How To Find The Interval Of Increase

Finding the interval of increase for a function is a fundamental concept in calculus, providing valuable insights into the behavior of the function. By determining the intervals where a function is increasing, we can better understand its graph, identify local extrema, and solve optimization problems. This article will provide a comprehensive guide on how to find the interval of increase for a given function, along with examples and practical applications.

Understanding Increasing Functions

A function is said to be increasing on an interval if its value increases as the input variable increases. More formally, a function f(x) is increasing on an interval (a, b) if for any two points x₁ and x₂ in (a, b), where x₁ < x₂, we have f(x₁) < f(x₂). In simpler terms, as you move from left to right along the graph of the function within that interval, the graph goes uphill.

Prerequisites

Before diving into the steps to find the interval of increase, it's essential to have a solid understanding of the following concepts:

• Derivatives: The derivative of a function, denoted as f'(x), represents the instantaneous rate of change of the function with respect to its input variable.
• Critical Points: Critical points are the points where the derivative of the function is either zero or undefined. These points are crucial in determining the intervals of increase and decrease.
• Interval Notation: Familiarity with interval notation is necessary to express the intervals of increase and decrease accurately. For example, (a, b) represents the open interval between a and b, while [a, b] represents the closed interval including a and b.

Steps to Find the Interval of Increase

Here's a step-by-step guide on how to find the interval of increase for a given function f(x):

Step 1: Find the Derivative of the Function

The first step is to find the derivative of the function f(x), denoted as f'(x). The derivative represents the slope of the tangent line to the function at any given point. This step typically involves applying various differentiation rules, such as the power rule, product rule, quotient rule, and chain rule, depending on the complexity of the function.

Example:

Let's consider the function f(x) = x³ - 3x² + 2x - 1.

To find the derivative f'(x), we apply the power rule to each term:

• The derivative of x³ is 3x².
• The derivative of -3x² is -6x.
• The derivative of 2x is 2.
• The derivative of -1 is 0.

Therefore, the derivative of the function is f'(x) = 3x² - 6x + 2.

Step 2: Find the Critical Points

Critical points are the points where the derivative of the function is either zero or undefined. These points are essential in determining the intervals of increase and decrease. To find the critical points, we need to solve the equation f'(x) = 0 for x. Additionally, we should also consider any points where f'(x) is undefined.

Example (continued):

To find the critical points of the function f(x) = x³ - 3x² + 2x - 1, we need to solve the equation f'(x) = 3x² - 6x + 2 = 0.

This is a quadratic equation, and we can solve it using the quadratic formula:

x = (-b ± √(b² - 4ac)) / (2a)

In this case, a = 3, b = -6, and c = 2. Plugging these values into the quadratic formula, we get:

x = (6 ± √((-6)² - 4 * 3 * 2)) / (2 * 3)

x = (6 ± √(36 - 24)) / 6

x = (6 ± √12) / 6

x = (6 ± 2√3) / 6

x = 1 ± √3 / 3

So, the critical points are x₁ = 1 - √3 / 3 and x₂ = 1 + √3 / 3.

Step 3: Create a Sign Chart

A sign chart is a visual tool used to determine the sign of the derivative f'(x) in different intervals. To create a sign chart, we first mark the critical points on a number line. These critical points divide the number line into several intervals. Then, we choose a test value within each interval and evaluate the derivative f'(x) at that test value. The sign of the derivative at the test value indicates the sign of the derivative throughout that interval.

Example (continued):

For the function f(x) = x³ - 3x² + 2x - 1, we have two critical points: x₁ = 1 - √3 / 3 ≈ 0.423 and x₂ = 1 + √3 / 3 ≈ 1.577.

We create a sign chart with these critical points marked on the number line:

-----|-----|-----
      x₁    x₂

Now, we choose test values in each interval:

• Interval 1: x < 1 - √3 / 3 (e.g., x = 0)
• Interval 2: 1 - √3 / 3 < x < 1 + √3 / 3 (e.g., x = 1)
• Interval 3: x > 1 + √3 / 3 (e.g., x = 2)

We evaluate the derivative f'(x) = 3x² - 6x + 2 at these test values:

• f'(0) = 3(0)² - 6(0) + 2 = 2 (positive)
• f'(1) = 3(1)² - 6(1) + 2 = -1 (negative)
• f'(2) = 3(2)² - 6(2) + 2 = 2 (positive)

We add the signs to the sign chart:

 +   |  -  |  +
-----|-----|-----
      x₁    x₂

Step 4: Determine the Intervals of Increase

The intervals of increase are the intervals where the derivative f'(x) is positive. From the sign chart, we can identify the intervals where f'(x) > 0. These intervals correspond to the intervals where the function f(x) is increasing.

Example (continued):

From the sign chart, we can see that f'(x) > 0 in the intervals x < 1 - √3 / 3 and x > 1 + √3 / 3.

Therefore, the intervals of increase for the function f(x) = x³ - 3x² + 2x - 1 are:

• (-∞, 1 - √3 / 3)
• (1 + √3 / 3, ∞)

Additional Considerations

Endpoints of Intervals

When determining the intervals of increase, it's important to consider whether to include the endpoints of the intervals. If the function is continuous at the endpoint and the derivative is zero or undefined at that point, we may or may not include the endpoint in the interval of increase, depending on the specific context.

Functions with Discontinuities

For functions with discontinuities, such as rational functions or piecewise-defined functions, we need to be careful when determining the intervals of increase. We should exclude any points of discontinuity from the intervals and analyze the behavior of the function around those points.

Using the Second Derivative Test

The second derivative test can be used to confirm whether a critical point corresponds to a local maximum or a local minimum. If the second derivative f''(x) is positive at a critical point, then the function has a local minimum at that point. If the second derivative is negative at a critical point, then the function has a local maximum at that point.

Examples

Let's go through a few more examples to illustrate the process of finding the interval of increase:

Example 1:

Find the interval of increase for the function f(x) = x² - 4x + 3.

1. Find the derivative: f'(x) = 2x - 4
2. Find the critical points: 2x - 4 = 0 => x = 2
3. Create a sign chart:

  -  |  +
-----|-----
      2

4. Determine the intervals of increase: (2, ∞)

Example 2:

Find the interval of increase for the function f(x) = -x³ + 6x² - 5.

1. Find the derivative: f'(x) = -3x² + 12x
2. Find the critical points: -3x² + 12x = 0 => -3x(x - 4) = 0 => x = 0, x = 4
3. Create a sign chart:

  -  |  +  |  -
-----|-----|-----
      0    4

4. Determine the intervals of increase: (0, 4)

Example 3:

Find the interval of increase for the function f(x) = x / (x² + 1).

1. Find the derivative using the quotient rule: f'(x) = (1 - x²) / (x² + 1)²
2. Find the critical points: (1 - x²) / (x² + 1)² = 0 => 1 - x² = 0 => x = -1, x = 1
3. Create a sign chart:

  -  |  +  |  -
-----|-----|-----
     -1    1

4. Determine the intervals of increase: (-1, 1)

Applications of Intervals of Increase

Finding the intervals of increase has numerous applications in various fields:

• Optimization Problems: In optimization problems, we often need to find the maximum or minimum value of a function. By identifying the intervals of increase and decrease, we can determine the critical points that correspond to local maxima or minima.
• Graphing Functions: Knowing the intervals of increase and decrease helps us sketch the graph of a function more accurately. We can identify where the graph is going uphill (increasing) and where it is going downhill (decreasing).
• Economics: In economics, the concept of increasing functions is used to analyze production costs, revenue, and profit. For example, a cost function is typically increasing, as the cost of producing goods increases with the quantity produced.
• Physics: In physics, intervals of increase can be used to analyze the motion of objects. For example, the velocity of an object may be increasing over a certain interval, indicating that the object is accelerating.

Common Mistakes to Avoid

• Forgetting to Find Critical Points Where the Derivative Is Undefined: Make sure to consider all points where the derivative is undefined, as these can also be critical points.
• Incorrectly Applying Differentiation Rules: Double-check that you are using the correct differentiation rules when finding the derivative of the function.
• Not Creating a Sign Chart: A sign chart is a crucial tool for determining the sign of the derivative in different intervals. Failing to create a sign chart can lead to incorrect conclusions.
• Assuming That a Critical Point Always Corresponds to a Local Maxima or Minima: While critical points are potential locations for local maxima or minima, further analysis (e.g., using the second derivative test) is needed to confirm this.

Conclusion

Finding the interval of increase is a fundamental concept in calculus that provides valuable insights into the behavior of functions. By following the steps outlined in this article, you can confidently determine the intervals where a function is increasing. This knowledge can be applied to solve optimization problems, graph functions accurately, and analyze various real-world phenomena. Understanding the nuances of increasing functions and avoiding common mistakes will further enhance your problem-solving skills in calculus and related fields.