Find the tangent line to the Witch of Agnesi (graphed here) at the point (2,1).

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Identify the function of the Witch of Agnesi, which is given as \( y = \frac{8}{x^2 + 4} \).

To find the tangent line at the point (2,1), first calculate the derivative of the function \( y = \frac{8}{x^2 + 4} \) to find the slope of the tangent line. Use the quotient rule for differentiation.

The quotient rule states that if you have a function \( \frac{u}{v} \), its derivative is \( \frac{u'v - uv'}{v^2} \). Here, \( u = 8 \) and \( v = x^2 + 4 \). Calculate \( u' \) and \( v' \).

Substitute \( u' = 0 \) and \( v' = 2x \) into the quotient rule formula to find the derivative \( y' = \frac{0 \cdot (x^2 + 4) - 8 \cdot 2x}{(x^2 + 4)^2} \). Simplify this expression to find \( y' \).

Evaluate the derivative \( y' \) at \( x = 2 \) to find the slope of the tangent line at the point (2,1). Use the point-slope form of a line, \( y - y_1 = m(x - x_1) \), where \( m \) is the slope and \( (x_1, y_1) = (2, 1) \), 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.

Tangent Line

A tangent line to a curve at a given point is a straight line that touches the curve at that point without crossing it. The slope of the tangent line represents the instantaneous rate of change of the function at that point, which can be found using the derivative of the function.

Derivative

The derivative of a function measures how the function's output value changes as its input value changes. It is defined as the limit of the average rate of change of the function over an interval as the interval approaches zero. For a function f(x), the derivative f'(x) gives the slope of the tangent line at any point x.

Witch of Agnesi

The Witch of Agnesi is a specific type of curve defined by the equation y = 8/(x² + 4). It is a smooth, bell-shaped curve that approaches the x-axis asymptotically. Understanding its properties, including its maximum point and symmetry, is essential for analyzing its behavior and finding tangent lines.