Tangent Line Slope Calculation For F(x) = 3x² - 7 At X = 2

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In the realm of calculus, understanding the concept of a tangent line is paramount. A tangent line, in simple terms, is a straight line that touches a curve at a single point, mirroring the curve's direction at that precise location. This exploration delves into the function f(x) = 3x² - 7, aiming to dissect and comprehend the tangent line's slope at a specific point. We will embark on a journey through the function's values, unravel the intricacies of the difference quotient, and ultimately, illuminate the path to finding the tangent line's slope.

Understanding the Function's Values

To begin our exploration, let's first grasp the function's behavior at specific points. We are given two crucial pieces of information:

  • f(2 + Δx) = 5 + 12Δx + 3(Δx)²
  • f(2) = 5

These values provide a glimpse into how the function changes around the point x = 2. The expression f(2 + Δx) represents the function's value at a point slightly shifted from x = 2 by an amount Δx. The term Δx, often referred to as the increment or change in x, plays a pivotal role in understanding the function's local behavior. By evaluating f(2 + Δx), we gain insight into how the function responds to small perturbations around x = 2. This is crucial for determining the slope of the tangent line, which captures the instantaneous rate of change of the function at that point.

The value f(2) = 5 simply tells us the function's output when x is 2. This serves as our reference point, the anchor around which we explore the function's behavior. Together, f(2 + Δx) and f(2) form the foundation for calculating the difference quotient, a cornerstone concept in calculus that helps us bridge the gap between average rates of change and instantaneous rates of change. The difference quotient, in essence, measures the average rate of change of the function over a small interval, and as we shrink this interval, it converges to the instantaneous rate of change, which is precisely what the slope of the tangent line represents.

Unveiling the Slope of the Tangent Line

The slope of the tangent line at a point embodies the function's instantaneous rate of change at that specific location. To find this slope, we employ the concept of the difference quotient. The difference quotient is defined as:

(f(x + Δx) - f(x)) / Δx

This expression calculates the average rate of change of the function over the interval [x, x + Δx]. Geometrically, it represents the slope of the secant line passing through the points (x, f(x)) and (x + Δx, f(x + Δx)) on the function's graph. As Δx approaches zero, this secant line pivots around the point (x, f(x)) and progressively morphs into the tangent line at that point. Therefore, the limit of the difference quotient as Δx approaches zero gives us the slope of the tangent line.

In our specific case, we want to find the slope of the tangent line at x = 2. Thus, we need to evaluate the following limit:

lim (Δx→0) [f(2 + Δx) - f(2)] / Δx

We already have the values for f(2 + Δx) and f(2). Substituting these values into the expression, we get:

lim (Δx→0) [(5 + 12Δx + 3(Δx)²) - 5] / Δx

Simplifying the numerator, the 5s cancel out, leaving us with:

lim (Δx→0) [12Δx + 3(Δx)²] / Δx

Now, we can factor out Δx from the numerator:

lim (Δx→0) Δx(12 + 3Δx) / Δx

Canceling out Δx from the numerator and denominator (since Δx is approaching 0 but not equal to 0), we obtain:

lim (Δx→0) (12 + 3Δx)

Finally, as Δx approaches 0, the expression 3Δx also approaches 0, leaving us with:

12

Therefore, the slope of the tangent line to the graph of f(x) = 3x² - 7 at x = 2 is 12. This means that at the point (2, 5) on the graph, the function is changing at an instantaneous rate of 12 units for every 1 unit change in x. The tangent line at this point is a straight line with a slope of 12, closely approximating the function's behavior in the immediate vicinity of (2, 5).

Delving Deeper: Connecting Concepts

The process we undertook to find the tangent line's slope is a fundamental application of differential calculus. The limit of the difference quotient is, in essence, the derivative of the function at a point. The derivative, denoted as f'(x), provides a powerful tool for analyzing a function's behavior. It not only gives us the slope of the tangent line at any point but also reveals valuable information about the function's increasing and decreasing intervals, local maxima and minima, and concavity.

In our example, we found the derivative of f(x) = 3x² - 7 at x = 2. The general derivative of this function can be found using the power rule of differentiation, which states that if f(x) = axⁿ, then f'(x) = nax^(n-1). Applying this rule to our function:

  • The derivative of 3x² is 2 * 3x^(2-1) = 6x
  • The derivative of -7 (a constant) is 0

Therefore, f'(x) = 6x. Evaluating f'(x) at x = 2, we get f'(2) = 6 * 2 = 12, which confirms our earlier calculation using the limit definition of the derivative.

Practical Applications and Significance

The concept of the tangent line and its slope has far-reaching applications across various fields, including:

  • Physics: Determining the instantaneous velocity of an object. Velocity is the rate of change of position with respect to time, which can be represented as the slope of the tangent line on a position-time graph.
  • Engineering: Optimizing designs and processes. Engineers use derivatives to find maximum and minimum values, crucial for tasks like maximizing the strength of a structure or minimizing the cost of production.
  • Economics: Analyzing marginal cost and revenue. Marginal cost, for example, represents the change in cost resulting from a one-unit change in production, which can be approximated by the slope of the tangent line on a cost function graph.
  • Computer Graphics: Creating smooth curves and animations. Tangent lines are used to ensure that curves connect smoothly, avoiding abrupt changes in direction.

In essence, the tangent line provides a powerful tool for understanding and analyzing the behavior of functions. Its slope, representing the instantaneous rate of change, captures the function's essence at a specific point. By mastering the concept of the tangent line, we unlock a deeper understanding of calculus and its profound impact on various aspects of our world.

Conclusion

Our journey into the world of tangent lines has illuminated the process of finding the slope of the tangent line to the graph of f(x) = 3x² - 7 at x = 2. By carefully evaluating the function's values, employing the difference quotient, and taking the limit as Δx approaches zero, we successfully determined the slope to be 12. This exploration has not only provided a concrete example but also highlighted the broader significance of tangent lines and derivatives in calculus and their applications in diverse fields. The tangent line, a seemingly simple concept, serves as a gateway to a deeper understanding of change and rates of change, empowering us to analyze and model the world around us with greater precision and insight.

By understanding the concept of a tangent line, especially how to compute its slope, unlocks doors to various real-world applications, bridging the gap between theoretical mathematics and practical problem-solving. This journey into the tangent line is a fundamental step towards mastering the power of calculus and its ability to illuminate the intricacies of change and motion.