Problem 88 - Calculus for Biology and Medicine

Question

Show that if \(0

Step-by-Step Solution

Answer

If \(0

Step 1: Understand the properties of the function

The given function is a power function where the base \(a\) is between 0 and 1, i.e., \(0

Step 2: Re-expressing the function in terms of the natural exponential function

We know that for any positive real number \(a\), we can express it in terms of the natural exponential function using the relation \(a = e^{atural log(a)}\). Thus, \(y = a^x = (e^{atural log(a)})^x = e^{x atural log(a)}\).

Step 3: Introduce a new expression for the exponent

Since \(0 < a < 1\), the natural logarithm of \(a\) (i.e., \(atural log(a)\)) will be negative. We can denote \(-atural log(a)\) as \(u\) so that it remains a positive constant. Therefore, \(y = e^{x atural log(a)} = e^{-x u}\).

Step 4: Define \( u\) in terms of \(a\) and identify \( u\) as \( extbackslash mu\)

Define \(u = -atural log(a)\) such that \(u\) is positive. Therefore, the expression \(y = e^{-x u}\) can be rewritten as \(y = e^{-u x}\) with \(u\) representing \( extbackslash mu\). Thus, \( extbackslash mu = -atural log(a)\) since it is positive when \(0 < a < 1\).

Key Concepts

Power Function

A power function is a type of mathematical function that is expressed in the form \(y = a^x\). Here, \(a\) is the base and \(x\) is the exponent. The base \(a\) can be any real number, and it is raised to the power of \(x\), which is the variable. Power functions exhibit a wide range of behaviors:

• If \(a > 1\), as \(x\) increases, the function value \(y\) also increases.
• If \(0 < a < 1\), as \(x\) increases, the function value \(y\) decreases, creating a descending trend.

In this context, where \(0 < a < 1\), the power function diminishes as \(x\) grows. Understanding how such functions behave is crucial in various fields like physics and economics, where diminishing returns are often modeled.

Natural Logarithm

The natural logarithm, denoted as \(\ln(x)\), is the inverse function of the exponential function \(e^x\). It provides a way to express powers in terms of multiplication, making it a valuable tool in algebra and calculus. When dealing with the natural logarithm:

• \(\ln(e) = 1\) because \(e\) is the base of natural logs.
• \(\ln(1) = 0\) because any number raised to zero power is 1.
• The natural logarithm of a number less than 1, such as \(a\) where \(0 < a < 1\), results in a negative value, indicating the reverse process of exponentiation.

Natural logarithms are extensively used in compound interest calculations, population growth models, and more. Here, they provide a way to convert a power function into an exponential form, simplifying complex algebraic expressions.

Exponential Form

The exponential form is a way to write expressions involving exponents neatly and systematically using the base of the natural logarithm \(e\). This is practical because \(e\) is a constant approximately equal to 2.71828, and growth processes often are described using \(e\). To express our power function \(y = a^x\) in exponential form, we apply the relation:

• Convert the base \(a\) using \(a = e^{\ln(a)}\).
• Raise \(e^{\ln(a)}\) to the power of \(x\), yielding \(y = e^{x \cdot \ln(a)}\).
• Given \(0 < a < 1\), \(\ln(a)\) is negative, hence \(x \cdot \ln(a)\) is re-expressed as \(-\mu x\), with \(\mu\) being positive.

This conversion simplifies operations and is widely applied in scientific fields to model exponential decay or growth processes, such as radioactive decay or cooling phenomena.

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Key Concepts

Power Function

A power function is a type of mathematical function that is expressed in the form \(y = a^x\). Here, \(a\) is the base and \(x\) is the exponent. The base \(a\) can be any real number, and it is raised to the power of \(x\), which is the variable. Power functions exhibit a wide range of behaviors:

• If \(a > 1\), as \(x\) increases, the function value \(y\) also increases.
• If \(0 < a < 1\), as \(x\) increases, the function value \(y\) decreases, creating a descending trend.

In this context, where \(0 < a < 1\), the power function diminishes as \(x\) grows. Understanding how such functions behave is crucial in various fields like physics and economics, where diminishing returns are often modeled.

Natural Logarithm

The natural logarithm, denoted as \(\ln(x)\), is the inverse function of the exponential function \(e^x\). It provides a way to express powers in terms of multiplication, making it a valuable tool in algebra and calculus. When dealing with the natural logarithm:

• \(\ln(e) = 1\) because \(e\) is the base of natural logs.
• \(\ln(1) = 0\) because any number raised to zero power is 1.
• The natural logarithm of a number less than 1, such as \(a\) where \(0 < a < 1\), results in a negative value, indicating the reverse process of exponentiation.

Natural logarithms are extensively used in compound interest calculations, population growth models, and more. Here, they provide a way to convert a power function into an exponential form, simplifying complex algebraic expressions.

Exponential Form

The exponential form is a way to write expressions involving exponents neatly and systematically using the base of the natural logarithm \(e\). This is practical because \(e\) is a constant approximately equal to 2.71828, and growth processes often are described using \(e\). To express our power function \(y = a^x\) in exponential form, we apply the relation:

• Convert the base \(a\) using \(a = e^{\ln(a)}\).
• Raise \(e^{\ln(a)}\) to the power of \(x\), yielding \(y = e^{x \cdot \ln(a)}\).
• Given \(0 < a < 1\), \(\ln(a)\) is negative, hence \(x \cdot \ln(a)\) is re-expressed as \(-\mu x\), with \(\mu\) being positive.

This conversion simplifies operations and is widely applied in scientific fields to model exponential decay or growth processes, such as radioactive decay or cooling phenomena.