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)).

70. y= x³/(x²+1), -1 ≤ x ≤ 1, x_0=1/2

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First, identify the function given: \(f(x) = \frac{x^3}{x^2 + 1}\) and the point at which we want the tangent line: \(x_0 = \frac{1}{2}\).

Calculate the value of the function at \(x_0\) to find the point on the curve: compute \(f\left(\frac{1}{2}\right) = \frac{\left(\frac{1}{2}\right)^3}{\left(\frac{1}{2}\right)^2 + 1}\).

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, \(u(x) = x^3\) and \(v(x) = x^2 + 1\).

Compute \(u'(x) = 3x^2\) and \(v'(x) = 2x\), then substitute into the quotient rule formula to get \(f'(x)\).

Evaluate the derivative at \(x_0 = \frac{1}{2}\) to find the slope of the tangent line: \(m = f'\left(\frac{1}{2}\right)\). Then use the point-slope form of a line: \(y - f\left(\frac{1}{2}\right) = m \left(x - \frac{1}{2}\right)\) 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.

Derivative and Tangent Line

The derivative of a function at a point gives the slope of the tangent line to the curve at that point. To find the tangent line equation, compute the derivative at x₀ to get the slope, then use the point-slope form y - f(x₀) = f'(x₀)(x - x₀). This line approximates the function near x₀.

Quotient Rule for Derivatives

When differentiating a function expressed as a quotient of two functions, use the quotient rule: (u/v)' = (u'v - uv') / v². Here, u = x³ and v = x² + 1, so find u' and v' separately, then apply the rule to find f'(x).

Evaluating Functions and Derivatives at a Point

After finding the derivative function, substitute the given x₀ value to find the slope at that point. Also, evaluate the original function at x₀ to get the point coordinates. These values are essential to write the tangent line equation accurately.