Difference Quotient And Limit Calculation For F(x) = X At C = 15

July 6, 2025 by Scholario Team 65 views

Unveiling the Difference Quotient and Limit for f(x) = x at c = 15

In the realm of calculus, the difference quotient serves as a fundamental building block for understanding the concept of the derivative. It provides a measure of the average rate of change of a function over a small interval. This article delves into the process of constructing the difference quotient for a specific function, f(x) = x, at a particular point, c = 15, and subsequently evaluating the limit of this quotient as the interval shrinks towards zero. This exploration will illuminate the essence of the derivative and its connection to the instantaneous rate of change.

Demystifying the Difference Quotient

The difference quotient, denoted as F(h), is defined as:

F(h) = [f(c + h) - f(c)] / h

where:

f(x) represents the function under consideration.
c signifies the point at which we are evaluating the rate of change.
h denotes a small change or increment in the input variable x.

The difference quotient essentially calculates the slope of the secant line passing through two points on the graph of the function: (c, f(c)) and (c + h, f(c + h)). As h approaches zero, this secant line progressively approximates the tangent line at the point (c, f(c)), and the slope of the tangent line represents the instantaneous rate of change, which is the derivative.

Constructing the Difference Quotient for f(x) = x at c = 15

For the given function f(x) = x and the specified value c = 15, we can construct the difference quotient as follows:

Evaluate f(c + h): f(c + h) = f(15 + h) = 15 + h
Evaluate f(c): f(c) = f(15) = 15

Substitute into the difference quotient formula:

F(h) = [f(c + h) - f(c)] / h
F(h) = [(15 + h) - 15] / h
F(h) = h / h

Simplify the expression: For h ≠ 0, we can simplify the expression by canceling out the h terms:
```
F(h) = 1
```

Therefore, the difference quotient for f(x) = x at c = 15 is F(h) = 1, for all h ≠ 0.

Evaluating the Limit of the Difference Quotient

To find the instantaneous rate of change, we need to evaluate the limit of the difference quotient as h approaches zero:

lim (h→0) F(h) = lim (h→0) 1

Since F(h) = 1 for all h ≠ 0, the limit as h approaches zero is simply 1.

lim (h→0) F(h) = 1

This result signifies that the instantaneous rate of change of the function f(x) = x at the point c = 15 is 1. In other words, the slope of the tangent line to the graph of f(x) = x at x = 15 is 1. This aligns with the understanding that the derivative of f(x) = x is a constant function equal to 1, indicating a constant rate of change across all points.

Significance of the Limit of the Difference Quotient

The limit of the difference quotient holds immense significance in calculus as it represents the derivative of the function at a specific point. The derivative, denoted as f'(c), provides the instantaneous rate of change of the function at x = c. It essentially captures how the function's output changes in response to an infinitesimally small change in its input.

In this case, the derivative of f(x) = x at c = 15 is 1, implying that for every minuscule change in x around x = 15, the function's output f(x) changes by the same amount. This linear relationship is characteristic of the function f(x) = x, where the rate of change is constant across its entire domain.

Furthermore, the derivative has profound geometrical implications. It represents the slope of the tangent line to the graph of the function at the point (c, f(c)). The tangent line provides the best linear approximation of the function at that particular point, making the derivative a crucial tool for analyzing the local behavior of functions.

Applications of the Difference Quotient and its Limit

The concepts of the difference quotient and its limit, which gives us the derivative, are central to a wide array of applications in mathematics, science, and engineering. Some notable applications include:

Optimization: Finding the maximum or minimum values of a function, which is crucial in various fields like economics, physics, and engineering.
Motion Analysis: Determining the velocity and acceleration of an object given its position as a function of time.
Curve Sketching: Understanding the shape and behavior of a function's graph, including its increasing and decreasing intervals, concavity, and points of inflection.
Related Rates: Solving problems that involve the rates of change of related quantities.
Approximation: Approximating the value of a function at a point using its tangent line or other linear approximations.

In summary, the difference quotient and its limit provide a powerful framework for understanding and analyzing the behavior of functions. By calculating the instantaneous rate of change, the derivative opens doors to a vast landscape of applications, making it a cornerstone of calculus and its related fields.

Conclusion

In conclusion, by meticulously constructing the difference quotient for the function f(x) = x at c = 15 and evaluating its limit as h approaches zero, we have successfully determined the instantaneous rate of change of the function at that specific point. The result, 1, signifies the slope of the tangent line to the graph of f(x) = x at x = 15, which aligns perfectly with the derivative of the function. This exercise serves as a compelling illustration of the fundamental principles of calculus and the power of the difference quotient in unraveling the intricacies of functional behavior. The ability to compute difference quotients and limits is essential for anyone seeking a deep understanding of calculus and its vast applications in various disciplines.