Problem 48 - Precalculus with Limits

Question

Graphing a Natural Exponential Function In Exercises \(45-50\) , use a graphing utility to graph the exponential function. $$s(t)=3 e^{-0.2 t}$$

Step-by-Step Solution

Answer

The function graph starts at an initial value of 3 and shows exponential decay, with the curve decreasing by roughly 20% for each unit increase in time. As time increases, the function value approaches 0 without ever reaching it, creating an asymptote along the x-axis.

Step 1: Understanding the function

The function s(t) = 3e^{-0.2t} is an exponential decay function where 3 is the initial amount (or the y-intercept) and -0.2 is the decay factor. 't' corresponds to time. So, the value of s(t) decreases as t increases.

Step 2: Identify key attributes of the function

It's important to note that the initial amount of the function is 3 (when t = 0) and because the exponent -0.2t is negative, we have an exponential decay. The decay rate of 0.2 means the function value decreases by roughly 20% each unit time increase.

Step 3: Graphing the function

To graph this function, plot the y-intercept at (0,3) on your graphing tool. Then, use the fact that the function value decreases by roughly 20% each unit time to plot the curve. The curve should start off high (at y = 3) and asymptotically approach the x-axis as t increases.

Step 4: Verification

To verify, plug in some values for t (e.g., t = 1, t = 2, t = 3, etc.) to ensure the values of s(t) are decreasing and approaching 0 as t increases. This will ensure the graph is correct.

Key Concepts

Exponential Decay

Exponential decay is a process that decreases at a rate proportional to its current value. When graphing an exponential decay function like the one in our exercise, \( s(t) = 3e^{-0.2t} \), you can see this phenomenon in action.

The graph will display a rapid decrease in value from the y-intercept, leveling off as it approaches the x-axis without ever touching it, known as an asymptote. This reflects a common occurrence in natural processes, such as radioactive decay or the cooling of a warm object in a cooler environment.

Understanding that the variable 't' often represents time can help you to visualize why an exponential decay graph slopes downward. As time progresses, the quantity represented by \( s(t) \) diminishes, mirroring how many real-world quantities dissipate over time.

Natural Exponential Function

The natural exponential function is a special exponential function often denoted as \( e^x \), where \( e \) (approximately equal to 2.71828) is the base of the natural logarithm. The function you are asked to graph, \( s(t) = 3e^{-0.2t} \), utilizes this natural base.

This function has a continuous growth rate and is highly significant in calculus and complex number theory because of its unique properties. For example, its derivative is equal to itself. Moreover, the natural exponential function is used in compounding interest, population growth models, and even in the physics of waves.

Graphing this type of function provides a visual representation of growth or decay that is happening continuously and proportionally—a key concept in many areas of mathematics and science.

Graphing Utility

A graphing utility, such as a graphing calculator or software, is a tool that enables you to visualize functions and their behavior. This is particularly helpful with exponential functions, given their non-linear nature.

To graph \( s(t) \) using a graphing utility, you usually input the equation and set an appropriate viewing window. The utility takes over from there, plotting points and rendering the curve. Using a graphing utility allows for more precision and ease in understanding complex behaviors of functions—especially when finding exact points is difficult or if the function includes terms not easily graphed by hand.

By leveraging the power of technology, you can also explore how changing different parameters of the function affects its graph, enhancing your grasp of the concept behind the mathematical representation.

Decay Factor

In the context of our function \( s(t) = 3e^{-0.2t} \), the decay factor is represented by -0.2. The decay factor determines the rate at which the exponential function decreases. The negative sign indicates that the function is indeed experiencing decay, rather than growth.

Understanding the decay factor is crucial when interpreting or predicting the function's behavior over time. A larger absolute value of the decay factor implies a quicker rate of decay, impacting how steeply the graph descends as 't' increases. Thus, for the given function, the quantity represented by the function \( s(t) \) will decrease by about 20% for each unit increase in time, which is essentially what the decay factor tells us.

This concept is vital in fields like finance, physics, and environmental science, where it's used to model depreciation, radioactive decay, or concentration of substances in a medium over time.