2
Multiplica -11 por 22.
100(2t2+15t+2ht+15h-2t2-12th-15t)(2t+15)(2t+2h+15)h100(2t2+15t+2ht+15h2t212th15t)(2t+15)(2t+2h+15)h
Paso 4.1.5.6.3
Multiplica -11 por 22.
100(2t2+15t+2ht+15h-2t2-2th-15t)(2t+15)(2t+2h+15)h100(2t2+15t+2ht+15h2t22th15t)(2t+15)(2t+2h+15)h
100(2t2+15t+2ht+15h-2t2-2th-15t)(2t+15)(2t+2h+15)h100(2t2+15t+2ht+15h2t22th15t)(2t+15)(2t+2h+15)h
Paso 4.1.5.7
Resta 2t22t2 de 2t22t2.
100(15t+2ht+15h+0-2th-15t)(2t+15)(2t+2h+15)h100(15t+2ht+15h+02th15t)(2t+15)(2t+2h+15)h
Paso 4.1.5.8
Suma 15t15t y 00.
100(15t+2ht+15h-2th-15t)(2t+15)(2t+2h+15)h100(15t+2ht+15h2th15t)(2t+15)(2t+2h+15)h
Paso 4.1.5.9
Resta 15t15t de 15t15t.
100(2ht+15h-2th+0)(2t+15)(2t+2h+15)h100(2ht+15h2th+0)(2t+15)(2t+2h+15)h
Paso 4.1.5.10
Suma 2ht+15h-2th2ht+15h2th y 00.
100(2ht+15h-2th)(2t+15)(2t+2h+15)h100(2ht+15h2th)(2t+15)(2t+2h+15)h
Paso 4.1.5.11
Resta 2th2th de 2ht2ht.
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Paso 4.1.5.11.1
Mueve hh.
100(15h+2th-2th)(2t+15)(2t+2h+15)h100(15h+2th2th)(2t+15)(2t+2h+15)h
Paso 4.1.5.11.2
Resta 2th2th de 2th2th.
100(15h+0)(2t+15)(2t+2h+15)h100(15h+0)(2t+15)(2t+2h+15)h
100(15h+0)(2t+15)(2t+2h+15)h100(15h+0)(2t+15)(2t+2h+15)h
Paso 4.1.5.12
Suma 15h15h y 00.
10015h(2t+15)(2t+2h+15)h10015h(2t+15)(2t+2h+15)h
Paso 4.1.5.13
Multiplica 100100 por 1515.
1500h(2t+15)(2t+2h+15)h1500h(2t+15)(2t+2h+15)h
1500h(2t+15)(2t+2h+15)h1500h(2t+15)(2t+2h+15)h
1500h(2t+15)(2t+2h+15)h1500h(2t+15)(2t+2h+15)h
Paso 4.2
Multiplica el numerador por la recíproca del denominador.
1500h(2t+15)(2t+2h+15)1h1500h(2t+15)(2t+2h+15)1h
Paso 4.3
Cancela el factor común de hh.
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Paso 4.3.1
Factoriza hh de 1500h1500h.
h1500(2t+15)(2t+2h+15)1hh1500(2t+15)(2t+2h+15)1h
Paso 4.3.2
Cancela el factor común.
h1500(2t+15)(2t+2h+15)1h
Paso 4.3.3
Reescribe la expresión.
1500(2t+15)(2t+2h+15)
1500(2t+15)(2t+2h+15)
1500(2t+15)(2t+2h+15)
Paso 5
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