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)).
67. y= √(3x-2), 2/3 ≤ x ≤ 4, x_0=3

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Identify the function given: \(y = \sqrt{3x - 2}\), and the point at which to find the tangent line is \(x_0 = 3\).

Calculate the value of the function at \(x_0\): find \(f(3) = \sqrt{3(3) - 2} = \sqrt{9 - 2}\) to get the point \((3, f(3))\) on the curve.

Find the derivative of the function \(f(x)\) to get the slope of the tangent line. Use the chain rule: \(f(x) = (3x - 2)^{1/2}\), so \(f'(x) = \frac{1}{2}(3x - 2)^{-1/2} \times 3\).

Evaluate the derivative at \(x_0 = 3\) to find the slope of the tangent line: \(m = f'(3)\).

Use the point-slope form of the line equation with the point \((3, f(3))\) and slope \(m\): \(y - f(3) = m(x - 3)\) to write the equation of the tangent line.

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주요 개념

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

Inverse Functions

An inverse function reverses the effect of the original function, swapping inputs and outputs. Understanding inverses helps in analyzing how functions behave and is essential when exploring derivatives of inverse functions or their tangent lines.

Derivative and Differentiation

The derivative measures the rate of change or slope of a function at a given point. Calculating the derivative of f(x) at x_0 provides the slope of the tangent line, which is crucial for finding the tangent line equation.

Equation of a Tangent Line

The tangent line to a function at a point touches the curve without crossing it locally. Its equation is found using the point-slope form: y - f(x_0) = f'(x_0)(x - x_0), where f'(x_0) is the slope from the derivative.