Use a graphing utility to graph the curve and the tangent line on the same set of axes.
y = (x + 5) / (x - 1); a = 3

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Step 1: Identify the function and the point of tangency. The function given is \( y = \frac{x + 5}{x - 1} \) and the point of tangency is at \( x = 3 \).

Step 2: Find the derivative of the function to determine the slope of the tangent line. Use the quotient rule: if \( y = \frac{u}{v} \), then \( y' = \frac{u'v - uv'}{v^2} \). Here, \( u = x + 5 \) and \( v = x - 1 \).

Step 3: Calculate the derivative \( y' \) at \( x = 3 \) to find the slope of the tangent line. Substitute \( x = 3 \) into the derivative expression obtained in Step 2.

Step 4: Determine the y-coordinate of the point of tangency by substituting \( x = 3 \) into the original function \( y = \frac{x + 5}{x - 1} \).

Step 5: Use the point-slope form of the equation of a line, \( y - y_1 = m(x - x_1) \), where \( m \) is the slope from Step 3 and \( (x_1, y_1) \) is the point from Step 4, to write the equation of the tangent line.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Graphing Rational Functions

A rational function is a ratio of two polynomials. To graph such functions, it's essential to identify key features like intercepts, asymptotes, and the overall shape of the curve. For the function y = (x + 5) / (x - 1), understanding its behavior near the vertical asymptote at x = 1 and the horizontal asymptote as x approaches infinity is crucial for accurate graphing.

Tangent Lines

A tangent line to a curve at a given point represents the instantaneous rate of change of the function at that point. To find the equation of the tangent line at a specific point, you need to calculate the derivative of the function and evaluate it at that point. For the function given, evaluating the derivative at x = 3 will provide the slope needed to write the tangent line's equation.

Using Graphing Utilities

Graphing utilities, such as graphing calculators or software, allow for the visualization of functions and their properties. These tools can plot both the curve of the function and the tangent line simultaneously, making it easier to analyze their relationship. Familiarity with the utility's features, such as inputting functions and adjusting viewing windows, is essential for effective graphing.

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Textbook Question

Derivatives of inverse functions from a table Use the following tables to determine the indicated derivatives or state that the derivative cannot be determined. <IMAGE>

b. (f^-1)'(6)

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Textbook Question

Witch of Agnesi Let y(x²+4)=8 (see figure). <IMAGE>

b. Find equations of all lines tangent to the curve y(x²+4)=8 when y=1.

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Textbook Question

79–82. {Use of Tech} Visualizing tangent and normal lines

b. Graph the tangent and normal lines on the given graph.

(x²+y²)² = 25/3 (x²-y²); (x0,y0) = (2,-1) (lemniscate of Bernoulli)