Problem 7 - Calculus for Biology and Medicine

Question

Let $$ f(x)=\frac{1}{x}, \quad x>0. $$ (a) Graph \(y=f(x)\) for \(0

Step-by-Step Solution

Answer

(b) \( x < \frac{1}{5} \) for \( f(x) > 5 \).

Step 1: Understand the function

The function given is \( f(x) = \frac{1}{x} \). This is a hyperbola, which is defined for all \( x > 0 \). As \( x \) increases, \( f(x) \) decreases. The domain for the function over which we are graphing is \( 0 < x \leq 4 \).

Step 2: Graph the function

To graph \( y = f(x) \) for \( 0 < x \leq 4 \), plot points for several values of \( x \), such as 0.5, 1, 2, 3, and 4. Note that at \( x = 1 \), \( y = 1 \), and at \( x = 4 \), \( y = \frac{1}{4} \). The function decreases from left to right and approaches the x-axis but never touches it.

Step 3: Solve for \( f(x) > 5 \)

We need to find the values of \( x \) where \( f(x) = \frac{1}{x} > 5 \). This can be solved by the inequality \( \frac{1}{x} > 5 \). To solve, rearrange to get \( x < \frac{1}{5} \). However, since the graph starts from \( x > 0 \), and nominally close to 0, it only means \( x < \frac{1}{5} \).

Step 4: Illustrate on the graph

On the graph created in Step 2, shade the region corresponding to \( x < \frac{1}{5} \) to indicate where \( y > 5 \). This region is to the left of the point where \( x = \frac{1}{5} \). Ensure this shaded area includes values approaching zero only (as they satisfy the condition \( x > 0 \)).

Key Concepts

Graphical representation

Graphical representation is an essential tool in calculus for visualizing functions and their behaviors. For the function \( f(x)=\frac{1}{x} \) defined over \( x>0 \), we need to represent it graphically to understand its characteristics and properties better. The nature of this function reveals a hyperbolic curve termed a hyperbola, where the function is only defined for positive values of \( x \).

When plotting the graph of \( y=f(x) \) for \( 0

• When \( x = 0.5 \), \( f(x) = 2 \)
• When \( x = 1 \), \( f(x) = 1 \)
• When \( x = 2 \), \( f(x) = 0.5 \)
• When \( x = 3 \), \( f(x) \approx 0.333 \)
• When \( x = 4 \), \( f(x) = 0.25 \)

Graphing these points provides a visual representation of the function. Notice how the graph approaches the x-axis but never touches or crosses it; this reflects the asymptotic nature of the hyperbolic function, indicating the value of \( f(x) \) gets closer to zero as \( x \) increases substantially.

Inequality solving

Inequalities are a mathematical expression of two values being unequal and are critical in solving various problems in calculus. In the exercise above, part (b) involves solving the inequality \( f(x) = \frac{1}{x} > 5 \). Let's break it down to understand how we can pinpoint which values of \( x \) lead to this condition being satisfied.
To solve this problem, we rearrange the inequality to find the boundary of \( x \):

• The inequality can be rewritten as \( \frac{1}{x} > 5 \).
  
• By reciprocating the terms, it becomes \( x < \frac{1}{5} \).
  

Since the domain where our function is valid (considering it’s defined for \( x > 0 \) from the original exercise) starts very close to 0, solutions for this inequality exist within the interval \( 0 < x < \frac{1}{5} \).

This demonstrates that the values of \( x \) which make the function \( f(x) \) greater than 5 will all be less than \( \frac{1}{5} \), representing an open interval—since both ends are not included.

Hyperbolic functions

Hyperbolic functions can sometimes seem a bit abstract and complex, yet they bear significant similarities to trigonometric functions. However, in our exercise with \( f(x)=\frac{1}{x} \), we are primarily looking at a hyperbolic function in the basic form, resulting in graphical characteristics distinct from trigonometric curves.
One primary feature of this hyperbolic function is its asymptotic behavior. As the function approaches a certain line, the curve does not intersect it:

• For \( f(x) \), this line is the x-axis, as \( f(x) \) never reaches zero.
• Hyperbolas display asymptotic behaviors both horizontally (along the x-axis) and vertically (along the y-axis, which is not observed here as our domain is only \( x>0 \)).

Understanding this intrinsic behavior helps when graphing or computing inequalities, as seen by the decreasing values of \( f(x) \) alongside increasing \( x \). Thus, the characteristics of hyperbolic functions include:

• A relationship between the function and its inverse \( f(x) \),
• An ever-decreasing or increasing trend for positive and negative inputs respectively.

This exercise especially benefits students by enhancing their understanding of how values of \( x \) impact the function's behavior, which could often lead to increased intuition in calculus applications.