91Ó°ÊÓ

In calculus, an antiderivative is a function whose derivative is the original function given. For the integral \(\[\begin{equation} \int_{a}^{b} -e^{t} dt \end{equation}\]\), we need to find a function whose derivative is \(-e^{t}\). When you differentiate \-e^{t}\, you get back \(-e^{t}\). Thus, \-e^{t}\ is an antiderivative of \-e^{t}\. Finding antiderivatives is crucial for solving integrals. This process basically reverses differentiation, allowing us to compute areas under curves, among other applications.

The Fundamental Theorem of Calculus links differentiation and integration and provides a way to evaluate definite integrals.
It consists of two main parts:

• The first part allows us to compute the derivative of an integral function, revealing that differentiation and integration are inverse processes.
• The second part allows us to calculate definite integrals using antiderivatives.

This is stated as:
If \(F\) is an antiderivative of \(f(t)\), then \(\[\begin{equation} F(b) - F(a) \end{equation}\]\), gives the value of the integral \(\[\begin{equation} \int_{a}^{b} f(t) dt \end{equation}\]\). In our problem, we evaluate the antiderivative \(-e^{t}\) at the bounds \(a\) and \(b\): \(-e^{b} - (-e^{a}) \). This simplifies to \(-e^{b} + e^{a} \).

Exponential functions have the form \(e^{x}\), where \(e\) is Euler's number (approximately 2.718). They are unique because their rate of change (derivative) is proportional to their value.
In the integral \(\[\begin{equation}\int_{a}^{b} -e^{t} dt \end{equation}\]\), the function \(-e^{t}\) is an exponential function. This means it grows (or decays, since it’s negative) exponentially.
Knowing how exponential functions behave helps to analyze their antiderivatives and evaluate integrals. For example, since \(\[\begin{equation}\frac{d[-e^{t}]}{dt} = -e^{t} \end{equation}\]\), integrating and evaluating using bounds allows us to find areas under curves that are exponential in nature.