Problem 53 - Calculus an Applied Approach

Question

Use a graphing utility to graph \(f\) and \(f^{\prime}\) on the interval \([-2,2] .\) $$ f(x)=x(x+1)(x-1) $$

Step-by-Step Solution

Answer

The original function \(f(x)\) is a cubic function, creating a curve that crosses the x- axis at points x=-1, x=0 and x=1. The derivative \(f'(x) = 3x^2 -1\), a quadratic function, creates a parabola that touches the x-axis at points \(x = -\sqrt{1/3}\) and \(x = \sqrt{1/3}\). The difference in shape shows how the original function's rate of change varies across the interval [-2,2].

Step 1: Compute the derivative

The original function is a product of three simple functions \(x\), \(x+1\), and \(x-1\), for which you can use the product rule for differentiation. The derivative \(f'(x)\) of function \(f(x)\) is computed as \(f'(x) = (x+1)(x-1) + x(x-1) + x(x+1)\). Simplify to get \(f'(x) = 3x^2 - 1\).

Step 2: Graphing the original function

To graph the original function \(f(x)\), put the range of values from -2 to 2 into the original function \(f(x)= x(x+1)(x-1)\). This will give you the corresponding y-values. Connect the points to get the function graph.

Step 3: Graphing the derivative

For the derivative function \(f'(x) = 3x^2-1\), again put the same range of values from -2 to 2. The resulting y-values would be different, and you can connect these points to get the derivative graph. Notice the differences and similarities between the original function and its derivative.

Key Concepts

Derivative of a Function

The derivative of a function is a powerful mathematical tool that gives us insight into the behavior of a function. In simple terms, it represents the rate at which a function changes with respect to its variable. For example, if you have a function expressed as \( f(x) \), its derivative, \( f'(x) \), will tell you how \( f(x) \) changes as \( x \) changes.

To find the derivative of the given function \( f(x) = x(x+1)(x-1) \), we used the product rule, since this function is a product of three simpler functions. The derivative \( f'(x) \) was calculated and simplified to \( f'(x) = 3x^2 - 1 \). This tells us that the rate of change of the function is determined by this quadratic expression.

Understanding the derivative helps us to know not just the slope of the function at various points but also to identify critical points, where the function might have a maximum, minimum, or a point of inflection.

Product Rule

The product rule is an essential rule in calculus used for finding the derivative of products of two or more functions. Imagine two functions \( u(x) \) and \( v(x) \) that you need to multiply to form a new function \( u(x)v(x) \). The product rule tells us how to differentiate this, and it states:

• The derivative of \( u(x)v(x) \) is \( u'(x)v(x) + u(x)v'(x) \).

This method is applied to our function \( f(x) = x(x+1)(x-1) \), which is a product of three smaller functions. We differentiate each part according to the product rule:

• Differentiating \( x(x+1) \), \( x+1)(x-1) \) and \( x(x-1) \) separately and then adding them gives the derivative \( f'(x) = (x+1)(x-1) + x(x-1) + x(x+1) \).

This results in \( f'(x) = 3x^2 - 1 \) after simplification. Understanding and applying the product rule correctly is crucial, especially when dealing with polynomial functions, as it allows us to calculate derivatives efficiently and accurately.

Graphical Analysis

Graphical analysis involves looking at the graphs of functions to understand their properties visually. It's a convenient way to explore how the function behaves and changes over a particular interval. In our exercise, we need to graph both the original function \( f(x) = x(x+1)(x-1) \) and its derivative \( f'(x) = 3x^2 - 1 \) on the interval \([-2, 2]\).

By graphing \( f(x) \), you observe the shape and traits, such as where it crosses the x-axis, which reflects the roots of the equation. Graphing \( f'(x) \) shows us where the original function is increasing or decreasing:

• When \( f'(x) > 0 \), the function \( f(x) \) is increasing.
• When \( f'(x) < 0 \), the function \( f(x) \) is decreasing.
• Where \( f'(x) = 0 \), \( f(x) \) may have local maxima, minima or points of inflection.

Graphical analysis lets us bridge the numeric and visual understanding, clarifying how the derivative relates to the behavior of the original function.

Polynomial Functions

Polynomial functions are expressions consisting of variables raised to whole number powers, combined using addition or subtraction. In our problem, \( f(x) = x(x+1)(x-1) \) is a polynomial function, expanded as \( f(x) = x^3 - x \).

Polynomial functions have important characteristics:

• The degree of the polynomial tells us the maximum number of roots and turning points it can have.
• For example, a third-degree polynomial like ours will have exactly three roots, depending on multiplicities, and up to two turning points.

Studying these functions often involves solving for roots, finding derivatives, and analyzing their behavior. These functions are important in modeling real-world phenomena, given their robust properties and ease of calculation. Derivatives of polynomial functions are straightforward to find as each term can be differentiated independently, based on power rule.