Finding Derivatives Using The Definition And Evaluating F'(1) F'(2) F'(3)

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In calculus, the derivative of a function is a fundamental concept that describes the instantaneous rate of change of the function. It provides valuable information about the function's behavior, such as its increasing or decreasing nature, concavity, and critical points. One way to find the derivative is by using the definition of the derivative, which involves evaluating a limit. This article will guide you through the process of finding the derivative of the function f(x) = -x² + 2x - 2 using the definition of the derivative. Then, we will evaluate the derivative at specific points, namely x = 1, x = 2, and x = 3, to understand the function's behavior at these locations.

Understanding the Definition of the Derivative

The derivative of a function f(x), denoted as f'(x), is defined as the limit:

f'(x) = lim (h -> 0) [f(x + h) - f(x)] / h

This definition represents the slope of the tangent line to the graph of f(x) at a given point x. It measures how much the function's output changes in response to a tiny change in its input. The limit is crucial because it allows us to consider the slope of the secant line between two points that get infinitesimally close, effectively giving us the instantaneous rate of change.

The process of finding the derivative using this definition involves the following steps:

  1. Substitute (x + h) into the function: Replace every instance of x in the function f(x) with (x + h).
  2. Calculate f(x + h) - f(x): Subtract the original function f(x) from the expression obtained in step 1.
  3. Divide by h: Divide the result from step 2 by h.
  4. Evaluate the limit as h approaches 0: Find the limit of the expression obtained in step 3 as h gets closer and closer to 0. This step often involves algebraic manipulation to eliminate h from the denominator.

Finding the Derivative of f(x) = -x² + 2x - 2

Let's apply the definition of the derivative to find f'(x) for the function f(x) = -x² + 2x - 2. Following the steps outlined above:

  1. Substitute (x + h) into the function: f(x + h) = -(x + h)² + 2(x + h) - 2

  2. Calculate f(x + h) - f(x): f(x + h) - f(x) = [-(x + h)² + 2(x + h) - 2] - [-x² + 2x - 2] Expand and simplify:

    • = -(x² + 2xh + h²) + 2x + 2h - 2 + x² - 2x + 2*
    • = -x² - 2xh - h² + 2x + 2h - 2 + x² - 2x + 2*
    • = -2xh - h² + 2h*

  3. Divide by h: [f(x + h) - f(x)] / h = (-2xh - h² + 2h) / h Factor out h and simplify:

    • = h(-2x - h + 2) / h*
    • = -2x - h + 2*

  4. Evaluate the limit as h approaches 0: f'(x) = lim (h -> 0) (-2x - h + 2) As h approaches 0, the term -h vanishes, leaving: f'(x) = -2x + 2

Therefore, the derivative of f(x) = -x² + 2x - 2 is f'(x) = -2x + 2. This expression gives us the slope of the tangent line to the graph of f(x) at any point x.

Evaluating the Derivative at Specific Points

Now that we have found the derivative f'(x) = -2x + 2, we can evaluate it at specific points to determine the slope of the tangent line at those points. This will give us insights into the function's instantaneous rate of change at x = 1, x = 2, and x = 3.

Evaluating at x = 1

Substitute x = 1 into the derivative f'(x) = -2x + 2:

f'(1) = -2(1) + 2 = -2 + 2 = 0

This means that at x = 1, the slope of the tangent line to the graph of f(x) is 0. Geometrically, this indicates that the function has a horizontal tangent at this point, suggesting a possible local maximum or minimum.

Evaluating at x = 2

Substitute x = 2 into the derivative f'(x) = -2x + 2:

f'(2) = -2(2) + 2 = -4 + 2 = -2

At x = 2, the slope of the tangent line is -2. This negative slope indicates that the function is decreasing at this point. For every unit increase in x, the function's value decreases by approximately 2 units.

Evaluating at x = 3

Substitute x = 3 into the derivative f'(x) = -2x + 2:

f'(3) = -2(3) + 2 = -6 + 2 = -4

At x = 3, the slope of the tangent line is -4. This steeper negative slope compared to f'(2) suggests that the function is decreasing more rapidly at this point. The function's value decreases by approximately 4 units for every unit increase in x.

Interpreting the Results

By evaluating the derivative at x = 1, x = 2, and x = 3, we have gained valuable information about the behavior of the function f(x) = -x² + 2x - 2:

  • f'(1) = 0: Indicates a horizontal tangent at x = 1, suggesting a potential local maximum or minimum. A more in-depth analysis, such as the first or second derivative test, would be needed to confirm the nature of this critical point.
  • f'(2) = -2: Indicates that the function is decreasing at x = 2. The slope of -2 tells us the rate of decrease.
  • f'(3) = -4: Indicates that the function is decreasing more rapidly at x = 3 compared to x = 2. The steeper slope of -4 signifies a faster rate of decrease.

Conclusion

Using the definition of the derivative, we successfully found the derivative of the function f(x) = -x² + 2x - 2 to be f'(x) = -2x + 2. Evaluating the derivative at specific points x = 1, x = 2, and x = 3 provided insights into the function's behavior at those locations. The zero derivative at x = 1 suggests a critical point, while the negative derivatives at x = 2 and x = 3 indicate that the function is decreasing, with a faster rate of decrease at x = 3. This process demonstrates the power of the derivative in understanding the instantaneous rate of change and overall behavior of a function. Understanding derivatives is crucial for various applications in calculus and related fields.

After successfully determining the derivative f'(x) = -2x + 2, our next step involves evaluating the derivatives at specific points: x = 1, x = 2, and x = 3. This process allows us to pinpoint the instantaneous rate of change of the function f(x) = -x² + 2x - 2 at these particular locations. By substituting these x-values into the derivative function, we can observe how the slope of the tangent line varies, providing a detailed insight into the function's behavior. The value of the derivative at a point tells us whether the function is increasing, decreasing, or stationary at that point, and also the steepness of the graph. This information is invaluable for sketching the graph of the function and for applications such as optimization problems.

Understanding the Significance of f'(1)

The first critical point we examine is x = 1. By substituting this value into the derivative function, we get f'(1) = -2(1) + 2 = 0. This result is quite significant, as a derivative value of zero suggests that the function has a horizontal tangent line at this point. Geometrically, this implies that the function is neither increasing nor decreasing at x = 1. Such points are often referred to as stationary points or critical points, and they can represent local maxima, local minima, or inflection points. To further classify the nature of this point, we would need to perform additional tests, such as the first derivative test or the second derivative test. However, the fact that f'(1) = 0 immediately highlights x = 1 as a point of interest in the function's behavior. Finding critical points is a key step in analyzing the function.

Analysis at x = 2: Interpreting f'(2)

Next, we consider the point x = 2. Evaluating the derivative at this point, we find f'(2) = -2(2) + 2 = -2. Here, the negative value of the derivative is particularly informative. A negative f'(x) indicates that the function is decreasing at x = 2. The magnitude of the derivative, in this case, 2, tells us the rate at which the function is decreasing. Specifically, for a small increase in x near x = 2, the function f(x) will decrease by approximately 2 times that amount. This gives us a clear picture of the function's local behavior: as we move slightly to the right from x = 2, the function's value will decrease. Understanding the sign of f'(x) is crucial for determining intervals of increasing and decreasing behavior.

Examining the Derivative at x = 3: Decoding f'(3)

Finally, let's examine the derivative at x = 3. Substituting this value into the derivative function yields f'(3) = -2(3) + 2 = -4. Similar to the case at x = 2, the negative value indicates that the function is decreasing at x = 3. However, the magnitude of the derivative is now 4, which is greater than the magnitude at x = 2. This signifies that the function is decreasing at a faster rate at x = 3 than it is at x = 2. The steeper negative slope of the tangent line at x = 3 provides a visual representation of this more rapid decrease. By comparing the derivatives at different points, we can gain insights into how the rate of change varies across the function's domain.

Synthesizing the Results: A Comprehensive View

By analyzing the derivatives at x = 1, x = 2, and x = 3, we've pieced together a more complete picture of the function f(x) = -x² + 2x - 2. At x = 1, the function has a horizontal tangent, suggesting a potential local extremum. As we move to x = 2 and x = 3, the function is decreasing, and the rate of decrease is greater at x = 3 than at x = 2. This suggests that the function's graph is likely to be concave downwards in this region. This type of analysis, combining the values of the derivative at specific points, is a powerful tool in understanding and visualizing the behavior of functions. This understanding is essential in various fields, including physics, engineering, and economics, where functions are used to model real-world phenomena. Interpreting derivatives in these contexts allows professionals to make predictions and optimize processes.

Conclusion: Derivative Insights

In summary, evaluating the derivatives at the points x = 1, x = 2, and x = 3 for the function f(x) = -x² + 2x - 2 has provided significant insights into the function's behavior. The zero derivative at x = 1 signals a stationary point, while the negative derivatives at x = 2 and x = 3 indicate that the function is decreasing, with the rate of decrease being more pronounced at x = 3. This detailed analysis exemplifies how the derivative can be used to understand a function's rate of change and behavior at specific points. Applying derivative concepts like these is fundamental to the study of calculus and its applications.