Net Change Of F(t) = 6/t Between T = A And T = A+h

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In the realm of mathematical analysis, understanding the concept of net change is paramount. This article delves into the intricacies of determining net change, specifically focusing on the function f(t) = 6/t. We will explore how to calculate the net change between two given values of the variable, t = a and t = a + h. This exploration will not only solidify your understanding of net change but also enhance your ability to apply this concept to various mathematical and real-world scenarios.

Understanding the Function f(t) = 6/t

At the heart of our exploration lies the function f(t) = 6/t. This function represents a reciprocal relationship, where the output f(t) is inversely proportional to the input t. In simpler terms, as t increases, f(t) decreases, and vice versa. This type of function is commonly encountered in various scientific and engineering applications, making its understanding crucial.

To truly grasp the behavior of this function, let's consider a few examples:

  • When t = 1, f(t) = 6/1 = 6
  • When t = 2, f(t) = 6/2 = 3
  • When t = 3, f(t) = 6/3 = 2

These examples clearly demonstrate the inverse relationship. As t doubles from 1 to 2, f(t) is halved from 6 to 3. Similarly, when t triples from 1 to 3, f(t) is reduced to one-third of its original value. This fundamental understanding of the function's behavior is essential for calculating the net change.

Visualizing the Function

A graphical representation of the function f(t) = 6/t further enhances our understanding. The graph is a hyperbola, with two distinct branches. One branch lies in the first quadrant (where both t and f(t) are positive), and the other lies in the third quadrant (where both t and f(t) are negative). The function is undefined at t = 0, as division by zero is not permissible.

The graph visually reinforces the inverse relationship. As we move along the positive t-axis, the curve approaches the t-axis but never touches it, indicating that f(t) gets closer to zero but never actually reaches it. Similarly, as we move along the negative t-axis, the curve approaches the t-axis from the negative side.

The steepness of the curve also provides valuable insights. The curve is steeper for smaller values of t, indicating a more rapid change in f(t). As t increases, the curve becomes less steep, suggesting a slower rate of change. This visual understanding will be crucial when we delve into the calculation of net change.

Defining Net Change

Before we proceed with the calculations, it's crucial to establish a clear understanding of what net change represents. In mathematical terms, the net change of a function f(t) between two points, t = a and t = b, is simply the difference in the function's values at those points. Mathematically, it is expressed as:

Net Change = f(b) - f(a)

In essence, net change quantifies the overall change in the function's output as the input varies from a to b. It's a fundamental concept in calculus and is closely related to the idea of average rate of change.

Interpreting Net Change

The net change can be positive, negative, or zero, each conveying a specific meaning:

  • Positive Net Change: Indicates that the function's value has increased between t = a and t = b. In other words, f(b) > f(a).
  • Negative Net Change: Indicates that the function's value has decreased between t = a and t = b. This means f(b) < f(a).
  • Zero Net Change: Indicates that the function's value has remained the same between t = a and t = b. Therefore, f(b) = f(a).

Net Change in Context

The concept of net change finds applications in various real-world scenarios. For instance, in physics, it can represent the change in position of an object over a certain time interval. In economics, it can represent the change in profit or revenue over a period. In biology, it can represent the change in population size.

Understanding the sign and magnitude of the net change provides valuable information about the behavior of the function and the system it represents. It allows us to quantify and interpret changes, making it a crucial tool in mathematical analysis and problem-solving.

Calculating Net Change for f(t) = 6/t

Now that we have a solid understanding of the function f(t) = 6/t and the concept of net change, we can proceed with calculating the net change between the given values of the variable, t = a and t = a + h.

Recall the formula for net change:

Net Change = f(b) - f(a)

In our case, a is simply a, and b is a + h. Substituting these values into the formula, we get:

Net Change = f(a + h) - f(a)

Now, we need to evaluate f(a + h) and f(a) using the function f(t) = 6/t:

  • f(a + h) = 6 / (a + h)
  • f(a) = 6 / a

Substituting these expressions back into the net change formula, we obtain:

Net Change = (6 / (a + h)) - (6 / a)

To simplify this expression, we need to find a common denominator, which is a(a + h). Multiplying the first term by a/a and the second term by (a + h)/(a + h), we get:

Net Change = (6a / a(a + h)) - (6(a + h) / a(a + h))

Now we can combine the fractions:

Net Change = (6a - 6(a + h)) / a(a + h)

Distribute the -6 in the numerator:

Net Change = (6a - 6a - 6h) / a(a + h)

Simplify the numerator:

Net Change = -6h / a(a + h)

This is the simplified expression for the net change of the function f(t) = 6/t between t = a and t = a + h.

Interpreting the Result

The result, Net Change = -6h / a(a + h), provides valuable insights into the behavior of the function. Let's analyze this expression:

  • The Negative Sign: The negative sign indicates that the net change is negative, meaning that the function's value decreases as t increases from a to a + h. This is consistent with the inverse relationship we observed earlier.
  • The Magnitude: The magnitude of the net change depends on the values of a and h. The larger the value of h (the change in t), the larger the magnitude of the net change. This means that a larger change in t results in a more significant change in f(t). The presence of a in the denominator indicates that the net change is also influenced by the initial value of t. For smaller values of a, the net change is larger in magnitude, while for larger values of a, the net change is smaller.

Practical Examples

To further solidify our understanding, let's consider a few practical examples:

  • Example 1: Let a = 1 and h = 1. Then, the net change is:

    Net Change = -6(1) / 1(1 + 1) = -6 / 2 = -3

    This means that as t increases from 1 to 2, f(t) decreases by 3 units.

  • Example 2: Let a = 2 and h = 1. Then, the net change is:

    Net Change = -6(1) / 2(2 + 1) = -6 / 6 = -1

    This means that as t increases from 2 to 3, f(t) decreases by 1 unit.

These examples illustrate how the net change depends on both the initial value of t (a) and the change in t (h). They also reinforce the negative net change, indicating a decrease in f(t) as t increases.

Conclusion

In this comprehensive exploration, we have delved into the concept of net change, specifically focusing on the function f(t) = 6/t. We have learned how to calculate the net change between two given values of the variable, t = a and t = a + h. Through detailed analysis and practical examples, we have gained a deeper understanding of the function's behavior and the meaning of net change.

The key takeaways from this discussion are:

  • The function f(t) = 6/t represents an inverse relationship.
  • Net change quantifies the overall change in a function's output as the input varies.
  • The net change of f(t) = 6/t between t = a and t = a + h is given by -6h / a(a + h).
  • The negative sign indicates a decrease in f(t) as t increases.
  • The magnitude of the net change depends on the values of a and h.

By mastering the concept of net change and its application to various functions, you will be well-equipped to tackle more complex mathematical problems and analyze real-world phenomena. This understanding forms a crucial foundation for further exploration in calculus and related fields.

This exploration serves as a stepping stone towards a more profound understanding of mathematical concepts and their applications. By continuing to delve into these topics, you will expand your problem-solving abilities and gain a deeper appreciation for the beauty and power of mathematics.