Problem 6 - Calculus Early Transcendentals: Pearson New International Edition

Question

Find \(F(f(t), g(t))\) if \(F(x, y)=e^{x}+y^{2}\) and \(f(t)=\ln t^{2}\) \(g(t)=e^{t / 2} .\)

Step-by-Step Solution

Answer

The expression is \( t^2 + e^{t} \).

Step 1: Identify the Functions

Begin by clearly identifying the functions you're working with. We have a composite function \( F(x, y) = e^{x} + y^{2} \) that we're applying to \( f(t) = \ln(t^2) \) and \( g(t) = e^{t/2} \). The task is to find \( F(f(t), g(t)) \).

Step 2: Substitute Functions into Composite Function

Substitute \( f(t) = \ln(t^2) \) and \( g(t) = e^{t/2} \) into \( F(x, y) \). This gives us: \( F(\ln(t^2), e^{t/2}) = e^{\ln(t^2)} + (e^{t/2})^2 \).

Step 3: Simplify the Expression using Logarithmic and Exponential Rules

Use the property that \( e^{\ln(a)} = a \) and simplify the expression: \( e^{\ln(t^2)} = t^2 \). Also, calculate \( (e^{t/2})^2 = e^{t} \). Thus, the expression becomes \( t^2 + e^{t} \).

Step 4: Finalize the Expression

Combine the simplified results to write the final expression: \( F(f(t), g(t)) = t^2 + e^{t} \).

Key Concepts

Exponential Functions

Exponential functions are mathematical expressions involving an exponent that is a variable. In simpler terms, they have the form \( f(x) = a^{x} \), where \( a \) is a constant and \( x \) is the exponent, which is typically a variable. These functions are vital in various fields, as they model growth and decay processes, such as population growth or radioactive decay.

Some key characteristics of exponential functions include:

• Growth or Decay: If the base \( a > 1 \), the function represents exponential growth; if \( 0 < a < 1 \), it signifies exponential decay.
• Horizontal Asymptote: Typically, exponential functions have a horizontal asymptote along the x-axis, indicating a stable endpoint as \( x \) approaches infinity or negative infinity.
• Base "e": The natural exponential function involves the mathematical constant \( e \approx 2.71828\). It's frequently encountered in calculus and complex exponential functions, like \( e^x \).

In our problem, the exponential expression \( e^{x} \) is a central element of the composite function \( F(x, y) = e^{x} + y^2 \), demonstrating how exponential functions can be involved in more intricate mathematical models.

Logarithmic Functions

Logarithmic functions are the inverses of exponential functions. They can be represented as \( f(x) = \log_a(x) \), where \( a \) is the base, and they allow the exponent to be the subject of the equation. Essentially, if \( a^y = x \), then \( y = \log_a(x) \).

The natural logarithm function specifically uses \( e \) as its base and is denoted as \( \ln(x) \). This function is crucial in calculus and many applications involving time and growth rates. Here’s why logarithmic functions are important:

• Inverse Nature: They solve equations involving exponents, making them invaluable for equations where the exponent is unknown.
• Properties: Use them to simplify multiplication, division, and exponentiation into addition, subtraction, and multiplication, respectively.
• Natural Logarithms: Since \( \ln(e^x) = x \), natural logs help in undoing exponential functions, as shown in our solution where \( e^{\ln(t^2)} = t^2 \).

In the problem, the function \( f(t) = \ln(t^2) \) exemplifies how logarithmic functions can transform terms within composite functions, making them easier to handle.

Function Composition

Function composition involves combining two functions to create a new one by plugging the output of one function into the input of another. If you have two functions, \( f(x) \) and \( g(x) \), their composition is usually denoted as \( (f \circ g)(x) = f(g(x)) \).

In the given problem, we're asked to find \( F(f(t), g(t)) \), which is an example of a more complex form of function composition involving multiple variables. This process allows us to build more sophisticated models by combining simpler linear or non-linear functions. Here is why function composition is useful:

• Complex Modeling: By combining simple functions, it becomes possible to model complex real-world situations more accurately.
• Flexibility: Offers flexibility in transforming inputs through various layers of functions, achieving multi-stage transformations.
• Simplification: Depending on function properties, sometimes compositions simplify to very manageable forms.

In the exercise, the use of function composition enables us to substitute \( f(t) = \ln(t^2) \) and \( g(t) = e^{t/2} \) into \( F(x, y) \), culminating in the final expression \( t^2 + e^t \), demonstrating how layers of different functions interact and resolve into a simpler form.