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05/08/2023 · Middle School
1. Sea \( f \) una función definida en el intervalo abierto \( (a, b) \). (a) Dar una expresión para la pendiente de la recta secante a la gráfica de \( f \) que pasa por \( \left(x_{0}, f\left(x_{0}\right)\right) \) y \( \left(x_{0}+h, f\left(x_{0}+h\right)\right) \). (b) ¿Cómo se podría definir la pendiente de la recta tangente a la gráfica de \( f \) en el punto \( \left(x_{0}, f\left(x_{0}\right)\right) \) a partir de las pendientes de las rectas secantes? (c) Dar la definición formal de derivada en el valor \( x_{0} \).
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(a) La pendiente de la recta secante entre \( \left(x_{0}, f\left(x_{0}\right)\right) \) y \( \left(x_{0}+h, f\left(x_{0}+h\right)\right) \) es:
\[
m_{\text{secante}} = \frac{f\left(x_{0}+h\right) - f\left(x_{0}\right)}{h}
\]
(b) La pendiente de la recta tangente en \( \left(x_{0}, f\left(x_{0}\right)\right) \) se define como el límite de \( m_{\text{secante}} \) cuando \( h \) se hace infinitesimalmente pequeño.
(c) La derivada de \( f \) en \( x_{0} \) es:
\[
f'(x_{0}) = \lim_{h \to 0} \frac{f\left(x_{0}+h\right) - f\left(x_{0}\right)}{h}
\]
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