Increasing and Decreasing Functions
Definition
- Increasing Function: A function f(x) is said to be increasing on an interval if, for any two numbers x1 and x2 in the interval, x1 < x2 implies f(x1) < f(x2).
- Decreasing Function: A function f(x) is decreasing on an interval if, for any two numbers x1 and x2 in the interval, x1 < x2 implies f(x1) > f(x2).
Identifying Increasing and Decreasing Intervals
- First Derivative Test: If f'(x) > 0 on an interval, then f(x) is increasing on that interval. If f'(x) < 0, then f(x) is decreasing. To fully grasp this concept, it's helpful to review the basics of differentiation.
Example 1:
Consider the function f(x) = x3 - 3x2 - 9x + 5. Find the intervals where the function is increasing and decreasing.
Solution:
- Find the first derivative: f'(x) = 3x2 - 6x - 9.
- Set f'(x) = 0 to find critical points: x = -1, 3.
- Use a number line to test the sign of f'(x) in each interval determined by the critical points.
- The function is increasing where f'(x) > 0 and decreasing where f'(x) < 0. Understanding differentiation rules can provide deeper insights into solving such problems.
Concavity and Points of Inflection
Definition
- Concave Up: A function f(x) is concave up on an interval if its graph opens upwards, like a parabola.
- Concave Down: A function f(x) is concave down on an interval if its graph opens downwards.
- Point of Inflection: A point where the graph of f(x) changes concavity.
Identifying Concavity
- Second Derivative Test: If f''(x) > 0, f(x) is concave up. If f''(x) < 0, f(x) is concave down. The introduction to integrals page can help in understanding how to calculate the second derivative.
Example 2:
Consider f(x) = x4 - 4x3. Determine the concavity of the function.
Solution:
- Find the second derivative: f''(x) = 12x2 - 24x.
- Set f''(x) = 0 to find potential points of inflection: x = 0, 2.
- Use a number line to test the sign of f''(x) in each interval determined by these points.
- The function is concave up where f''(x) > 0 and concave down where f''(x) < 0. Delving into how areas between curves are computed can enrich your understanding of concavity.
Integrating Concepts in Curve Sketching
Steps for Curve Sketching
- Find Critical Points: Determine where f'(x) = 0 or is undefined.
- Analyse Intervals: Use the first derivative test to find where the function is increasing or decreasing.
- Determine Concavity: Use the second derivative test to find where the function is concave up or down.
- Identify Points of Inflection: Find where the concavity changes.
- Evaluate Limits: Determine the end behaviour of the function as x approaches infinity or negative infinity.
- Plot Points: Use the above information to sketch the graph. For further guidance on creating accurate graphs, see the section on applications of differentiation.
Example 3:
Sketch the curve of f(x) = x3 - 3x2 - 9x + 5.
Solution:
- Critical Points: Found in Example 1 as x = -1, 3.
- Increasing/Decreasing: Use the intervals and signs of f'(x) from Example 1.
- Second Derivative: f''(x) = 6x - 6.
- Concavity: Test intervals determined by f''(x) = 0 (i.e., x = 1).
- End Behaviour: As x approaches infinity, f(x) also approaches infinity, and vice versa.
- Sketch: Combine all information to sketch the curve, ensuring it reflects the identified properties. The process of curve sketching elaborates on these steps to sketch more complex functions effectively.
Here is the graph of the function f(x) = x3 - 3x2 - 9x + 5:
The graph provides a visual representation of the function across the interval x in [-4, 4]. You can observe the behavior, roots, and turning points of the function within this interval.
FAQ
The second derivative in curve sketching, which is related to concavity, has substantial real-world implications, particularly in physics and economics. In physics, the second derivative of a position function with respect to time gives acceleration, providing insights into the object's motion dynamics. In economics, the second derivative of a profit function can indicate whether the profit is experiencing increasing or decreasing rates of change, which can inform strategic decision-making regarding production levels, pricing, and other relevant variables.
Points of inflection, while not indicative of local maxima or minima, are crucial in understanding the overall shape and behaviour of the graph of a function. At a point of inflection, the concavity of the graph changes, which can be pivotal in interpreting and predicting the function’s behaviour, especially in applied contexts like physics and economics. For instance, in a scenario describing velocity and acceleration, a point of inflection in the position-time graph might indicate a transition from increasing to decreasing acceleration, which can be crucial information in practical applications.
Yes, a function can have multiple points of inflection. Each point of inflection indicates a change in concavity of the graph, transitioning from concave up to concave down, or vice versa. Multiple points of inflection imply that the graph changes its concavity multiple times across its domain. Understanding these points and the intervals of different concavities between them is vital for accurately sketching the curve and interpreting the various phases or states of the function, especially in scenarios where the function models dynamic systems or processes in real-world applications.
Symmetry in a function can significantly simplify the curve sketching process. To determine if a function is even (symmetric about the y-axis), check if f(x) = f(-x) for all x in the domain of f. If a function is odd (symmetric about the origin), then f(-x) = -f(x) for all x in the domain of f. Recognising symmetry can reduce the workload when identifying critical points, intervals of increase/decrease, and concavity, as you only need to consider half of the function and reflect it across the axis or origin to complete the graph.
Understanding limits is fundamental to curve sketching as it helps to determine the end behaviour of a function. Specifically, limits describe the behaviour of a function as the input (or variable) approaches a particular value. In the context of curve sketching, evaluating the limit of a function as x approaches infinity or negative infinity can provide insights into the horizontal asymptotes of the function, if they exist. This information is crucial for sketching the graph accurately, especially in the tails of the function. Additionally, limits can help identify vertical asymptotes and holes in the graph by exploring the behaviour of the function as it approaches particular x-values.
Practice Questions
The first derivative of the function, f'(x), gives us the rate of change of the function and helps us identify where the function is increasing or decreasing. Calculating f'(x), we get f'(x) = 3x2 - 12x + 9. To find the critical points, we set f'(x) = 0 and solve for x. Factoring, we get f'(x) = 3(x - 1)(x - 3), so x = 1 and x = 3 are critical points. Using the first derivative test, we test values in the intervals x < 1, 1 < x < 3, and x > 3 to determine where f'(x) is positive (increasing) or negative (decreasing). We find that f(x) is decreasing on (-∞, 1), increasing on (1, 3), and decreasing on (3, ∞). Therefore, f(x) has a local minimum at x = 1 and a local maximum at x = 3.
To determine the concavity of the function g(x), we need to find the second derivative, g''(x). Calculating, we get g''(x) = 12x2 - 24x. Setting g''(x) = 0, we find the potential points of inflection: x = 0 and x = 2. To determine the concavity in each interval created by these points, we test a value in each interval in the second derivative. We find that g(x) is concave up (U-shaped) on (-∞, 0) and (2, ∞) and concave down (∩-shaped) on (0, 2). Therefore, g(x) has points of inflection at x = 0 and x = 2, where the graph changes concavity.
