Problem 250 - Calculus Volume 1

Question

Use the information in the following table to find \(h^{\prime}(a)\) at the given value for \(a\). $$\begin{array}{|c|c|c|c|c|} \hline x & f(x) & f^{\prime}(x) & g(x) & g^{\prime}(x) \\ \hline 0 & 2 & 5 & 0 & 2 \\ \hline 1 & 1 & -2 & 3 & 0 \\ \hline 2 & 4 & 4 & 1 & -1 \\ \hline 3 & 3 & -3 & 2 & 3 \\ \hline \end{array}$$ $$ h(x)=(1+g(x))^{3} ; a=2 $$

Step-by-Step Solution

Answer

The derivative \( h'(2) = -12 \).

Step 1: Identify the Function to Differentiate

We are given the function \( h(x) = (1 + g(x))^3 \). Our task is to find the derivative \( h'(a) \) at \( a = 2 \).

Step 2: Apply the Chain Rule

To differentiate \( h(x) = (1 + g(x))^3 \), we use the chain rule. The chain rule states that if \( h(x) = [u(x)]^n \), then \( h'(x) = n[u(x)]^{n-1} \cdot u'(x) \). Here, \( u(x) = 1 + g(x) \), and therefore \( h'(x) = 3(1 + g(x))^2 \cdot g'(x) \).

Step 3: Substitute Values for \( g(x) \) and \( g'(x) \)

We need to substitute the given values from the table for \( a = 2 \). From the table, \( g(2) = 1 \) and \( g'(2) = -1 \).

Step 4: Calculate \( h'(2) \)

Substitute the values from Step 3 into the derivative expression from Step 2:\[ h'(2) = 3(1 + g(2))^2 \cdot g'(2) = 3(1 + 1)^2 \cdot (-1) \] This simplifies to:\[ h'(2) = 3(2)^2 \cdot (-1) = 3 \cdot 4 \cdot (-1) = -12 \].

Step 5: Conclusion

So, the value of \( h'(a) \) at \( a = 2 \) is \(-12\).

Key Concepts

Chain Rule

The Chain Rule is a fundamental technique in calculus for differentiating composite functions. Imagine you have a function nested within another function, like a matryoshka doll. That's where the Chain Rule comes into play, enabling you to break down this complex structure into manageable pieces.

In mathematical terms, when you have a composite function such as \( h(x) = [u(x)]^n \), it can be differentiated using the Chain Rule. The Chain Rule states that \( h'(x) = n[u(x)]^{n-1} \cdot u'(x) \).

Essentially, it requires you to:

• Differentiate the outer function, treating the inner function as a single unit.
• Multiply this by the derivative of the inner function.

This efficient rule is crucial because many functions we encounter are in the form of composites and can become cumbersome to differentiate without this method.

Derivative

A derivative, in calculus, signifies the rate at which a function is changing at any given point. Think of it as a tool that tells you how steep a curve is, or how quickly a stock price is rising or falling at a particular moment.

Mathematically, the derivative of a function \( f(x) \) is represented as \( f'(x) \).

The process involves:

• Finding how a small change in \( x \) influences the function \( f(x) \).
• Using this information to predict the slope or direction of the function at that point.

Differentiation is the act of computing a derivative. Differentiating functions allows us to solve many real-world problems by predicting rates of change, optimizing solutions, and more.

Function Differentiation

Function differentiation is the process of finding the derivative of a function, essentially determining how the value of the function changes as its input changes. Understanding this concept is crucial for solving many problems in calculus.

To perform differentiation, follow these basic steps:

• Identify the function you're working with.
• Apply differentiation rules such as the power rule, product rule, or chain rule based on the structure of the function.

For instance, when differentiating a function like \( h(x) = (1 + g(x))^3 \), you will use the chain rule, as it deals with a function raised to a power and involving another function nested inside. This process often transforms complex equations into simpler derivatives that describe the function's rate of change.
This differentiation is essential in fields such as physics for computing acceleration or in economics for finding maximum profit or cost.

Mathematical Problem Solving

Effective mathematical problem solving is about understanding the problem, selecting the right tools, and applying them systematically to find a solution. This structured approach helps to solve exercises like finding derivatives, where various calculus rules are applied together.

Key steps for successful problem solving include:

• Carefully identify what is being asked in the problem.
• Select the appropriate mathematical techniques, such as the Chain Rule for differentiation.
• Break the problem down into smaller, manageable steps.
• Verify your solution by substituting values back into the derived formula or by checking against known values.

In the exercise given, applying these systematic strategies ensures we accurately determine \(h'(a)\). These skills are invaluable not just in academics but in tackling real-world problems where logical analysis is needed.