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An exponential function is a mathematical function of the form \( f(x) = a \cdot e^{bx} \), where \( e \) is the base of the natural logarithm, approximately equal to 2.71828, and \( a \) and \( b \) are constants. In the context of the integral that we are solving, we are primarily interested in the function \( e^{-2x} \), which is an exponential decay function due to the negative sign in the exponent.

• The graph of an exponential decay function slopes downward as you move from left to right, getting progressively closer to zero.
• This behavior is key in many applications, such as modeling decay processes or cooling.

Understanding exponential functions is crucial in calculus, as they often appear in integrals and derivatives, forming the backbone of many problems and solutions. Here, our task is to find the area under the curve of the exponential function, which is achieved by calculating the definite integral of \( 5e^{-2x} \).

An antiderivative, also known as an indefinite integral, is a function \( F(x) \) such that \( F'(x) = f(x) \), where \( f(x) \) is the original function. Finding an antiderivative is the reverse process of differentiation.

• The antiderivative of \( e^{ax} \) is \( \frac{1}{a}e^{ax} \) plus a constant.
• Thus, for \( e^{-2x} \), the antiderivative is \( -\frac{1}{2}e^{-2x} \).

In the integral \( \int 5e^{-2x} \, dx \), we multiply the antiderivative by the constant 5, giving \( -\frac{5}{2} e^{-2x} \). Finding the correct antiderivative is essential for proceeding with evaluating a definite integral, especially when utilizing the Fundamental Theorem of Calculus.

The Fundamental Theorem of Calculus is a powerful tool in mathematics linking differentiation and integration. It states that if \( F \) is an antiderivative of \( f \) on an interval \([a, b]\), then: \[ \int_{a}^{b} f(x) \, dx = F(b) - F(a) \]This means the definite integral of a function over an interval can be found by evaluating its antiderivative at the upper and lower limits.

• Helps convert the process of integration into simple subtraction.
• Used here to find the definite integral of \( 5e^{-2x} \) from 0 to 4.

By applying this theorem to the antiderivative \( -\frac{5}{2} e^{-2x} \), we evaluate at 4 and 0, simplifying the problem to: \[ -\frac{5}{2} e^{-8} + \frac{5}{2} \times 1 \]Understanding this theorem is vital because it links the concept of area under a curve to antiderivatives, streamlining calculations significantly.

In definite integrals, limits of integration define the interval over which the area under a curve is evaluated. These are the bounds \( a \) and \( b \) in the integral \( \int_{a}^{b} f(x) \, dx \).

• For this problem, the limits of integration are 0 and 4.
• They specify that we calculate the area from \( x = 0 \) to \( x = 4 \).

They allow us to calculate specific areas rather than indefinite areas, making results applicable to real-world scenarios.When calculating a definite integral, the antiderivative is evaluated at both the upper and lower limits, and the two results are subtracted. Here, the subtraction process is represented by \(-\frac{5}{2} e^{-8} - \left(-\frac{5}{2} e^{0}\right)\), which considers the function's behavior between these bounds. Understanding limits of integration is crucial as they confine the region over which an integral is computed, ensuring precision and application.