Problem 54 - Calculus

Question

Are the statements true or false? Give an explanation for your answer. The function \(y=5-3 e^{-4 t}\) has a horizontal asymptote of \(y=5\)

Step-by-Step Solution

Answer

True. The function approaches a horizontal asymptote at \(y=5\).

Step 1: Identify the form of the function

Examine the given function, \(y = 5 - 3 e^{-4t}\). This function includes an exponential term \(e^{-4t}\), which will help in determining the behavior as \(t\) approaches positive or negative infinity.

Step 2: Analyze the behavior of the exponential term

The term \(e^{-4t}\) contains the exponential function. As \(t\rightarrow \infty\), \(e^{-4t}\rightarrow 0\), because the exponent \(-4t\) becomes very large negative, making the output of the exponential term approach zero.

Step 3: Evaluate the function limit as t approaches infinity

Determine the limit of \(y\) as \(t\rightarrow \infty\):\[\lim_{{t \to \infty}} y = \lim_{{t \to \infty}} \left(5 - 3e^{-4t}\right) = 5 - 3 \cdot 0 = 5\]. Thus, as \(t\) approaches infinity, \(y\) approaches 5.

Step 4: Conclusion of horizontal asymptote

Since the limit of the function \(y\) as \(t\rightarrow \infty\) is 5, the function approaches 5 for very large values of \(t\). This confirms that the horizontal asymptote of the function \(y = 5 - 3 e^{-4t}\) is indeed \(y=5\).

Key Concepts

Horizontal Asymptote

The term "horizontal asymptote" might sound complex, but it's quite a simple concept to grasp. It refers to a particular value that the function approaches as the input goes to positive or negative infinity. In simpler words, it's the "line" the graph of a function gets very close to but never quite touches.

In the case of the function \( y = 5 - 3e^{-4t} \), to find the horizontal asymptote, we observe the behavior of the function's output \( y \) as \( t \) becomes very large. When this happens, the exponential part \( e^{-4t} \) tends to zero because \(-4t\) turns into a large negative number making \( e^{-4t} \) really small. Hence, the formula simplifies to \( y = 5 - 3 \cdot 0 = 5 \).

• The horizontal asymptote is a line the function approaches as \( t \) goes to infinity.
• For the given function, this line, or horizontal asymptote, is \( y = 5 \).

Limits at Infinity

The concept of "limits at infinity" is a fundamental idea in calculus. It involves evaluating what a function does when its variable heads towards either positive or negative infinity. Essentially, we're interested in discovering what value the function gets closer to as the variable increases or decreases without bound.

For the function \( y = 5 - 3e^{-4t} \), as \( t \) goes to infinity, the term \( e^{-4t} \) decreases towards zero. This is because the exponent \( -4t \) becomes a large negative number, driving the value of \( e^{-4t} \) close to zero. Therefore, the limit of the function as \( t \) reaches infinity can be calculated as follows:

• \( \lim_{{t \to \infty}} (5 - 3e^{-4t}) = 5 - 3 \cdot 0 = 5 \)

This illustrates how the function approaches \( y = 5 \) as \( t \) continues to grow larger and larger.

Exponential Decay

"Exponential Decay" is a term used to describe situations where something decreases at a rate proportional to its current value. It's a concept often observed in processes like cooling, radioactive decay, and more.

In analyzing \( y = 5 - 3e^{-4t} \), we see an example of exponential decay. The function \( e^{-4t} \) represents the exponential decay portion of the equation. As \( t \) increases, \( e^{-4t} \) diminishes towards zero. This decay causes the influence of \(-3e^{-4t}\) on \(y\) to vanish over time.

• Exponential decay involves a rapid decrease that slows over time.
• For large \( t \), the effect of the decay, through \( e^{-4t} \), becomes negligible.

This behavior is why the function stabilizes around the horizontal asymptote of \( y = 5 \), as the influence of the exponential term disappears.