Problem 5 - Calculus Early Transcendentals

Question

Given the following table of values, find the indicated derivatives in parts (a) and (b). $$ \begin{array}{|c|c|c|c|c|}\hline x & {f(x)} & {f^{\prime}(x)} & {g(x)} & {g^{\prime}(x)} \\ \hline 3 & {5} & {-2} & {5} & {7} \\ \hline 5 & {3} & {-1} & {12} & {4} \\ \hline\end{array} $$ (a) \(F^{\prime}(3),\) where \(F(x)=f(g(x))\) (b) \(G^{\prime}(3),\) where \(G(x)=g(f(x))\)

Step-by-Step Solution

Answer

(a) \( F'(3) = -7 \). (b) \( G'(3) = -8 \).

Step 1: Interpret the Function F(x)

Given that \( F(x) = f(g(x)) \), this means that \( F(x) \) is a composite function where \( f \) is applied to \( g(x) \). We need to find \( F'(x) \) using the chain rule.

Step 2: Set Up the Chain Rule for F'(x)

The derivative \( F'(x) \) of the composite function \( F(x) = f(g(x)) \) can be found using the chain rule: \[ F'(x) = f'(g(x)) \cdot g'(x). \]

Step 3: Substitute x=3 into the Expression for F'(x)

Substitute \( x = 3 \) into \( F'(x) = f'(g(x)) \cdot g'(x) \). First, find \( g(3) \), which is given as 5. So, \( f'(g(3)) = f'(5) \) and \( g'(3) \) is 7 from the table.

Step 4: Compute F'(3)

Using the table, \( f'(5) = -1 \). Then \( F'(3) = f'(5) \cdot g'(3) = -1 \cdot 7 = -7. \)

Step 5: Interpret the Function G(x)

Given that \( G(x) = g(f(x)) \), this means that \( G(x) \) is a composite function where \( g \) is applied to \( f(x) \). We need to find \( G'(x) \) using the chain rule.

Step 6: Set Up the Chain Rule for G'(x)

The derivative \( G'(x) \) of the composite function \( G(x) = g(f(x)) \) can be found using the chain rule: \[ G'(x) = g'(f(x)) \cdot f'(x). \]

Step 7: Substitute x=3 into the Expression for G'(x)

Substitute \( x = 3 \) into \( G'(x) = g'(f(x)) \cdot f'(x) \). Find \( f(3) \), which is given as 5. So, \( g'(f(3)) = g'(5) \) and \( f'(3) \) is -2 from the table.

Step 8: Compute G'(3)

Using the table, \( g'(5) = 4 \). Hence, \( G'(3) = g'(5) \cdot f'(3) = 4 \cdot -2 = -8. \)

Key Concepts

Composite Function

When you're working with functions, you might encounter a situation where a function is applied to the result of another function. This type of setup is known as a composite function. Simply put, if you have two functions, say \( f(x) \) and \( g(x) \), forming a new function like \( f(g(x)) \), you are dealing with a composite function.

To better understand, think of it as a two-step process: first, compute \( g(x) \), and then use that result as the input for \( f(x) \). This is where the chain concept becomes apparent, as the output of the first function hinges directly on the second function's input.

• The outer function: this changes what the result is based on the inner function. In our example, \( f \) acts as the outer function.
• The inner function: this prepares the value to be fed into the outer function. In the example, this is \( g(x) \).

Understanding composite functions is crucial for solving problems that require breaking down intricate relationships between different mathematical expressions.

Derivative Calculation

The derivative is a tool that helps us understand how functions change. When dealing with a composite function, the chain rule is used to find the derivative. This rule is like a formula that tells us how to combine the derivatives of the inner and outer functions.

In the original exercise, we needed to find the derivatives of two specific composite functions: \( F(x) = f(g(x)) \) and \( G(x) = g(f(x)) \). Here's how we approached it:

For \( F'(x) \):

• Use the chain rule: \( F'(x) = f'(g(x)) \cdot g'(x) \).
• Find \( g(3) \), leading to \( g(3) = 5 \), and use it to determine \( f'(5) \).
• The table tells us \( f'(5) = -1 \) and \( g'(3) = 7 \).
• Multiply these values: \( F'(3) = -1 \times 7 = -7 \).

For \( G'(x) \):

• Again, use the chain rule: \( G'(x) = g'(f(x)) \cdot f'(x) \).
• Determine \( f(3) = 5 \), and therefore \( g'(5) \) must be found.
• The table gives \( g'(5) = 4 \) and \( f'(3) = -2 \).
• Combine them: \( G'(3) = 4 \times -2 = -8 \).

By carefully applying the chain rule, we unlock insights into how these functions behave and change.

Table of Values

A table of values is a handy tool that summarizes function evaluations and their derivatives at specific points. Think of it as a mathematical reference guide that helps streamline the problem-solving process.

In our task, the table provided values for \( f(x) \), \( f'(x) \), \( g(x) \), and \( g'(x) \) at particular \( x \) values, like 3 and 5. These not only guided our calculations but also served as shortcuts for finding needed values quickly.

• At \( x = 3 \), the table tells us \( f(3) = 5 \), \( f'(3) = -2 \), \( g(3) = 5 \), and \( g'(3) = 7 \).
• At \( x = 5 \), we see \( f(5) = 3 \), \( f'(5) = -1 \), \( g(5) = 12 \), and \( g'(5) = 4 \).

By extracting these values, we're able to apply the chain rule efficiently to calculate the necessary derivatives. This helps us focus on understanding the functions' behavior without re-computing basic calculations each time, enhancing both speed and understanding.