Problem 8 - Calculus for Biology and Medicine

Question

Let $$ f(x)=e^{-x}, \quad x \geq 0 $$ (a) Graph \(y=f(x)\) for \(0 \leq x \leq 6\). (b) For which values of \(x\) is \(y=f(x)\) less than 0.1? (c) Illustrate your result in (b) on the graph that you obtained in (a).

Step-by-Step Solution

Answer

\( y=f(x) \) is less than 0.1 for \( x > 2.302 \).

Step 1: Understanding the Function

The function given is \( f(x) = e^{-x} \) for \( x \geq 0 \). This is an exponential decay function. As \( x \) increases, \( f(x) \) decreases towards zero.

Step 2: Graphing the Function

To graph \( y = f(x) = e^{-x} \) for \( 0 \leq x \leq 6 \), create a table of values for \( x = 0, 1, 2, 3, 4, 5, 6 \) and compute the corresponding \( f(x) \). Plot these points and sketch the curve starting at \( y = 1 \) when \( x = 0 \) and approaching \( y = 0 \) as \( x \) increases.

Step 3: Determine Values Where \( f(x) < 0.1 \)

To find where \( f(x) < 0.1 \), solve the inequality \( e^{-x} < 0.1 \). Rewrite it as \( -x < \ln(0.1) \) which simplifies to \( x > \-\ln(0.1) \). Using a calculator, find that \( x > 2.302 \). Thus, for \( x > 2.302 \), \( f(x) < 0.1 \).

Step 4: Illustrating the Result on the Graph

On the graph obtained in Step 2, draw a horizontal line at \( y = 0.1 \). The region where the graph of \( y = f(x) \) is below this line represents the portion where \( f(x) < 0.1 \). Highlight the portion of the \( x \)-axis where \( x > 2.302 \) to illustrate this result.

Key Concepts

Graphing Exponential Functions

Understanding how to graph exponential functions, like the given function \( f(x) = e^{-x} \), is crucial in mathematics. This particular function is known for its exponential decay nature. In this type of decay, as the input value \( x \) increases, the output \( f(x) \) decreases. To visualize this graphically, you can start by selecting a range of \( x \) values, such as \( 0 \leq x \leq 6 \), and calculate the corresponding \( f(x) \) values. Here are some sample computations for \( y = f(x) \):

• When \( x = 0 \), \( f(x) = 1 \)
• When \( x = 1 \), \( f(x) \approx 0.3679 \)
• When \( x = 2 \), \( f(x) \approx 0.1353 \)
• When \( x = 3 \), \( f(x) \approx 0.0498 \)
• Continue this for \( x = 4, 5, 6 \)

By plotting these \( (x, f(x)) \) points on a graph and sketching the curve, you observe that the graph starts at \( (0, 1) \) which is the y-intercept, and slowly descends towards \( y = 0 \) as \( x \) increases. This graphical representation helps in visualizing the behavior of exponential decay as the line approaches but never quite meets the x-axis.
It becomes a powerful tool for predicting values and understanding the nature of the function.

Inequality Solving

Solving inequalities involving exponential functions like \( f(x) = e^{-x} \) involves determining the range of x values that satisfy the inequality. In the problem, we want to find when \( f(x) < 0.1 \). The inequality \( e^{-x} < 0.1 \) can be rewritten using logarithmic transformations, which assists in solving for \( x \). By taking the natural logarithm on both sides, we have: \[ -x < \ln(0.1) \] Rearranging gives us: \[ x > -\ln(0.1) \] Calculating \( \ln(0.1) \), which is approximately \(-2.302\), we get: \[ x > 2.302 \] This solution tells us that \( f(x) \) will be less than 0.1 for any \( x \) greater than approximately 2.302. This process demonstrates a vital skill in inequality solving, namely the transformation technique, which is particularly useful when dealing with exponential equations.

Function Behavior Analysis

Analyzing the behavior of functions helps in understanding its different traits or characteristics under various conditions. With the exponential decay function \( f(x) = e^{-x} \), the behavior can be analyzed in multiple ways:

• Decay Rate: The rate of decay for the function is persistent and constant, meaning that \( f(x) \) continually decreases as \( x \) increases, which characterizes exponential decay. This decrease slows down over time as the value of the function approaches zero asymptotically.
• Intercepts: This function has a y-intercept at 1 when \( x = 0 \). There are no x-intercepts, as \( f(x) \) never actually reaches zero.
• Graphical Behavior: Visually, as we plot the graph from earlier, you identify that the curve approaches the x-axis closely but never touches it completely. This is a hallmark of exponential functions as they exhibit asymptotic behavior, approaching a line infinitely without crossing it.

Analyzing these aspects provides insights into the real-world application and interpretation of exponential functions, especially in fields like biology and finance where such functions often model growth or decay phenomena. Understanding these characteristics helps students grasp how exponential functions behave and why they are so useful in modeling realistic situations.