Problem 5 - Calculus - AP Edition

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) \(-7\), (b) \(-8\).

Step 1: Understand the Function for F'(3)

For part (a), we need to find the derivative of the composite function \(F(x) = f(g(x))\). We'll need to use the chain rule, which states that the derivative of \(F(x)\) is \(F'(x) = f'(g(x)) \cdot g'(x)\).

Step 2: Apply the Chain Rule for F'(3)

Since we are asked to find \(F'(3)\), the chain rule tells us to calculate \(f'(g(3)) \cdot g'(3)\). First, find \(g(3)\) from the table, which is 5. Then find \(f'(5)\) and \(g'(3)\) from the table.

Step 3: Substitute Known Values in F'(3)

From the table, \(g(3) = 5\), \(f'(5) = -1\), and \(g'(3) = 7\). Substituting these values in, we have:\[ F'(3) = f'(5) \cdot g'(3) = (-1) \cdot 7 = -7. \]

Step 4: Understand the Function for G'(3)

For part (b), we aim to find \(G'(3)\) where \(G(x) = g(f(x))\). Again, we apply the chain rule: \(G'(x) = g'(f(x)) \cdot f'(x)\).

Step 5: Apply the Chain Rule for G'(3)

To find \(G'(3)\), substitute \(f(3)\) from the table into the formula, and use the values of \(g'(f(3))\) and \(f'(3)\) that we obtain.

Step 6: Substitute Known Values in G'(3)

From the table, \(f(3) = 5\), \(g'(5) = 4\), and \(f'(3) = -2\). Plug these into the formula:\[ G'(3) = g'(5) \cdot f'(3) = 4 \cdot (-2) = -8. \]

Key Concepts

Composite Functions

When we talk about composite functions, we mean a combination of two functions. Essentially, you input one function's output into another function. This can be visualized as a function operating within a function, kind of like Russian nesting dolls! For instance, suppose we have two functions, \( f(x) \) and \( g(x) \). Then, a composite function \( F(x) = f(g(x)) \) means you first apply \( g(x) \) to \( x \), and subsequently apply \( f(x) \) to the result of \( g(x) \).

Here's a breakdown to better understand using our exercise:

• For part (a), we're dealing with \( F(x) = f(g(x)) \). So, we find \( g(x) \) first and then feed that result into \( f(x) \).
• For part (b), we have \( G(x) = g(f(x)) \), reversing the order by applying \( f(x) \) first before \( g(x) \).

Understanding the order of operations in composite functions is crucial because each step affects the subsequent result.

Derivative Calculation

Calculating derivatives often involves applying the chain rule, especially when dealing with composite functions. The chain rule is a formula to compute the derivative of a composite function.

For the composite function \( F(x) = f(g(x)) \), the derivative \( F'(x) \) is found using the chain rule: \[ F'(x) = f'(g(x)) \cdot g'(x). \]Here’s how it works for our exercise:

• First, find the inner function's value, which is \( g(x) \).
• Next, find the derivative of \( f(x) \) evaluated at \( g(x) \), noted as \( f'(g(x)) \).
• Finally, multiply this by the derivative of \( g(x) \), which is \( g'(x) \).

For \( G(x) = g(f(x)) \), the process is similar but inverted:

• Evaluate \( f(x) \) first.
• Find \( g'(f(x)) \), the derivative of \( g(x) \) evaluated at \( f(x) \).
• Multiply by the derivative of \( f(x) \), noted as \( f'(x) \).

This method allows us to effectively and accurately find derivatives of complex functions.

Table of Values

A table of values is a handy tool when working with derivatives and functions, especially in exercises requiring substitution of known values. It provides quick access to specific values of functions and their derivatives, aiding in efficiently solving derivative problems.

In our context, the table lists values for \( x \), \( f(x) \), \( f'(x) \), \( g(x) \), and \( g'(x) \). When determining the indicated derivatives:

• For \( F'(3) \), our first task was to find \( g(3) \) using the table, providing a value we then used to determine \( f'(g(3)) \) and multiply by \( g'(3) \).
• When calculating \( G'(3) \), we looked up \( f(3) \) to find \( g'(f(3)) \) and multiplied it with \( f'(3) \).

Access to such a table streamlines the calculation process, as it avoids frequent recalculations and potential errors. It functions like a cheat sheet, giving you reliable data at a glance, ensuring you make quick and correct substitutions in your formulas.