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Derivadas y funciones
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Ejercicio de deriadas y funciones
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Guía de actividades y rúbrica de evaluación – Tarea 3 -Derivadas
Anexo 3 - Ejercicios Tarea 3
EJERCICIOS
1. De acuerdo con la definición de derivada de una función, es:
𝒇(𝒙 + 𝒉) − 𝒇(𝒙)
𝒇´(𝒙) = 𝐥𝐢𝐦
𝒉→𝟎 𝒉
Calcular la derivada de las siguientes funciones aplicando la definición
de derivada, es decir, siguiendo el proceso del límite, luego evaluar en
un punto x (escogido por el estudiante) y, finalmente mediante
GeoGebra graficar la recta tangente a la función original y su pendiente
en el punto x escogido, realizar su comprobación y análisis gráfico.
Recuerde que uno de los elementos a evaluar en la actividad es al
análisis gráfico en GeoGebra.
C 𝑓(𝑥) = −2.5𝑥 2 − 4.3𝑥 − 1.2
Solución
𝑓(𝑥 + ℎ) = −2.5 (𝑥 + ℎ)2 − 4.3(𝑥 + ℎ) − 1.2
Simplificamos
𝑓(𝑥 + ℎ) = −2.5 (𝑥 2 + 2𝑥ℎ + ℎ2 ) − 4.3𝑥 − 4.3ℎ − 1.2
𝑓(𝑥 + ℎ) = −2.5𝑥 2 − 5𝑥ℎ − 2.5ℎ2 − 4.3𝑥 − 4.3ℎ − 1.2
Calculamos
𝑓(𝑥 + ℎ) − 𝑓(𝑥) = (−2.5𝑥 2 − 5𝑥ℎ − 2.5ℎ2 − 4.3𝑥 − 4.3ℎ − 1.2) −
(−2.5𝑥 2 − 4.3𝑥 − 1.2)
𝑓(𝑥 + ℎ) − 𝑓(𝑥) = −5𝑥ℎ − 2.5ℎ2 − 4.3ℎ
, Dividimos por h y tomamos el límite de cuando ℎ → 0
−5𝑥ℎ − 2.5ℎ2 − 4.3ℎ
𝒇´(𝒙) = 𝐥𝐢𝐦
𝒉→𝟎 𝒉
𝒇´(𝒙) = 𝐥𝐢𝐦(−5𝑥 − 2.5ℎ − 4.3)
𝒉→𝟎
𝒇´(𝒙) = −5x − 4.3
Evaluamos derivada
𝒇´(𝟐) = −5(2) − 4.3 = −10 − 4.3 = −14.3
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