Match The Function Shown Below With Its Derivative

Matching a function with its derivative is a fundamental skill in calculus, bridging the gap between a function's behavior and its rate of change. Understanding this relationship allows you to glean insights into the function's increasing/decreasing intervals, concavity, and critical points, all by analyzing its derivative.

Introduction

The derivative of a function, often denoted as f'(x) or dy/dx, represents the instantaneous rate of change of the function f(x) with respect to its input x. Geometrically, the derivative at a point gives the slope of the tangent line to the function's graph at that point. Matching functions with their derivatives involves recognizing how specific features of a function, such as its slope, concavity, and turning points, translate into corresponding features in its derivative. This process relies on a deep understanding of differentiation rules and graphical analysis.

Basic Differentiation Rules: A Quick Review

Before delving into the matching process, let's refresh some fundamental differentiation rules:

• Power Rule: If f(x) = xⁿ, then f'(x) = nx^n-1.
• Constant Multiple Rule: If f(x) = cf(x), then f'(x) = cf'(x), where c is a constant.
• Sum/Difference Rule: If f(x) = u(x) ± v(x), then f'(x) = u'(x) ± v'(x).
• Product Rule: If f(x) = u(x)v(x), then f'(x) = u'(x)v(x) + u(x)v'(x).
• Quotient Rule: If f(x) = u(x)/v(x), then f'(x) = [u'(x)v(x) - u(x)v'(x)] / [v(x)]².
• Chain Rule: If f(x) = g(h(x)), then f'(x) = g'(h(x)) * h'(x).
• Derivatives of Trigonometric Functions:
  • (d/dx) sin(x) = cos(x)
  • (d/dx) cos(x) = -sin(x)
  • (d/dx) tan(x) = sec²(x)
• Derivatives of Exponential and Logarithmic Functions:
  • (d/dx) e^x = e^x
  • (d/dx) ln(x) = 1/x

Step-by-Step Guide to Matching Functions with Derivatives

Here's a comprehensive approach to matching functions with their respective derivatives:

1. Analyze the Original Function (f(x))

• Identify Key Features: Look for critical points (where the slope is zero), intervals of increase and decrease, concavity (where the function is curving up or down), inflection points (where concavity changes), and asymptotes.
• Critical Points: These occur where f'(x) = 0 or is undefined. They correspond to local maxima, local minima, or saddle points on the graph of f(x).
• Intervals of Increase and Decrease:
  • If f'(x) > 0 on an interval, then f(x) is increasing on that interval.
  • If f'(x) < 0 on an interval, then f(x) is decreasing on that interval.
• Concavity:
  • If f''(x) > 0 on an interval, then f(x) is concave up (shaped like a U) on that interval.
  • If f''(x) < 0 on an interval, then f(x) is concave down (shaped like an upside-down U) on that interval.
• Inflection Points: These occur where the concavity of f(x) changes. They correspond to points where f''(x) = 0 or is undefined.
• Asymptotes:
  • Vertical Asymptotes: Occur where the function approaches infinity or negative infinity as x approaches a certain value. These often arise in rational functions where the denominator approaches zero.
  • Horizontal Asymptotes: Describe the function's behavior as x approaches positive or negative infinity.

2. Predict the Behavior of the Derivative (f'(x))

Based on the analysis of f(x), predict the key features of f'(x):

• Where f(x) has a horizontal tangent (critical point), f'(x) will be zero. The x-values of the critical points of f(x) will be the x-intercepts of f'(x).
• Where f(x) is increasing, f'(x) will be positive. The intervals where f(x) is increasing will correspond to intervals where f'(x) is above the x-axis.
• Where f(x) is decreasing, f'(x) will be negative. The intervals where f(x) is decreasing will correspond to intervals where f'(x) is below the x-axis.
• The steepness of f(x) influences the magnitude of f'(x). Steeper slopes on f(x) correspond to larger absolute values of f'(x).
• If f(x) is a polynomial of degree n, then f'(x) will be a polynomial of degree n-1. For example, if f(x) is a quadratic (degree 2), then f'(x) will be linear (degree 1).
• Concavity of f(x) relates to the slope of f'(x).
  • If f(x) is concave up, f'(x) is increasing.
  • If f(x) is concave down, f'(x) is decreasing.

3. Analyze the Potential Derivatives

Examine the graphs or equations of the potential derivatives and identify their key features:

• X-intercepts: Where f'(x) = 0.
• Positive and Negative Intervals: Where f'(x) > 0 and f'(x) < 0.
• Increasing and Decreasing Intervals: Where f'(x) is increasing and decreasing.
• Asymptotes: Vertical and horizontal asymptotes.
• General Shape: Is it linear, quadratic, cubic, exponential, trigonometric, etc.?

4. Match the Features

Compare the predicted features of f'(x) with the actual features of the potential derivatives. Look for a match in:

• X-intercepts: Do the x-intercepts of f'(x) correspond to the critical points of f(x)?
• Sign: Does the sign of f'(x) match the increasing/decreasing behavior of f(x)?
• Shape: Does the general shape of f'(x) make sense given the shape of f(x) (e.g., linear derivative for a quadratic function)?
• End Behavior: How does f'(x) behave as x approaches infinity or negative infinity? Does this align with the behavior of f(x)?

5. Use Examples and Test Points (If Necessary)

If you're still unsure, pick a specific x-value and calculate the approximate slope of f(x) at that point. Then, see if that slope matches the value of the potential derivative at that same x-value.

Examples and Applications

Let's illustrate this process with several examples:

Example 1: f(x) = x²

• Analysis of f(x): f(x) = x² is a parabola opening upwards with a vertex at (0, 0). It is decreasing for x < 0 and increasing for x > 0. It has a critical point at x = 0. It is concave up everywhere.
• Prediction of f'(x): f'(x) should be zero at x = 0, negative for x < 0, and positive for x > 0. It should be a linear function.
• Matching: The derivative of f(x) = x² is f'(x) = 2x. This is a linear function with a zero at x = 0, negative for x < 0, and positive for x > 0. It matches our prediction perfectly.

Example 2: f(x) = sin(x)

• Analysis of f(x): f(x) = sin(x) is a periodic function oscillating between -1 and 1. It has critical points at x = π/2 + nπ (where n is an integer). It is increasing on intervals like (-π/2, π/2) and decreasing on intervals like (π/2, 3π/2).
• Prediction of f'(x): f'(x) should be zero at x = π/2 + nπ. It should be positive where sin(x) is increasing and negative where sin(x) is decreasing. It should also be periodic.
• Matching: The derivative of f(x) = sin(x) is f'(x) = cos(x). This matches our prediction as cos(x) is zero at x = π/2 + nπ, positive where sin(x) is increasing, and negative where sin(x) is decreasing.

Example 3: f(x) = e^x

• Analysis of f(x): f(x) = e^x is an exponential function that is always increasing and always positive. It has no critical points and is concave up everywhere. It has a horizontal asymptote at y = 0 as x approaches negative infinity.
• Prediction of f'(x): f'(x) should be always positive and always increasing. It should also approach zero as x approaches negative infinity.
• Matching: The derivative of f(x) = e^x is f'(x) = e^x. This matches our prediction perfectly.

Example 4: f(x) = x³ - 3x

• Analysis of f(x): This is a cubic function. By finding the derivative and setting it to zero, we can find the critical points: f'(x) = 3x² - 3 = 0 x² = 1 x = ±1 So, there are critical points at x = -1 and x = 1. Analyzing the intervals:
  • x < -1: f'(x) > 0 (increasing)
  • -1 < x < 1: f'(x) < 0 (decreasing)
  • x > 1: f'(x) > 0 (increasing) This tells us that there's a local maximum at x = -1 and a local minimum at x = 1.
• Prediction of f'(x): f'(x) should be a quadratic function (degree 2) with roots at x = -1 and x = 1. It should be positive for x < -1 and x > 1, and negative for -1 < x < 1.
• Matching: The derivative is f'(x) = 3x² - 3. This matches our prediction. It's a quadratic with roots at x = -1 and x = 1.

Example 5: f(x) = 1/x

• Analysis of f(x): f(x) = 1/x is a rational function with a vertical asymptote at x = 0 and a horizontal asymptote at y = 0. It is decreasing for all x except at x = 0 where it is undefined.
• Prediction of f'(x): f'(x) should be always negative (except where it's undefined). It should have a vertical asymptote at x = 0.
• Matching: The derivative of f(x) = 1/x is f'(x) = -1/x². This matches our prediction. It is always negative (except at x = 0 where it's undefined) and has a vertical asymptote at x = 0.

Common Mistakes and How to Avoid Them

• Confusing f(x) with f'(x): Remember that f(x) represents the function itself, while f'(x) represents its rate of change. Don't assume that features of f(x) will directly translate to f'(x) without considering the derivative rules.
• Ignoring Critical Points: Critical points of f(x) are crucial for identifying the x-intercepts of f'(x).
• Misinterpreting Intervals of Increase and Decrease: Ensure you understand that f'(x) > 0 implies f(x) is increasing, and f'(x) < 0 implies f(x) is decreasing.
• Neglecting Concavity: Concavity of f(x) provides information about the slope of f'(x). If f(x) is concave up, f'(x) is increasing, and vice versa.
• Not Considering Asymptotes: Asymptotes of f(x) can provide clues about the behavior of f'(x), especially vertical asymptotes.

Advanced Techniques and Considerations

• Second Derivative Test: The second derivative f''(x) can be used to determine the concavity of f(x) and to classify critical points as local maxima or minima. If f'(c) = 0 and f''(c) > 0, then f(x) has a local minimum at x = c. If f'(c) = 0 and f''(c) < 0, then f(x) has a local maximum at x = c.
• L'Hôpital's Rule: This rule can be used to evaluate limits of indeterminate forms (e.g., 0/0 or ∞/∞) by taking the derivatives of the numerator and denominator. While not directly related to matching functions and derivatives, it's a useful tool in calculus for analyzing function behavior.
• Taylor and Maclaurin Series: These series provide polynomial approximations of functions, which can be helpful for understanding the local behavior of functions and their derivatives.
• Numerical Differentiation: When an analytical expression for f(x) is not available, numerical methods can be used to approximate the derivative at a point.

Practice Problems

Here are a few practice problems to test your understanding:

1. Match the following functions with their derivatives:

   • f(x) = x³
   • f(x) = cos(x)
   • f(x) = ln(x)
   • f(x) = 3x²
   • f(x) = -sin(x)
   • f(x) = 1/x

2. Given the graph of a function f(x), sketch a possible graph of its derivative f'(x).

3. Determine which of the following functions could be the derivative of f(x) = x⁴ - 2x² + 1:

   • g(x) = 4x³ - 4x
   • h(x) = 12x² - 4
   • k(x) = x³ - x

Conclusion

Matching a function with its derivative is a valuable exercise that reinforces your understanding of calculus concepts. By carefully analyzing the features of the original function and predicting the behavior of its derivative, you can successfully link a function's graphical representation to its rate of change. Remember to pay attention to critical points, intervals of increase and decrease, concavity, and asymptotes. With practice, this skill will become second nature and enhance your ability to analyze and interpret functions in various contexts. Mastering these techniques builds a strong foundation for more advanced topics in calculus and its applications in science, engineering, and economics.