For the function \( f(x) \), find the area from \( 0 \) to \( 1 \) where \( f(x) = e^x \). What can you conclude about its accumulation function?
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To find the area under the curve of \( f(x) = e^x \) from \( 0 \) to \( 1 \), you need to compute the definite integral: \[ \int_{0}^{1} e^x \, dx \] The antiderivative of \( e^x \) is \( e^x \) itself, so: \[ \int e^x \, dx = e^x + C \] Evaluating the definite integral, we get: \[ \left[ e^x \right]_{0}^{1} = e^1 - e^0 = e - 1 \] Thus, the area under the curve from \( 0 \) to \( 1 \) is \( e - 1 \), approximately \( 1.718 \). Regarding the accumulation function, since the function \( f(x) = e^x \) continuously increases, the accumulation function reflects the total area under \( f(x) \) as \( x \) increases. This means that \( F(x) = \int_{0}^{x} e^t \, dt \) will also be an increasing function, showcasing the exponential growth characteristic of \( e^x \). Therefore, the accumulation function \( F(x) \) provides a powerful insight into the behavior of areas under the curve of exponential functions!
