In Exercises 67–72, you will explore some functions and their inverses together with their derivatives and tangent line approximations at specified points. Perform the following steps using your CAS:
c. Find the equation for the tangent line to f at the specified point (x_0, f(x_0)).
68. y= (3x+2)/(2x-11), -2 ≤ x ≤ 2, x_0=1/2

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Identify the function given: \(y = \frac{3x + 2}{2x - 11}\) and the point \(x_0 = \frac{1}{2}\) where we want to find the tangent line.

Calculate the value of the function at \(x_0\): find \(f\left(\frac{1}{2}\right) = \frac{3\left(\frac{1}{2}\right) + 2}{2\left(\frac{1}{2}\right) - 11}\) to get the point of tangency \((x_0, f(x_0))\).

Find the derivative \(f'(x)\) using the quotient rule: if \(f(x) = \frac{u(x)}{v(x)}\), then \(f'(x) = \frac{u'(x)v(x) - u(x)v'(x)}{(v(x))^2}\). Here, \(u(x) = 3x + 2\) and \(v(x) = 2x - 11\).

Evaluate the derivative at \(x_0\): compute \(f'\left(\frac{1}{2}\right)\) to find the slope of the tangent line at the point.

Use the point-slope form of the line equation: \(y - f(x_0) = f'(x_0)(x - x_0)\) to write the equation of the tangent line at \(x_0 = \frac{1}{2}\).

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Derivative and its Geometric Interpretation

The derivative of a function at a point measures the instantaneous rate of change or slope of the tangent line to the function's graph at that point. It is found by differentiating the function and evaluating at the given x-value, providing the slope needed for the tangent line equation.

Equation of a Tangent Line

The tangent line to a function at a point (x₀, f(x₀)) can be expressed using the point-slope form: y - f(x₀) = f'(x₀)(x - x₀). This line touches the curve at exactly one point and has the same slope as the function at that point.

Rational Functions and Their Differentiation

A rational function is a ratio of two polynomials. Differentiating such functions requires the quotient rule, which states that the derivative of f(x) = g(x)/h(x) is (g'(x)h(x) - g(x)h'(x)) / [h(x)]². This rule is essential for finding the slope of the tangent line to rational functions.