Find The Slope Of The Line Tangent To The Graph Of Y = 10x / (x - 3) At x = -2.
Understanding how to determine the slope of a tangent line to a curve at a specific point is a fundamental concept in calculus. The slope of the tangent line provides insight into the instantaneous rate of change of the function at that point. In this article, we will explore the process of finding this slope for the function \( y = \frac{10x}{x - 3} \) at \( x = -2 \). We will walk through the necessary steps, including differentiation techniques, simplifying expressions, and calculating the derivative's value at the given point.
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Understanding the Function and Its Significance
Overview of the Function \( y = \frac{10x}{x - 3} \)
The function \( y = \frac{10x}{x - 3} \) is a rational function, which means it is expressed as the ratio of two polynomials. Its behavior is characterized by potential asymptotes, discontinuities, and specific points where the slope can be evaluated. The function is undefined at \( x = 3 \), which causes a vertical asymptote, but our point of interest \( x = -2 \) is well within the domain of the function.
Why Find the Slope at a Specific Point?
The slope of the tangent line at a point \( x = a \) is the derivative \( y' \) evaluated at \( x = a \). This slope indicates how rapidly the function is increasing or decreasing at that precise point. For example, if the derivative is positive, the function is increasing; if negative, decreasing. Understanding this slope is crucial in many applications, from physics to economics, where the rate of change impacts decision-making and analysis.
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Step 1: Differentiating the Function \( y = \frac{10x}{x - 3} \)
Choosing a Differentiation Method
Since the function is a ratio of two functions, the quotient rule is the most appropriate method for differentiation. The quotient rule states:
\[
\frac{d}{dx}\left[\frac{u(x)}{v(x)}\right] = \frac{v(x) \cdot u'(x) - u(x) \cdot v'(x)}{[v(x)]^2}
\]
where \( u(x) \) is the numerator and \( v(x) \) is the denominator.
Identifying \( u(x) \) and \( v(x) \)
- \( u(x) = 10x \)
- \( v(x) = x - 3 \)
Calculating the Derivatives of \( u(x) \) and \( v(x) \)
- \( u'(x) = 10 \)
- \( v'(x) = 1 \)
Applying the Quotient Rule
Using the quotient rule:
\[
y' = \frac{(x - 3) \times 10 - 10x \times 1}{(x - 3)^2}
\]
Simplify numerator:
\[
y' = \frac{10(x - 3) - 10x}{(x - 3)^2}
\]
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Step 2: Simplifying the Derivative Expression
Expanding and Combining Like Terms
Calculate the numerator:
\[
10x - 30 - 10x
\]
Notice that \( 10x \) and \( -10x \) cancel out:
\[
(10x - 10x) - 30 = 0 - 30 = -30
\]
Therefore, the derivative simplifies to:
\[
y' = \frac{-30}{(x - 3)^2}
\]
This is a key result because it allows us to evaluate the slope directly at the given \( x \)-value.
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Step 3: Evaluating the Derivative at \( x = -2 \)
Substituting \( x = -2 \) into the Derivative
Plug in \( x = -2 \):
\[
y'(-2) = \frac{-30}{(-2 - 3)^2}
\]
Calculate the denominator:
\[
-2 - 3 = -5
\]
Square it:
\[
(-5)^2 = 25
\]
Now, compute the slope:
\[
y'(-2) = \frac{-30}{25} = -\frac{6}{5}
\]
Result:
The slope of the tangent line to the graph of \( y = \frac{10x}{x - 3} \) at \( x = -2 \) is \(-\frac{6}{5}\).
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Interpreting the Result
Significance of the Slope
The negative value indicates that at \( x = -2 \), the function is decreasing—meaning the graph is slanting downward as it passes through that point. The magnitude \( \frac{6}{5} \) shows the rate of change; for each unit increase in \( x \), \( y \) decreases approximately 1.2 units.
Implications in Applied Contexts
- In physics, if \( y \) represented position and \( x \) time, the slope would represent velocity. A negative slope indicates motion in the reverse direction.
- In economics, it could represent decreasing returns or diminishing marginal utility at that point.
- In calculus, this process exemplifies how derivatives enable the analysis of the behavior of functions at specific points.
Summary and Key Takeaways
- Identifying the correct differentiation rule (quotient rule) is crucial when dealing with rational functions.
- Simplifying the derivative expression makes evaluation straightforward.
- Evaluating the derivative at a specific point involves substituting the \( x \)-value directly into the simplified derivative expression.
- The resulting slope provides immediate insights into the function's behavior at that point.
Final Answer:
The slope of the tangent line to the graph of \( y = \frac{10x}{x - 3} \) at \( x = -2 \) is \(-\frac{6}{5}\).
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Additional Considerations
Verifying the Point on the Graph
While the primary goal was to find the slope, confirming the exact point on the graph can deepen understanding. To do this, substitute \( x = -2 \) into the original function:
\[
y = \frac{10 \times (-2)}{-2 - 3} = \frac{-20}{-5} = 4
\]
So, the point of tangency is \( (-2, 4) \).
Graphical Interpretation
Plotting the function and the tangent line at \( (-2, 4) \) with the calculated slope will visually demonstrate the tangent's orientation. The tangent line equation can be written as:
\[
y - 4 = -\frac{6}{5}(x + 2)
\]
which can be used for further visualization or analysis.
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Conclusion
Calculating the slope of the tangent line to a rational function involves understanding the quotient rule, simplifying derivatives, and carefully evaluating at the specific point. For the function \( y = \frac{10x}{x - 3} \), the derivative simplifies elegantly to \( y' = \frac{-30}{(x - 3)^2} \). Evaluating this at \( x = -2 \) yields a slope of \( -\frac{6}{5} \), indicating a decreasing function at that point with a moderate rate of change. Mastery of these techniques is essential in calculus, providing invaluable tools for analyzing the behavior of complex functions across various disciplines.
