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Understanding the first derivative test in calculus is essential for identifying relative extrema, which are the peaks and valleys on the graph of a function. This test uses the derivative of a function to determine whether a given point is a relative maximum or minimum.

As illustrated in the exercise with the function g(t) = e^{-t^2}, we begin by finding the derivative and its critical points. A critical point is where the first derivative is zero or undefined. After locating these points, we apply the first derivative test by examining the sign of the derivative before and after each critical point. If the sign changes from positive to negative, we have a relative maximum; if it changes from negative to positive, a relative minimum.

Thus, for our function g(t), the critical point at t = 0 shows a change from increasing (positive derivative) to decreasing (negative derivative), confirming a relative maximum at this point.

Exponential functions, like e^{-t^2} in our exercise, have unique characteristics that set them apart from other types of functions. Defined by the formula f(x) = a^{x}, where a is a positive constant (often Euler's number e), these functions grow or decay at rates proportional to their current value.

An important property of exponential functions is that they are never zero, meaning the function e^{-t^2} is always positive, regardless of the value of t. This insight is crucial in solving calculus problems because it allows us to ignore the exponential part when equating the derivative to zero to find critical points. This characteristic also indicates that as t approaches infinity, e^{-t^2} approaches zero but never actually reaches it.

A relative extremum of a function is a point where the function takes on a local maximum or minimum value; in other words, it's higher or lower than all nearby points. Relative maxima and minima are not necessarily the highest or lowest points on the graph, but they are important in understanding the shape and behavior of the function.

In the context of the function g(t) = e^{-t^2}, the relative extremum is identified by using the first derivative test at the critical point. Since the derivative g'(t) changes from positive to negative at t = 0, it suggests that the function has a peak, or relative maximum, at this point. These points are particularly useful for identifying turning points on the curve of the graph.

The absolute extremum of a function refers to the highest or lowest point over its entire domain. In many cases, this will occur at a critical point or an endpoint of the domain.

Considering our function g(t), its domain extends to positive and negative infinity. To identify absolute extrema, we not only look at the critical points but also the endpoints of the domain (if they exist) and the behavior of the function as it approaches infinity. As we saw from the previous steps, g(t) approaches zero as t heads towards positive or negative infinity, but it never reaches zero, indicating there are no absolute maxima or minima. This knowledge is crucial for a comprehensive understanding of the function's global behavior beyond the local context of relative extremum.