Problem 47 - Calculus Early Transcendentals: Pearson New International Edition

Question

Use a graphing calculator or a computer to do. Let \(f^{\prime}(x)=x^{3}-5 x^{2}+2\) on \(I=[-2,4]\). Where on \(I\) is \(f\) increasing?

Step-by-Step Solution

Answer

The function \( f(x) \) is increasing on \([-2, 0) \cup (3, 4]\).

Step 1: Graph the function

Plot the derivative function \( f'(x) = x^3 - 5x^2 + 2 \) on the domain \( I = [-2, 4] \) using your graphing calculator or computer software. This gives a visual representation of where the function increases or decreases.

Step 2: Identify where \( f'(x) \) is positive or zero

On the graph, observe where \( f'(x) \) is above or at the x-axis. The function \( f(x) \) increases where \( f'(x) \geq 0 \). Analyze these regions within the interval \( I = [-2, 4] \).

Step 3: Determine intervals of increase

Check the specific points where \( f'(x) = 0 \) and analyze these along with where \( f'(x) > 0 \). For \( f(x) \) to be increasing, \( f'(x) \) should mostly be positive or zero but changing from negative at those points will define the intervals accurately.

Key Concepts

Graphing Calculators

Graphing calculators are powerful tools that help visualize mathematical functions and concepts. In calculus, they are particularly useful when dealing with functions and their derivatives. A graphing calculator can plot the graph of a derivative function, allowing you to see where the function increases or decreases by observing the graph's position relative to the x-axis.

By entering the derivative function into the calculator, you get an immediate visual representation. This visual aid is crucial, especially when handling complex functions like polynomials. Knowing how to effectively use a graphing calculator can make understanding calculus much easier. You can explore:

• The shape of the graph.
• The points where the graph intersects the axis.
• The intervals where the graph is above or below the axis.

Mastering the graphing calculator leads to better insight into problems that involve rates of change and areas of increase or decrease.

Derivatives

Derivatives are a fundamental concept in calculus, representing the rate of change or the slope of a function at any given point. In the context of the original exercise, we deal with the derivative of a function, denoted as \( f'(x) \).

When you have a function like \( f'(x) = x^3 - 5x^2 + 2 \), it describes how the original function \( f(x) \) changes at each point along the interval \( I = [-2, 4] \). Specifically:

• The derivative tells you how fast (and in which direction) the function is changing.
• Where \( f'(x) > 0 \), the function is increasing.
• Where \( f'(x) = 0 \), the function reaches a local maximum or minimum, or it possibly changes direction.
• Where \( f'(x) < 0 \), the function is decreasing.

Understanding derivatives is essential as they provide insights into the behavior of functions, helping identify patterns and intervals of increase and decrease.

Intervals of Increase

Intervals of increase are segments over which a function consistently climbs as you move from left to right along the graph. For the function \( f(x) \) to be increasing, its derivative \( f'(x) \) must be positive within those intervals.

To determine these intervals:

• Graph \( f'(x) \) and observe where it's above the x-axis on the interval \( I = [-2, 4] \).
• Find the points where \( f'(x) = 0 \), known as critical points, which could signify potential changes in the direction of the function.
• Analyze between these points to find where \( f'(x) > 0 \). This is where the function \( f(x) \) is increasing.

This approach helps pinpoint the behavior of \( f(x) \) without needing to crunch through complex algebraic calculations.

Mathematical Analysis

Mathematical analysis involves breaking down a mathematical problem into its core parts to understand its underlying behavior. Typically used in calculus, analysis can be the difference between simply solving a problem and truly grasping how solutions are developed.

For the exercise in question, mathematical analysis helps us by:

• Interpreting the graph and derivatives to find intervals of increase or decrease without calculating the full function \( f(x) \).
• Recognizing how changes in the function's growth rate appear in its graph and mathematical expression.
• Ensuring a deeper comprehension of critical points and their effect on overall graph behavior.

By practicing mathematical analysis, students develop a more solid foundation in calculus, equipping them with tools to tackle complex problems by breaking them down into manageable sections.