To solve this problem, we need to follow these steps:
- Define the function \( f(x) \) as given in the problem.
- Compute the derivative \( f'(x) \) using the Fundamental Theorem of Calculus and the Chain Rule.
- Evaluate \( f'(k) \) and express it in the form \( A e^B \).
- Extract the values of \( A \) and \( B \) and compute \( A + B \).
Nos piden encontrar el valor de \(A + B\) dado que \(f(x) = \int_{k}^{kx} e^{u^2} \, du\) y \(f'(k) = A e^B\), con \(k = 11\).
Primero, definimos la función \(f(x)\):
\[ f(x) = \int_{11}^{11x} e^{u^2} \, du \]
Para encontrar \(f'(x)\), utilizamos la regla de Leibniz para la derivada de una integral con límites variables:
\[ f'(x) = \frac{d}{dx} \left( \int_{11}^{11x} e^{u^2} \, du \right) = e^{(11x)^2} \cdot \frac{d}{dx}(11x) - e^{11^2} \cdot \frac{d}{dx}(11) \]
La derivada de \(11x\) con respecto a \(x\) es 11, y la derivada de 11 con respecto a \(x\) es 0:
\[ f'(x) = e^{(11x)^2} \cdot 11 - e^{121} \cdot 0 \]
\[ f'(x) = 11 e^{(11x)^2} \]
Ahora evaluamos \(f'(x)\) en \(x = k = 11\):
\[ f'(11) = 11 e^{(11 \cdot 11)^2} = 11 e^{121^2} = 11 e^{14641} \]
Comparando con \(f'(k) = A e^B\), tenemos:
\[ A = 11 \]
\[ B = 14641 \]
Sumamos \(A\) y \(B\):
\[ A + B = 11 + 14641 = 14652 \]
\[
\boxed{14652}
\]
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