91Ó°ÊÓ

When dealing with improper integrals, understanding whether they converge is essential. An improper integral has some unusual aspect, often involving infinity either as a limit or within the integrand itself. The integral in our problem \[ \int_{0}^{\infty} e^{-0.05 t} \, dt \]is improper because it has an infinite upper limit. To determine if such an integral converges, you need to evaluate it using a limit approach, essentially asking whether it adds up to a finite number.
In this case, we write:\[ \lim_{b \to \infty} \int_{0}^{b} e^{-0.05 t} \, dt \]The concept of convergence involves checking if this limit results in a finite real number as the parameter \(b\) goes to infinity. If it does, the integral is said to converge, showing that even an infinite sum can yield a perfectly finite result. This is crucial in various fields including physics and probability, where integrating over an infinite range is common.

Exponential functions are incredibly vital in calculus and many applications. The function \( e^{-0.05 t} \) is an example of a decreasing exponential function. These functions have a general form \( e^{kt} \),where \(k\) is a constant that can specify growth or decay depending on its sign.

• If \(k > 0\), the function models exponential growth.
• If \(k < 0\), as in our problem, the function models exponential decay.

A key feature is the rate of change it describes — it always changes at a rate proportional to its current value. This makes them crucial in modeling processes where the rate of change is a function of the current state, such as in population dynamics, radioactive decay, and cooling processes. Understanding how exponential functions behave is essential to working with integrals that contain them, as in this exercise.

Antiderivatives, also known as indefinite integrals, are the reverse process of differentiation. For a given function \(f(t)\), its antiderivative is a function \(F(t)\) such that \[ F'(t) = f(t) \].In our exercise, we needed the antiderivative of the function \( e^{-0.05 t} \).To find an antiderivative, you often look for a function whose derivative gives you back the original function. For exponential functions like \( e^{kt} \),we use the rule: \[ \int e^{kt} \, dt = \frac{1}{k} e^{kt} + C \].Applying this to our integrand gives: \[ \int e^{-0.05 t} \, dt = -\frac{1}{0.05}e^{-0.05 t} + C = -20e^{-0.05 t} + C \].The constant \(C\) represents the arbitrary constant of integration, which "disappears" when evaluating definite integrals, as it cancels out in the subtraction when you apply the upper and lower limits.