Problem 71 - Calculus an Applied Approach

Question

Use the given information to find \(f^{\prime}(2) .\) \(g(2)=3\) and \(g^{\prime}(2)=-2\) \(h(2)=-1 \quad\) and \(\quad h^{\prime}(2)=4\) $$ f(x)=g(x)+h(x) $$

Step-by-Step Solution

Answer

The value of \(f^{\prime}(2) = 2\).

Step 1: Identify the functions

The given functions are \(g(x)\) and \(h(x)\) and \(f(x)\) is the sum of these two functions: \(f(x) = g(x) + h(x)\). The derivatives of \(g(x)\) and \(h(x)\) at the point \(x = 2\) are \(g^{\prime}(2) = -2\) and \(h^{\prime}(2) = 4\) respectively.

Step 2: Apply the Sum Rule

According to the Sum Rule in differentiation, the derivative of \(f(x)\) at \(x = 2\) will be the sum of the derivatives of \(g(x)\) and \(h(x)\) at \(x = 2\). In mathematical terms, \(f^{\prime}(2) = g^{\prime}(2) + h^{\prime}(2)\).

Step 3: Substitute the known values

Substitute the known derivative values into the sum rule equation: \(f^{\prime}(2) = (-2) + 4 = 2\).

Key Concepts

Sum Rule in Differentiation

In calculus, the Sum Rule is a fundamental concept used to find the derivative of a sum of functions. If you are dealing with a function like \(f(x) = g(x) + h(x)\), the Sum Rule asserts that the derivative of this function, \(f'(x)\), is simply the sum of the derivatives of \(g(x)\) and \(h(x)\). This rule simplifies the differentiation process significantly.

For instance, consider the function \(f(x) = g(x) + h(x)\). According to the Sum Rule:

• \(f'(x) = g'(x) + h'(x)\)

This allows you to calculate the derivative at any point \(x\) as long as you know \(g'(x)\) and \(h'(x)\).

In the given exercise, you find \(f'(2)\) by identifying \(g'(2) = -2\) and \(h'(2) = 4\) and simply adding these two values. The solution becomes straightforward: \(f'(2) = -2 + 4 = 2\).

Thus, the Sum Rule basically breaks down a potentially complex operation into easier parts, helping you handle differentiation with ease.

Understanding Derivatives

Derivatives represent the rate at which a function is changing at any given point. Understanding derivatives is crucial for analyzing and predicting the behavior of functions. The derivative of a function \(f(x)\), denoted as \(f'(x)\) or \(\frac{df}{dx}\), can be interpreted as the slope of the tangent to the function's curve at a point.

Why are derivatives important?

• They help track rates of change, like velocity or growth rates.
• Derivatives optimize functions, essential for finding maximum and minimum values.
• They provide critical insights into the behavior of functions, making them crucial for calculus and analysis.

In our context, knowing \(g'(2)\) and \(h'(2)\) as given in the problem is all about understanding how each function \(g(x)\) and \(h(x)\) changes specifically at \(x = 2\).

The interplay between multiple functions through their derivatives at specific points is a powerful tool in calculus, allowing you not only to find instantaneous rates of change but also to build deeper insights into the dynamics of complex functions.

Function Analysis Using Derivatives

Function analysis involves exploring functions to understand their characteristics like growth, limits, and concavity. Analyzing functions using derivatives is a method that delves into these properties, offering insights that are not immediately visible from the function's basic expression.

How do derivatives aid in function analysis?

• Identify critical points where \(f'(x) = 0\), showing potential maxima, minima, or points of inflection.
• Evaluate function behavior, helping determine if a function is increasing or decreasing.
• Enhance understanding of concavity through the second derivative \(f''(x)\), identifying concave up or down regions.

In this exercise's context, although we primarily calculate \(f'(2)\) using the Sum Rule, the derivative provides more than just a rate of change. By knowing \(f'(x)\) across different values of \(x\), you glean insights into the rise and fall of the function \(f(x)\), forming a more complete picture of its graph.

By mastering function analysis via derivatives, you gain a robust toolkit for understanding and predicting functions' behaviors, empowering you to apply these principles across diverse mathematical problems.