Problem 2 - Biocalculus Calculus for the Life Sciences

Question

(a) Graph the functions \(f(x)=1 / x^{1.1}\) and \(g(x)=1 / x^{0.9}\) in the viewing rectangles \([0,10]\) by \([0,1]\) and \([0,100]\) by \([0,1] .\) (b) Find the areas under the graphs of \(f\) and \(g\) from \(x=1\) to \(x=b\) and evaluate for \(b=10,100,10^{4}, 10^{6}, 10^{10},\) and \(10^{20}\) . (c) Find the total area under each curve for \(x \geqslant 1,\) if it exists.

Step-by-Step Solution

Answer

Graph the functions, calculate definite integrals for given bounds, and assess total areas as b approaches infinity.

Step 1: Graphing the Functions

We need to graph both functions, \( f(x) = \frac{1}{x^{1.1}} \) and \( g(x) = \frac{1}{x^{0.9}} \), in two viewing rectangles: \([0,10]\) by \([0,1]\) and \([0,100]\) by \([0,1]\). This can be done using a graphing calculator or software. You should observe how the graphs behave as \(x\) increases, noting the faster decay of \(f(x)\) compared to \(g(x)\) due to their different exponents.

Step 2: Area Under the Curve for Given Bounds

For each given \(b\), calculate the area under the curve from \(x = 1\) to \(x = b\) for both functions. The integral of \(f(x)\) is \( \int_1^b \frac{1}{x^{1.1}} \, dx \), which equals \( \left[\frac{x^{-0.1}}{-0.1}\right]_1^b \). The result is \( \frac{10}{9} (1 - b^{-0.1}) \). Similarly, calculate the integral for \(g(x)\) as \( \int_1^b \frac{1}{x^{0.9}} \, dx \), which equals \( \left[\frac{x^{0.1}}{0.1}\right]_1^b \), resulting in \( 10(b^{0.1} - 1) \).

Step 3: Evaluate the Areas for Specific b Values

Compute the areas for \(b = 10, 100, 10^{4}, 10^{6}, 10^{10}, 10^{20}\):- For \(f(x)\): - \(b=10\): \( \frac{10}{9}(1 - 10^{-0.1}) \) - \(b=100\): \( \frac{10}{9}(1 - 100^{-0.1}) \) - \(b=10^4\): \( \frac{10}{9}(1 - 10^{-0.4}) \) - \(b=10^6\): \( \frac{10}{9}(1 - 10^{-0.6}) \) - \(b=10^{10}\): \( \frac{10}{9}(1 - 10^{-1}) \) - \(b=10^{20}\): \( \frac{10}{9}(1 - 10^{-2}) \)- For \(g(x)\): - \(b=10\): \(10(10^{0.1} - 1)\) - \(b=100\): \(10(100^{0.1} - 1)\) - \(b=10^4\): \(10(10^{0.4} - 1)\) - \(b=10^6\): \(10(10^{0.6} - 1)\) - \(b=10^{10}\): \(10(10 - 1)\) - \(b=10^{20}\): \(10(10^2 - 1)\)

Step 4: Determine the Total Area as b Approaches Infinity

Investigate the behavior of the integrals as \(b\) approaches infinity. For \(f(x)\), compute the limit of \( \frac{10}{9}(1 - b^{-0.1}) \) as \(b \to \infty\). The expression \(b^{-0.1}\) approaches 0, hence the area converges to \(\frac{10}{9}\).Meanwhile, for \(g(x)\), compute the limit of \( 10(b^{0.1} - 1) \) as \(b \to \infty\). Since \(b^{0.1}\) becomes infinite, the area is infinite. Thus, for \(g(x)\), the area does not exist.

Key Concepts

Graphing Functions

Graphing functions is a fundamental skill in calculus that helps visualize the behavior of different equations. When graphing the functions \( f(x) = \frac{1}{x^{1.1}} \) and \( g(x) = \frac{1}{x^{0.9}} \), it is crucial to use graphing software or calculators to view these graphs over specific intervals, such as \([0,10]\) by \([0,1]\) and \([0,100]\) by \([0,1]\).

Here are key observations to make:

• The decay rate of \( f(x) \) is steeper compared to \( g(x) \) because of the higher power in its denominator.
• Both functions asymptotically approach zero as \( x \) becomes very large.

Observing how these functions behave provides insights into their properties and potential applications, such as decay processes or distribution functions.

Area Under a Curve

The area under a curve is a crucial concept in calculus, providing insight into the integral of a function. For the functions \( f(x) = \frac{1}{x^{1.1}} \) and \( g(x) = \frac{1}{x^{0.9}} \), the task is to calculate the area from \( x = 1 \) to a variable endpoint \( x = b \).

This involves integration:

• For \( f(x) \), integrate to obtain \( \frac{10}{9} (1 - b^{-0.1}) \).
• For \( g(x) \), the integral results in \( 10(b^{0.1} - 1) \).

To find specific area values, substitute different \( b \) values such as \( 10, 100, 10^4, 10^6, 10^{10}, \text{and} 10^{20} \). Understanding how the area develops as \( b \) changes is critical in fields like physics and economics, where it represents quantitative measures like total energy or cost.

Limit of a Function

The limit of a function is a fundamental concept that determines the behavior of a function as it approaches a specific point or infinity. In this context, as \( b \to \infty \), limits reveal the total area under each curve starting from \( x = 1 \).

For \( f(x) \):

• The function \( \frac{10}{9}(1 - b^{-0.1}) \) converges to \( \frac{10}{9} \) as \( b \to \infty \). This means the total area is finite and equal to \( \frac{10}{9} \).

For \( g(x) \):

• The expression \( 10(b^{0.1} - 1) \) does not converge; it diverges to infinity, indicating that the total area under this curve is infinite.

This difference helps differentiate between functions that have a finite total area from those that do not, informing potential applications and expectations.