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Unidad 2
Actividad 2
Taller sobre derivadas
Estimado estudiante:
Para la elaboración del siguiente taller, tenga en cuenta los conceptos y ejercicios que ha
practicado en su proceso de lectura, tal como se indica en la actividad. Recuerde mostrar el
paso a paso de cada uno de los ejercicios planteados.
Muchos éxitos.
1. En cada una de las siguientes funciones, determine la razón de cambio promedio en el
valor de y al pasar de 𝑥 = −1 𝑎 𝑥 = 2. (Represéntelas gráficamente).
a) 𝑦 = 𝑓(𝑥) = 𝑥2
− 2𝑥 + 3
b) 𝑦 = 𝑓(𝑥) = 𝑥3
− 2𝑥2
+ 𝑥 + 2
c) 𝑦 = 𝑓(𝑥) = 4𝑥2
− 2
d) 𝑦 = 𝑓(𝑥) =
𝑥2
𝑥+4
2. Para los siguientes ejercicios:
 determine la derivada de f por el método del límite,
 calcule la pendiente cuando 𝑥 = −3 𝑦 𝑥 = 2 y
 determine en qué parte de la función la pendiente es 0.
Las funciones son:
a) 𝑓(𝑥) = 15𝑥2
+ 2𝑥 − 8
b) 𝑓(𝑥) = −𝑥2
− 3𝑥 + 5
c) 𝑓(𝑥) = 𝑥3
+ 3𝑥2
d) 𝑓(𝑥) = −4𝑥2
− 8𝑥
e) 𝑓(𝑥) =
𝑥2
2
+ 6𝑥
f) 𝑓(𝑥) =
3𝑥3
2
− 𝑥2
3. En los siguientes ejercicios:
 Encuentre la expresión 𝑓´(𝑥)
 Calcule 𝑓´(2)y 𝑓´(−1)
 Determine los valores de x para los cuales 𝑓´(𝑥) = 0
a) 𝑓(𝑥) = (𝑥3
− 2𝑥)(−𝑥 + 6𝑥2
)
b) 𝑓(𝑥) = (4 − 2𝑥)(−𝑥3
− 2𝑥2
+ 1)
c) 𝑓(𝑥) =
−3𝑥2
(𝑥2−2𝑥+1)
d) 𝑓(𝑥) =
−𝑥
𝑥−5
e) 𝑓(𝑥) = (−𝑥3
+ 1)(−2𝑥2
− 2𝑥)
f) 𝑓(𝑥) =
𝑥3
3
− 6𝑥2
+ 8𝑥 − 7
4. Resuelva los siguientes problemas e indique el proceso para llegar a la solución.
a) Un capital de $25.000.000 se invierte en una entidad financiera durante tres años, que
paga un interés trimestral del 6 %.
 Exprese el valor futuro y de este capital en función del tiempo x expresado en
trimestres.
 Halle la variación promedio de y con respecto a x durante el primer año y
durante el segundo año. Compare estos resultados y explique su diferencia.
b) Una población p de cierta ciudad, en función del tiempo x expresado en años, está
decreciendo a una tasa continua del 10 % anual.
 Exprese la población en función del tiempo x.
 Halle la variación promedio de p con respecto a x durante los primeros cinco
años y luego entre los cinco y diez años. Compare estos resultados.
c) El sistema de cable ha estado en crecimiento durante los últimos cuatro años, debido
a que la gente se ha quedado más tiempo en casa por cuestiones de la pandemia. La
siguiente figura resume el total de suscriptores a cable de televisión durante los
últimos 4 años:
 Determine la razón de cambio promedio del total de suscriptores a cable de televisión
entre 2017 y 2019, y entre 2017 y 2020.
93
164
212
267
Total de suscriptores a cablede televisión
2017 2018 2019 2020

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Taller cálculo diferencial

  • 1. Unidad 2 Actividad 2 Taller sobre derivadas Estimado estudiante: Para la elaboración del siguiente taller, tenga en cuenta los conceptos y ejercicios que ha practicado en su proceso de lectura, tal como se indica en la actividad. Recuerde mostrar el paso a paso de cada uno de los ejercicios planteados. Muchos éxitos. 1. En cada una de las siguientes funciones, determine la razón de cambio promedio en el valor de y al pasar de 𝑥 = −1 𝑎 𝑥 = 2. (Represéntelas gráficamente). a) 𝑦 = 𝑓(𝑥) = 𝑥2 − 2𝑥 + 3 b) 𝑦 = 𝑓(𝑥) = 𝑥3 − 2𝑥2 + 𝑥 + 2 c) 𝑦 = 𝑓(𝑥) = 4𝑥2 − 2 d) 𝑦 = 𝑓(𝑥) = 𝑥2 𝑥+4 2. Para los siguientes ejercicios:  determine la derivada de f por el método del límite,  calcule la pendiente cuando 𝑥 = −3 𝑦 𝑥 = 2 y  determine en qué parte de la función la pendiente es 0. Las funciones son: a) 𝑓(𝑥) = 15𝑥2 + 2𝑥 − 8 b) 𝑓(𝑥) = −𝑥2 − 3𝑥 + 5 c) 𝑓(𝑥) = 𝑥3 + 3𝑥2 d) 𝑓(𝑥) = −4𝑥2 − 8𝑥 e) 𝑓(𝑥) = 𝑥2 2 + 6𝑥 f) 𝑓(𝑥) = 3𝑥3 2 − 𝑥2 3. En los siguientes ejercicios:  Encuentre la expresión 𝑓´(𝑥)
  • 2.  Calcule 𝑓´(2)y 𝑓´(−1)  Determine los valores de x para los cuales 𝑓´(𝑥) = 0 a) 𝑓(𝑥) = (𝑥3 − 2𝑥)(−𝑥 + 6𝑥2 ) b) 𝑓(𝑥) = (4 − 2𝑥)(−𝑥3 − 2𝑥2 + 1) c) 𝑓(𝑥) = −3𝑥2 (𝑥2−2𝑥+1) d) 𝑓(𝑥) = −𝑥 𝑥−5 e) 𝑓(𝑥) = (−𝑥3 + 1)(−2𝑥2 − 2𝑥) f) 𝑓(𝑥) = 𝑥3 3 − 6𝑥2 + 8𝑥 − 7 4. Resuelva los siguientes problemas e indique el proceso para llegar a la solución. a) Un capital de $25.000.000 se invierte en una entidad financiera durante tres años, que paga un interés trimestral del 6 %.  Exprese el valor futuro y de este capital en función del tiempo x expresado en trimestres.  Halle la variación promedio de y con respecto a x durante el primer año y durante el segundo año. Compare estos resultados y explique su diferencia. b) Una población p de cierta ciudad, en función del tiempo x expresado en años, está decreciendo a una tasa continua del 10 % anual.  Exprese la población en función del tiempo x.  Halle la variación promedio de p con respecto a x durante los primeros cinco años y luego entre los cinco y diez años. Compare estos resultados. c) El sistema de cable ha estado en crecimiento durante los últimos cuatro años, debido a que la gente se ha quedado más tiempo en casa por cuestiones de la pandemia. La siguiente figura resume el total de suscriptores a cable de televisión durante los últimos 4 años:
  • 3.  Determine la razón de cambio promedio del total de suscriptores a cable de televisión entre 2017 y 2019, y entre 2017 y 2020. 93 164 212 267 Total de suscriptores a cablede televisión 2017 2018 2019 2020