Difference Quotient For F(x) = X³ + X + 7 Calculation And Explanation

Jul 13, 2025 by ADMIN 70 views

Understanding the Difference Quotient for f(x) = x³ + x + 7

The difference quotient is a fundamental concept in calculus that provides a way to calculate the average rate of change of a function over a small interval. It serves as the foundation for understanding the derivative, which represents the instantaneous rate of change of a function at a specific point. In simpler terms, the difference quotient helps us to see how much a function's output changes compared to the change in its input. This concept is crucial in various fields like physics, engineering, and economics, where understanding rates of change is essential. The difference quotient is defined as:

\frac{f(x + h) - f(x)}{h}

where:

f(x) is the function.
h is a small change in x.
f(x + h) is the function evaluated at x + h.

The difference quotient essentially calculates the slope of the secant line through two points on the function's graph: (x, f(x)) and (x + h, f(x + h)). As h approaches zero, this secant line approaches the tangent line, and the difference quotient approaches the derivative of the function. Understanding this concept is vital for grasping the core principles of calculus and its applications.

Applying the Difference Quotient to f(x) = x³ + x + 7

To find the difference quotient for the function f(x) = x³ + x + 7, we need to follow these steps:

Find f(x + h): This involves substituting x + h into the function wherever x appears. So, we have:
```
f(x + h) = (x + h)³ + (x + h) + 7
```
Expand and simplify f(x + h): Expanding (x + h)³ gives us x³ + 3x²h + 3xh² + h³. Thus,
```
f(x + h) = x³ + 3x²h + 3xh² + h³ + x + h + 7
```
Expanding and simplifying correctly is crucial, as any error here will propagate through the rest of the calculation. This step is a common area for mistakes, so it's important to be meticulous.
Calculate f(x + h) - f(x): Subtract the original function f(x) from the expanded f(x + h):
```
f(x + h) - f(x) = (x³ + 3x²h + 3xh² + h³ + x + h + 7) - (x³ + x + 7)
```
Simplify the expression: Notice that several terms cancel out: x³, x, and 7. This leaves us with:
```
f(x + h) - f(x) = 3x²h + 3xh² + h³ + h
```
Divide by h: Divide the entire expression by h to obtain the difference quotient:
```
\frac{f(x + h) - f(x)}{h} = \frac{3x²h + 3xh² + h³ + h}{h}
```
Simplify the final expression: Factor out h from the numerator and cancel it with the denominator:
```
\frac{h(3x² + 3xh + h² + 1)}{h} = 3x² + 3xh + h² + 1
```

Following these steps carefully ensures that we arrive at the correct difference quotient for the given function. Each step builds upon the previous one, making a clear and methodical approach essential for accuracy. The result, 3x² + 3xh + h² + 1, represents the average rate of change of the function f(x) = x³ + x + 7 over the interval [x, x + h]. The meticulous execution of each step is vital in arriving at the correct expression for the difference quotient, which forms the basis for understanding derivatives in calculus.

The Correct Difference Quotient

Based on the step-by-step calculation, the difference quotient for f(x) = x³ + x + 7 is:

3x² + 3xh + h² + 1

This matches option B from the provided choices. The other options are incorrect because they do not result from the proper application of the difference quotient formula to the given function. Understanding the algebraic manipulations and simplifications involved is key to arriving at the right answer. The difference quotient, in this case, provides a crucial insight into how the function f(x) = x³ + x + 7 changes over small intervals, setting the stage for more advanced calculus concepts such as derivatives and tangent lines.

Why Other Options Are Incorrect

To fully understand the solution, it's important to examine why the other options are incorrect. This reinforces the concept and the importance of following the correct procedure.

Option A: n² + 1

This option is incorrect because it does not arise from the difference quotient formula applied to the given function. There is no direct mathematical pathway from substituting x + h into f(x) = x³ + x + 7, simplifying, and dividing by h that would result in an expression like n² + 1. This answer choice appears to be a distractor, unrelated to the actual process of finding the difference quotient.
Option C: n

Similar to option A, this option is also incorrect. The difference quotient calculation involves several steps of algebraic manipulation, and the result should be a more complex expression that includes terms with x and h. A simple term like n does not reflect the changes in the function f(x) = x³ + x + 7 over a small interval. This choice likely serves as another distractor, as it lacks the components derived from the difference quotient formula.
Option D: 3x² + 3xh + h²

This option is closer to the correct answer but is missing a crucial component. When calculating the difference quotient, after substituting x + h into the function, expanding, subtracting f(x), and dividing by h, the constant term 1 remains from the original x term in f(x). The correct difference quotient should be 3x² + 3xh + h² + 1. Option D omits this 1, making it an incomplete and therefore incorrect answer. This highlights the importance of careful attention to detail in each step of the calculation.

Understanding why these options are incorrect reinforces the correct methodology and the specific steps involved in finding the difference quotient. Each incorrect option deviates from the correct process in different ways, emphasizing the necessity of a thorough and accurate application of the formula.

Significance of the Difference Quotient

The difference quotient is not just an algebraic exercise; it holds significant importance in calculus and related fields. Here's why:

Foundation for the Derivative: The difference quotient is the precursor to the derivative. The derivative, denoted as f'(x), represents the instantaneous rate of change of a function at a specific point. Mathematically, the derivative is defined as the limit of the difference quotient as h approaches zero:
```
f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h}
```
Understanding the difference quotient is crucial for grasping the concept of the derivative, which is a cornerstone of calculus.
Average Rate of Change: The difference quotient itself gives the average rate of change of a function over the interval [x, x + h]. This is useful in various applications. For example, in physics, if f(x) represents the position of an object at time x, then the difference quotient gives the average velocity of the object over the time interval [x, x + h]. Similarly, in economics, if f(x) represents the cost of producing x items, the difference quotient gives the average cost per item over a certain production range.
Tangent Lines: The difference quotient helps in finding the slope of the tangent line to a curve at a particular point. As h approaches zero, the secant line, whose slope is given by the difference quotient, approaches the tangent line. Thus, the limit of the difference quotient gives the slope of the tangent line, which is a fundamental concept in calculus and geometry.
Applications in Science and Engineering: The concept of the difference quotient and its limit, the derivative, are widely used in science and engineering. They are essential for modeling and analyzing systems where rates of change are important, such as motion, growth, decay, and optimization problems. For instance, in chemical kinetics, the rate of a chemical reaction can be described using derivatives, which are based on the difference quotient.
Numerical Methods: In numerical analysis, the difference quotient is used to approximate derivatives when analytical solutions are not possible. Finite difference methods, which are based on the difference quotient, are used to solve differential equations numerically. These methods are crucial in computational physics, engineering simulations, and other areas where numerical solutions are necessary.

In summary, the difference quotient is a fundamental concept with wide-ranging applications. It provides a foundation for understanding derivatives, rates of change, and tangent lines, and it is used extensively in various fields to model and analyze dynamic systems. Mastering the difference quotient is an essential step in learning calculus and its applications.