Answer The Following Questions About The Function Whose Derivative Is

Let's explore the characteristics of a function by examining its derivative. The derivative, often denoted as f'(x), reveals crucial information about the original function f(x), including where it's increasing, decreasing, has local extrema (maxima and minima), and the concavity of its graph. By analyzing f'(x), we can reconstruct a comprehensive understanding of f(x) without explicitly knowing the equation for f(x) itself.

Understanding the Relationship Between a Function and Its Derivative

Before diving into specific questions, let’s cement the foundational relationship between a function and its derivative:

• The Derivative as Slope: At any point 'x', the derivative f'(x) gives the slope of the tangent line to the graph of f(x) at that point. A positive f'(x) indicates f(x) is increasing, a negative f'(x) signifies f(x) is decreasing, and f'(x) = 0 suggests a potential local maximum or minimum.
• Increasing and Decreasing Intervals:
  • If f'(x) > 0 for all x in an interval, then f(x) is increasing on that interval.
  • If f'(x) < 0 for all x in an interval, then f(x) is decreasing on that interval.
  • If f'(x) = 0 for all x in an interval, then f(x) is constant on that interval.
• Critical Points: Points where f'(x) = 0 or f'(x) is undefined are called critical points. These points are crucial because they are the potential locations of local maxima, local minima, or saddle points.
• Concavity and the Second Derivative: The second derivative, f''(x), tells us about the concavity of f(x).
  • If f''(x) > 0, then f(x) is concave up (shaped like a cup).
  • If f''(x) < 0, then f(x) is concave down (shaped like a frown).
  • Points where the concavity changes are called inflection points. These points occur where f''(x) = 0 or f''(x) is undefined.

With this groundwork in place, we can now tackle specific questions about a function based solely on information about its derivative.

Questions and Answers About a Function Based on Its Derivative

Let's consider a scenario where you are given the derivative, f'(x), of a function f(x). The goal is to answer various questions about the behavior of f(x) using only the information gleaned from f'(x).

Scenario: Suppose we know that the derivative of a function f(x) is given by:

f'(x) = (x - 2)(x + 1) / x

We will now answer several common questions about the function f(x) based on this derivative.

Question 1: Find the intervals on which f(x) is increasing and decreasing.

Answer:

To determine where f(x) is increasing or decreasing, we need to analyze the sign of f'(x). This involves finding the critical points of f(x) (where f'(x) = 0 or is undefined) and then creating a sign chart.

• Find Critical Points:

  • f'(x) = 0 when (x - 2)(x + 1) = 0, which gives us x = 2 and x = -1.
  • f'(x) is undefined when x = 0 (due to the denominator).

• Create a Sign Chart: We'll create a number line and mark the critical points: -1, 0, and 2. These points divide the number line into intervals. We then test a value within each interval to determine the sign of f'(x) in that interval.

  Interval	Test Value	f'(x) = (x - 2)(x + 1) / x	Sign of f'(x)	Conclusion
  (-∞, -1)	x = -2	((-2) - 2)((-2) + 1) / (-2)	-	f(x) is decreasing
  (-1, 0)	x = -0.5	((-0.5) - 2)((-0.5) + 1) / (-0.5)	+	f(x) is increasing
  (0, 2)	x = 1	((1) - 2)((1) + 1) / (1)	-	f(x) is decreasing
  (2, ∞)	x = 3	((3) - 2)((3) + 1) / (3)	+	f(x) is increasing

• Conclusion:

  • f(x) is increasing on the intervals (-1, 0) and (2, ∞).
  • f(x) is decreasing on the intervals (-∞, -1) and (0, 2).

Question 2: Find the x-coordinates of any local maxima and local minima of f(x).

Answer:

Local maxima and minima occur at critical points where the derivative changes sign. We can use the First Derivative Test to determine the nature of these critical points.

• First Derivative Test:

  • At x = -1: f'(x) changes from negative to positive. Therefore, f(x) has a local minimum at x = -1.
  • At x = 0: f'(x) changes from positive to negative. Therefore, f(x) has a local maximum at x = 0.
  • At x = 2: f'(x) changes from negative to positive. Therefore, f(x) has a local minimum at x = 2.

• Conclusion:

  • f(x) has a local minimum at x = -1.
  • f(x) has a local maximum at x = 0.
  • f(x) has a local minimum at x = 2.

Question 3: Find the intervals on which the graph of f(x) is concave up and concave down. Also, find the x-coordinates of any inflection points.

Answer:

To determine concavity, we need to find the second derivative, f''(x), and analyze its sign.

• Find the Second Derivative:

  First, rewrite f'(x) to make differentiation easier: f'(x) = (x^2 - x - 2) / x = x - 1 - 2/x

  Now, find f''(x): f''(x) = 1 + 2/x^2 = (x^2 + 2) / x^2

• Find Critical Points of f''(x):

  • f''(x) = 0 has no solution because x^2 + 2 is always positive.
  • f''(x) is undefined when x = 0.

• Create a Sign Chart for f''(x):

  Interval	Test Value	f''(x) = (x^2 + 2) / x^2	Sign of f''(x)	Conclusion
  (-∞, 0)	x = -1	((-1)^2 + 2) / (-1)^2	+	f(x) is concave up
  (0, ∞)	x = 1	((1)^2 + 2) / (1)^2	+	f(x) is concave up

• Conclusion:

  • f(x) is concave up on the intervals (-∞, 0) and (0, ∞).
  • Since the concavity does not change at x = 0, there is no inflection point at x = 0. While f''(x) is undefined at x=0, the concavity remains consistently upward on either side.

Question 4: Does f(x) have any vertical asymptotes? If so, where?

Answer:

Vertical asymptotes of f(x) can occur where f'(x) is undefined and the limit of f(x) approaches infinity (or negative infinity). The fact that f'(x) contains a term divided by x hints at a potential asymptote at x = 0.

• Analysis: Since f'(x) = (x - 2)(x + 1) / x, we know that f'(x) is undefined at x = 0. This suggests a possible vertical asymptote for f(x) at x = 0. To confirm, we'd ideally want to understand the behavior of f(x) near x = 0. However, without the explicit equation for f(x), we can only infer based on the behavior of f'(x). The fact that f'(x) approaches infinity as x approaches 0 strongly suggests that f(x) will have a vertical asymptote at x = 0.

• Conclusion: There is likely a vertical asymptote at x = 0. Note: Without knowing the original function, we can't definitively prove this, but the structure of the derivative makes it highly probable. To confirm, we would need to analyze the limit of f(x) as x approaches 0 from both sides, but this requires knowing the explicit form of f(x).

Question 5: Sketch a possible graph of f(x) based on the information gathered.

Answer:

Based on our analysis, we can sketch a possible graph of f(x). Remember, the sketch will be qualitative, as we don't know the exact y-values of the function.

• Key Features:
  • Increasing on (-1, 0) and (2, ∞).
  • Decreasing on (-∞, -1) and (0, 2).
  • Local minimum at x = -1 and x = 2.
  • Local maximum at x = 0.
  • Concave up on (-∞, 0) and (0, ∞).
  • Vertical asymptote at x = 0 (likely).

The sketch would show a curve that decreases from negative infinity to a local minimum at x = -1, then increases to a local maximum at x = 0. As x approaches 0 from the negative side, the graph goes towards negative infinity (due to the vertical asymptote). On the positive side of the y-axis, the graph comes from positive infinity, decreasing to a local minimum at x = 2, and then increasing towards positive infinity. The entire curve is concave up.

Question 6: How would the graph change if we knew that f(1) = 0?

Answer:

Knowing that f(1) = 0 provides a crucial anchor point for our sketch. It tells us that the graph of f(x) passes through the point (1, 0). This information allows us to refine our sketch and place the curve more accurately in the coordinate plane. All the other qualitative features remain the same, but now the graph has a defined point.

• Impact on the Sketch:

  The graph, as described previously, is simply shifted vertically such that it includes the point (1, 0). This doesn't change the locations of the local minima or maxima, the intervals of increasing or decreasing behavior, or the concavity. It only specifies the function's value at one particular point.

Question 7: Determine the intervals where the rate of change of the slope of f(x) is positive.

Answer:

The rate of change of the slope of f(x) is described by the second derivative, f''(x). When f''(x) is positive, the slope of f(x) is increasing. We've already determined that f''(x) = (x^2 + 2) / x^2. As previously stated, this is positive for all x except x=0 where it is undefined.

• Conclusion: The rate of change of the slope of f(x) is positive on the intervals (-∞, 0) and (0, ∞).

Question 8: Explain how the original function f(x) can be found if f'(x) is known, but without using integration.

Answer:

While integration is the standard method to find f(x) from f'(x), it is, strictly speaking, not the only way. The process is more of an educated guess, and its practicality depends heavily on the complexity of f'(x). Here's the conceptual approach and its limitations:

1. Pattern Recognition and "Reverse Engineering": Look for recognizable derivative patterns in f'(x).
   • If f'(x) contains terms like x, it likely came from x^2 in f(x).
   • If f'(x) contains terms like 1/x, it likely came from ln|x| in f(x).
   • If f'(x) contains terms like e^x, it likely came from e^x in f(x).
   • The key is to think backward: "What function, when differentiated, would result in this term in f'(x)?"
2. Reconstructing Based on Known Rules: Knowing basic derivative rules in reverse lets you build a potential f(x).
   • Power Rule (in reverse): If f'(x) has ax^n, then f(x) likely had (a/(n+1))x^(n+1).
   • Constant Multiple Rule (in reverse): If f'(x) has c * g'(x), then f(x) likely had c * g(x).
   • Sum/Difference Rule (in reverse): If f'(x) is u'(x) + v'(x), then f(x) is likely u(x) + v(x).
3. Considering the Constant of Integration (Crucially): When you reverse a derivative, you always have a "+ C" (the constant of integration) because the derivative of a constant is zero. So, any constant term could have been present in f(x) and would have disappeared during differentiation. Include "+ C" in your reconstructed f(x).
4. Verification (Essential): The MOST IMPORTANT step is to differentiate your reconstructed f(x) to see if it matches the given f'(x). If it doesn't match perfectly, your "reverse engineering" was incorrect, and you need to adjust your guess.

Example (with our f'(x)):

f'(x) = (x - 2)(x + 1) / x = x - 1 - 2/x

• x likely came from (1/2)x^2
• -1 likely came from -x
• -2/x likely came from -2ln|x|

So, a possible f(x) = (1/2)x^2 - x - 2ln|x| + C

Differentiate to check:

f'(x) = x - 1 - 2/x = (x^2 - x - 2) / x = (x - 2)(x + 1) / x (This matches!)

Limitations:

• Complexity: This method becomes extremely difficult or impossible for more complex derivatives involving trigonometric functions, inverse trigonometric functions, products, quotients, or composite functions.
• Guaranteed Correctness: There's no guarantee that your "reverse engineering" will lead to the correct original function, especially if you miss a subtle pattern or make an incorrect assumption. You might find a function that has the given derivative, but it might not be the function you were looking for.
• Reliance on Intuition: It relies heavily on pattern recognition and intuition, which can be unreliable.

In summary, while theoretically possible in simple cases, "reverse engineering" derivatives without integration is generally impractical and prone to errors. Integration provides a systematic and reliable way to find the original function.

Question 9: Why is knowing the derivative useful? Provide real-world examples.

Answer:

The derivative is a cornerstone of calculus and provides powerful tools for understanding rates of change and optimization in a vast range of fields. Its usefulness stems from its ability to quantify instantaneous change, which is essential for modeling and predicting real-world phenomena.

Here are several real-world examples illustrating the utility of the derivative:

1. Physics:

   • Velocity and Acceleration: If you know the position of an object as a function of time (s(t)), the derivative s'(t) gives you the velocity of the object at any time t. The derivative of velocity, v'(t) = s''(t), gives you the acceleration. This is fundamental to understanding motion in classical mechanics, projectile motion, and orbital mechanics.
   • Heat Transfer: The derivative is used to model the rate of heat flow in materials. Understanding how temperature changes over time and space is critical in engineering design, materials science, and climate modeling.

2. Engineering:

   • Optimization: Engineers use derivatives to optimize designs. For example, minimizing the weight of a bridge while ensuring its structural integrity involves finding the minimum of a function representing the bridge's weight, subject to constraints on its strength. Derivatives help find the dimensions that minimize cost, maximize efficiency, or minimize waste.
   • Control Systems: Derivatives are used in control systems to predict how a system will respond to changes and to adjust parameters to maintain stability. For example, in cruise control, the derivative of the car's speed is used to adjust the throttle and maintain a constant speed.

3. Economics and Finance:

   • Marginal Analysis: Economists use derivatives to analyze marginal cost, marginal revenue, and marginal profit. These concepts help businesses make decisions about production levels, pricing, and investment. For example, marginal cost is the derivative of the total cost function, and it represents the cost of producing one additional unit.
   • Stock Market Analysis: While not directly a derivative application, understanding rates of change is crucial in finance. Traders analyze the rate of change of stock prices to identify trends and make informed decisions. More sophisticated models use calculus to price derivatives (financial instruments).

4. Biology and Medicine:

   • Population Growth: The derivative can model the rate of population growth (or decline). This helps ecologists understand how populations change over time and predict future population sizes.
   • Drug Dosage: Doctors use derivatives to model how drugs are absorbed and eliminated from the body. This helps them determine the optimal dosage to achieve a desired therapeutic effect while minimizing side effects.
   • Enzyme Kinetics: The rate of enzyme-catalyzed reactions can be modeled using derivatives. Understanding these rates is crucial in biochemistry and pharmacology.

5. Computer Science:

   • Machine Learning: Derivatives are the heart of many machine-learning algorithms, particularly those that use gradient descent. Gradient descent is an iterative optimization algorithm that uses derivatives to find the minimum of a loss function. This is essential for training neural networks and other machine-learning models.
   • Image Processing: Derivatives are used to detect edges and features in images. This is a fundamental step in many image processing applications, such as object recognition and image segmentation.

6. Weather Forecasting:

   • Atmospheric Modeling: Sophisticated weather models use derivatives to simulate the complex interactions of temperature, pressure, humidity, and wind. These models help meteorologists predict weather patterns and issue warnings about severe weather events.

In summary, the derivative is a fundamental tool for understanding and modeling change in a wide variety of fields. Its ability to quantify instantaneous rates of change and identify optimal solutions makes it an indispensable tool for scientists, engineers, economists, and many others. By understanding the derivative, we can gain deeper insights into the world around us and make more informed decisions.

Conclusion

Analyzing the derivative of a function allows us to understand the behavior of the original function without necessarily knowing its explicit formula. By identifying critical points, creating sign charts, and applying the First and Second Derivative Tests, we can determine intervals of increasing and decreasing behavior, locate local extrema, and describe the concavity of the function's graph. This knowledge is invaluable in various fields, from physics and engineering to economics and computer science, highlighting the importance of understanding the relationship between a function and its derivative. The ability to extract meaningful information from the derivative underscores its fundamental role in calculus and its wide-ranging applications.