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Objective:
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Digital Tools and AI for Teaching Learning and Research
3º mat pc_e_03_septiembre-alumno
1. 1
Cuadernillo
Matemática
Tercer Curso Bachillerato Científico y Técnico
Semana – Arapokõindy: 30 de agosto al 3 de septiembre
Estudiante – Temimbo’e: ………………………………………………………………………………………
Tema – Mbo’epyrã: Recta tangente y normal.
Actividad de inicio – Tembiapo ñepyrũrã
Recuerdo
“El problema de la recta tangente”
1. Para deducir la definición de la derivada de una función en un punto dado, hemos
estudiado uno de los problemas que desarrollo el cálculo: “El problema de la recta
tangente “
2. 2
La pendiente 𝑚 de la recta tangente es:
𝑚 = 𝑓′(𝑥) = lim
∆𝑥→0
𝒇(𝒙 + ∆𝒙) − 𝒇(𝒙)
∆𝒙
Por lo tanto, se dedujo su ecuación:
Ecuación de la recta tangente:
𝒚 − 𝒚𝟎 = 𝒎 ∙ (𝒙 − 𝒙𝟎 )
en el punto 𝐴 = [𝑥0, 𝑓(𝑥0)]
2. Si trazamos una recta perpendicular a
la recta tangente de la función 𝑓(𝑥)
en dicho punto, en donde 𝑚 es la
pendiente de la recta tangente (T) y
𝑚1 la pendiente de la recta
perpendicular (S) a dicha tangente.
Respondo las siguientes preguntas.
¿Cuál es la condición de perpendicular entre
dos rectas con respecto a sus pendientes?
………………………………………………………………
………………………………………………………………
¿Cómo se la nombra a la recta perpendicular
a la recta tangente de una curva?
………………………………………………………………
………………………………………………………………
Recta tangente a la curva
Si una función 𝑦 = 𝑓(𝑥) posee una derivada en el punto 𝑥0, la curva tiene una tangente en
𝑃 = (𝑥0, 𝑦0) cuya pendiente es 𝑚 = 𝑓′(𝑥) y la ecuación de la recta que pasa por un punto y
con una pendiente dada es:
𝒚 − 𝒚𝟎 = 𝒎 ∙ (𝒙 − 𝒙𝟎 )
S
3. 3
Por lo tanto, si se sustituye la pendiente por la derivada, la ecuación de la recta tangente en un
punto de una curva es:
Si 0
=
m tiene tangente horizontal a la curva
Si
=
m tiene tangente vertical a la curva.
Recta normal a la curva
Una recta normal a una curva en uno de sus puntos es la recta que pasando por dicho punto es
perpendicular a la recta tangente en él.
✓ La condición de perpendicular entre dos rectas es: 𝑚1 = −
1
𝑚
✓ Luego, la ecuación de la recta normal es: 𝑦 − 𝑦0 = −
1
𝑚
∙ (𝑥 − 𝑥0 )
✓ Es equivalente a expresarlo en la forma:
3. Resuelvo las siguientes situaciones
a. Determino la ecuación de la recta tangente y la recta normal en el punto 𝑃 = (2,4) de
la función 𝑓(𝑥) = 𝑥2
. Represento gráficamente.
b. Determino la ecuación de la recta tangente a la curva 8
2
2
=
+ y
x en el punto 𝑃(2,2).
𝒚 − 𝒚𝟎 = 𝑓′(𝑥) ∙ (𝒙 − 𝒙𝟎 )
𝑦 − 𝑦1 = −
1
𝑓´(𝑥1)
(𝑥 − 𝑥1)
4. 4
Actividades para enviar a través del Gestor de Tareas
Tarea correspondiente a la semana: 30 de agosto al 3 de septiembre
Tema: Recta tangente y normal.
Fecha de envío: ________________
Nombre del estudiante: _________________________________________
Encuentro las ecuaciones de las rectas tangente y
normal a la curva en los puntos indicados. Represento
en forma grafica
1. Hallo la ecuación de la recta tangente y normal a la
curva 6
)
( 2
−
−
= x
x
x
f en el punto )
6
,
4
(
P .
2. 2
x
y = en 𝑃 (3,9)
3. 80
16
5 2
2
=
+ y
x en 𝑃 (4,0)
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