Problem 1 - Calculus Early Transcendentals: Pearson New International Edition

Question

In Problems 1-10, use the Monotonicity Theorem to find where the given function is increasing and where it is decreasing. \(f(x)=3 x+3\)

Step-by-Step Solution

Answer

The function is increasing everywhere.

Step 1: Understanding the Function Type

The given function is a linear function, which can be expressed as \(f(x)=mx+b\). In this case, \(m=3\) and \(b=3\). Linear functions generally have a constant slope and, therefore, consistent monotonicity.

Step 2: Determine the Derivative

To apply the Monotonicity Theorem, we need to find the derivative of the function. For \(f(x) = 3x + 3\), the derivative \(f'(x)\) is simply the coefficient of \(x\), which is 3.

Step 3: Analyzing the Derivative

The value of the derivative \(f'(x) = 3\) is positive. According to the Monotonicity Theorem, if \(f'(x) > 0\) for all \(x\) in an interval, then \(f(x)\) is increasing on that interval.

Step 4: Conclusion About Monotonicity

Since \(f'(x) = 3 > 0\) for all \(x\), the function \(f(x) = 3x + 3\) is increasing over its entire domain \((-\infty, \infty)\).

Key Concepts

Linear Functions

Linear functions are one of the simplest types of functions you will encounter. They have the form \(f(x) = mx + b\) where:

• \(m\) is the slope of the line, representing the rate of change.
• \(b\) is the y-intercept, indicating where the line crosses the y-axis.

The beauty of a linear function lies in its simplicity and predictability.
The slope \(m\) tells you everything about the function's direction: if it's positive, the line rises as you move from left to right; if negative, it falls.
The constant slope means linear functions have a consistent rate of change across all values of \(x\). This allows for a straightforward analysis of their behavior over intervals, making them a foundational concept in calculus and beyond.

Derivatives

Derivatives are a fundamental concept in calculus, providing insights into the behavior of functions.
In simple terms, a derivative represents the rate at which a function is changing at any given point.
Mathematically, for a function \(f(x)\), its derivative \(f'(x)\) can be thought of as the slope of the tangent line to the curve at any point \(x\).

For linear functions like \(f(x) = 3x + 3\), finding the derivative \(f'(x)\) is straightforward because their rates of change are constant.
The derivative of \(f(x) = 3x + 3\) is simply \(f'(x) = 3\).
This constant value tells us that the function increases steadily across its domain.
Derivatives are crucial in understanding how functions behave and changing conditions affect them. They are key to solving problems involving rates, optimization, and understanding the motion of objects.

Function Analysis

Function analysis involves studying a function to understand its properties, such as where it increases, decreases, and what happens at critical points.
This analysis often starts with examining derivatives.
When analyzing a linear function like \(f(x) = 3x + 3\), we look at its derivative \(f'(x) = 3\).
A positive derivative consistently indicates that the function is increasing across its domain.
There are no critical points where the function could change its increasing or decreasing behavior.

When doing function analysis, recognizing these cues helps render a clear picture of the function's behavior at a glance. Analytical skills developed here can be applied to more complex functions, making this a valuable exercise in any calculus learning path.

Calculus Problem-Solving

Calculus problem-solving combines the understanding of derivatives, functions, and their behaviors to tackle a variety of mathematical challenges.
Solving calculus problems often involves:

• Identifying the type of function you're dealing with, such as linear or polynomial.
• Finding the derivative to understand how the function changes.
• Using the derivative to determine the function's behavior, like where it's increasing or decreasing, using the Monotonicity Theorem.

In the example of the function \(f(x) = 3x + 3\), applying these steps reveals the simplicity of linear functions.
The constant derivative helped conclude that the function increases over all \(x\), giving a complete picture of its monotonicity.
Mastering these skills and strategies makes tackling more difficult calculus problems manageable and insightful.