Given The Function F(x) = 2x + 3, What Is The Value Of [f(3+h) - F(3)]/h?

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In the realm of mathematics, functions serve as fundamental building blocks, mapping inputs to corresponding outputs. Understanding how functions behave and how their values change is crucial for tackling a wide range of mathematical problems. In this article, we delve into the intricacies of the function f(x) = 2x + 3 and embark on a step-by-step journey to calculate the value of the expression [f(3+h) - f(3)]/h. This expression, known as the difference quotient, plays a pivotal role in calculus, serving as the foundation for understanding derivatives, which measure the instantaneous rate of change of a function.

1. Grasping the Essence of Functions

Before we immerse ourselves in the calculations, let's take a moment to solidify our understanding of functions. At its core, a function is a rule that assigns a unique output to each input. Think of it as a mathematical machine that takes an input, processes it according to its internal rule, and spits out a corresponding output. In our case, the function f(x) = 2x + 3 takes an input 'x', multiplies it by 2, and then adds 3 to produce the output. For instance, if we input x = 2, the function would yield f(2) = 2 * 2 + 3 = 7.

Delving into the Realm of f(x) = 2x + 3

The function f(x) = 2x + 3 is a linear function, characterized by its straight-line graph. The coefficient 2, known as the slope, dictates the steepness of the line, while the constant term 3, known as the y-intercept, determines where the line crosses the vertical axis. Linear functions are ubiquitous in mathematics and real-world applications, modeling relationships that exhibit a constant rate of change.

2. Deciphering the Difference Quotient: [f(3+h) - f(3)]/h

The expression [f(3+h) - f(3)]/h is a cornerstone of calculus, known as the difference quotient. It provides a measure of the average rate of change of a function over a small interval. The difference quotient essentially calculates the slope of the secant line that connects two points on the function's graph. As 'h' approaches zero, this secant line morphs into the tangent line, and the difference quotient converges to the derivative, which represents the instantaneous rate of change at a specific point.

Unraveling the Components of the Difference Quotient

To fully grasp the difference quotient, let's break it down into its constituent parts:

  • f(3+h): This represents the value of the function when the input is (3+h). We substitute (3+h) for 'x' in the function's expression to obtain f(3+h) = 2(3+h) + 3.
  • f(3): This represents the value of the function when the input is 3. We substitute 3 for 'x' in the function's expression to obtain f(3) = 2 * 3 + 3.
  • h: This represents a small change in the input value. It's the difference between the two input values we're considering, (3+h) and 3.

3. Embarking on the Calculation Journey: A Step-by-Step Approach

Now that we've laid the groundwork, let's embark on the calculation journey, breaking down the process into manageable steps:

Step 1: Evaluating f(3+h)

To begin, we need to determine the value of f(3+h). We substitute (3+h) for 'x' in the function's expression:

f(3+h) = 2(3+h) + 3

Expanding the expression, we get:

f(3+h) = 6 + 2h + 3

Simplifying further, we arrive at:

f(3+h) = 2h + 9

Step 2: Evaluating f(3)

Next, we need to find the value of f(3). We substitute 3 for 'x' in the function's expression:

f(3) = 2 * 3 + 3

Performing the calculation, we get:

f(3) = 6 + 3

Therefore:

f(3) = 9

Step 3: Plugging into the Difference Quotient

Now that we've calculated f(3+h) and f(3), we can substitute these values into the difference quotient expression:

[f(3+h) - f(3)]/h = [(2h + 9) - 9]/h

Step 4: Simplifying the Expression

Simplifying the numerator, we get:

[f(3+h) - f(3)]/h = (2h)/h

Step 5: The Grand Finale: Canceling 'h'

Finally, we can cancel out the 'h' in the numerator and denominator, provided that h is not equal to zero:

[f(3+h) - f(3)]/h = 2

4. The Unveiling: The Value of the Difference Quotient

After our meticulous step-by-step journey, we've arrived at the solution: the value of the difference quotient [f(3+h) - f(3)]/h for the function f(x) = 2x + 3 is 2.

The Significance of the Result

This result holds profound significance in the realm of calculus. It tells us that the average rate of change of the function f(x) = 2x + 3 over any small interval around x = 3 is constant and equal to 2. In the context of derivatives, this implies that the instantaneous rate of change of the function at x = 3 is also 2. This aligns with the fact that the function f(x) = 2x + 3 is a linear function with a slope of 2, indicating a constant rate of change across its entire domain.

5. Expanding the Horizon: Applications of the Difference Quotient

The difference quotient is not merely an abstract mathematical concept; it has far-reaching applications in various fields:

  • Calculus: As we've seen, the difference quotient is the cornerstone of derivatives, which are essential for understanding rates of change, optimization problems, and curve sketching.
  • Physics: In physics, the difference quotient can be used to calculate average velocities and accelerations over time intervals.
  • Economics: In economics, the difference quotient can be used to determine marginal costs and marginal revenues, which are crucial for making optimal production decisions.
  • Engineering: In engineering, the difference quotient can be used to analyze the behavior of systems and design control mechanisms.

6. Conclusion: A Mathematical Odyssey Completed

In this article, we've embarked on a mathematical odyssey, unraveling the solution to the problem of finding the value of the difference quotient [f(3+h) - f(3)]/h for the function f(x) = 2x + 3. We've journeyed through the essence of functions, deciphered the meaning of the difference quotient, and meticulously executed the calculations, step by step. Along the way, we've highlighted the significance of the result and explored the diverse applications of the difference quotient in various fields.

This exploration serves as a testament to the power of mathematics in illuminating the intricacies of the world around us. By grasping fundamental concepts and honing our problem-solving skills, we can unlock a deeper understanding of the mathematical principles that govern our universe.

To enhance clarity and understanding, let's reword the original question and identify the core keywords:

Reworded Question: Given the function f(x) = 2x + 3, what is the value of the expression [f(3+h) - f(3)]/h?

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