Problem 9 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

How are the graphs of \(f^{\prime}\) and \(g^{\prime}\) related in each of the following situations? Explain your reasoning. (a) \(g(x)=f(x)+3\) (b) \(g(x)=f(x+3)\) (c) \(g(x)=3 f(x)\)

Step-by-Step Solution

Answer

For \(g(x) = f(x) + 3\), the graphs of \(g'(x)\) and \(f'(x)\) are the same. For \(g(x) = f(x + 3)\), the graph of \(g'(x)\) is the graph of \(f'(x)\) shifted to the left by 3 units. For \(g(x) = 3f(x)\), the graph of \(g'(x)\) is the graph of \(f'(x)\) vertically stretched by a factor of 3.

Step 1: Relationship Between \(g(x)=f(x)+3\) and \(f(x)\)

Start by taking the derivative of both sides of the equation \(g(x) = f(x) + 3\) to find \(g'(x)\). Utilize the sum rule, which states that the derivative of a sum of two functions is the sum of their derivatives. In this case, the derivative of a constant (3) is zero, so we simply have \(g'(x) = f'(x)\). Thus, the graph of \(g'(x)\) will be the same as the graph of \(f'(x)\) because adding a constant to a function does not change its derivative or the slope of its tangent lines.

Step 2: Relationship Between \(g(x)=f(x+3)\) and \(f(x)\)

Again, take the derivative of both sides of the equation \(g(x) = f(x + 3)\) to find \(g'(x)\). Use the chain rule, which in this case leads to the derivative being \(g'(x) = f'(x + 3)\). The chain rule states that the derivative of a composed function is the product of the derivative of the outer function and the derivative of the inner function evaluated at the outer function. Here, the graph of \(g'(x)\) is the graph of \(f'(x)\) shifted to the left by 3 units, or horizontally translated by -3.

Step 3: Relationship Between \(g(x)=3f(x)\) and \(f(x)\)

For \(g(x) = 3f(x)\), take the derivative on both sides to find \(g'(x)\). Use the constant rule, which states that the derivative of a constant times a function is the constant times the derivative of the function: \(g'(x) = 3f'(x)\). The graph of \(g'(x)\) is the graph of \(f'(x)\) vertically stretched by a factor of 3. This is due to the fact that multiplying a function by a constant affects the steepness of its graph.

Key Concepts

Understanding the Sum Rule in Derivatives

The sum rule is a fundamental concept in calculus, making the process of finding derivatives simpler when dealing with multiple terms added together. Consider two functions, say \(u(x)\) and \(v(x)\). If we have a new function defined as \(w(x) = u(x) + v(x)\), the sum rule helps us determine its derivative.
The sum rule states that the derivative of this new function \(w(x)\) is simply the sum of the derivatives of the individual functions:

• \(w'(x) = u'(x) + v'(x)\)

Applying this to the problem where \(g(x) = f(x) + 3\), we recognize 3 as a constant. Since the derivative of a constant is zero, the equation simplifies to \(g'(x) = f'(x)\).
Therefore, adding a constant does not affect the slope or the graph of the derivative. The graph of \(g'(x)\) therefore mirrors the graph of \(f'(x)\) perfectly.

Mastering the Chain Rule for Function Composition

The chain rule is especially useful when dealing with functions within functions, known as composite functions. Imagine we have a function \(h(x) = f(g(x))\). The chain rule provides a method to find the derivative \(h'(x)\) by taking into account both the outside and inside functions.
According to the chain rule, the derivative is:

• \(h'(x) = f'(g(x)) \cdot g'(x)\)

In simpler terms, you differentiate the outer function first while keeping the inner function intact, then multiply by the derivative of the inner function. In the scenario where \(g(x) = f(x+3)\), the chain rule tells us that:

• \(g'(x) = f'(x+3)\)

This causes the graph of the derivative to shift horizontally by 3 units to the left. Thus, while the shape of \(g'(x)\) remains similar to \(f'(x)\), its position changes along the x-axis.

Exploring the Constant Rule in Derivatives

The constant rule is one of the simplest rules when it comes to differentiation, often expressed as: the derivative of a constant times a function is the constant times the derivative of the function. If you have \(y = cu(x)\), where \(c\) is a constant, then the derivative is

• \(y' = c \cdot u'(x)\)

This means multiplying the entire derivative of the function by the constant. In the exercise where \(g(x) = 3f(x)\), this gives us:

• \(g'(x) = 3f'(x)\)

As a result, the graph of \(g'(x)\) is a vertical stretch of the graph of \(f'(x)\). The slopes become steeper by a factor of 3, changing how steep or flat the graph appears without altering its basic shape.

Understanding Function Transformation

Function transformation involves various manipulations of a function's graph, such as shifts or stretches. These changes can be vertical, horizontal, or a combination, impacting how the graph is positioned visually and statistically.
Consider the function transformation in \(g(x) = f(x+3)\) and \(g(x) = 3f(x)\):

• For \(g(x) = f(x+3)\), this is a horizontal transformation. More specifically, a shift to the left by 3 units.
• For \(g(x) = 3f(x)\), this involves a vertical transformation, as the graph becomes vertically stretched by a factor of 3.

These transformations affect the derivative graphs as explained in their respective rules (chain rule and constant rule), leading to changes in the position and steepness of the graph. Understanding these transformations helps in predicting and visualizing how the function and its derivative's graph will look after the manipulation.