{Use of Tech} The Witch of Agnesi The graph of y = a³ / (x² + a²) , where a is a constant, is called the witch of Agnesi (named after the 18th-century Italian mathematician Maria Agnesi).
Let a = 3 and find an equation of the line tangent to y = 27 / (x² + 9) at x = 2.

Verified step by step guidance

1. Step 1: Substitute a = 3 into the given function to get y = \frac{27}{x^2 + 9}.
2. Step 2: Find the derivative of y with respect to x, which will give us 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 = 27 and v = x^2 + 9.
3. Step 3: Calculate the derivative: u' = 0 (since 27 is a constant) and v' = 2x. Substitute these into the quotient rule formula to find y'.
4. Step 4: Evaluate the derivative at x = 2 to find the slope of the tangent line at this point.
5. Step 5: Use the point-slope form of a line, y - y_1 = m(x - x_1), where m is the slope found in Step 4 and (x_1, y_1) is the point on the curve at x = 2, to write the equation of the tangent line.

Key Concepts

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 is determined by the derivative of the function at that point. To find the equation of the tangent line, one needs the point of tangency and the slope, which can be calculated using the derivative.

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. In practical terms, the derivative provides the slope of the tangent line to the graph of the function at any given point.

Function Evaluation

Function evaluation involves substituting a specific value into a function to determine its output. In this context, evaluating the function y = 27 / (x² + 9) at x = 2 is necessary to find the y-coordinate of the point where the tangent line touches the curve. This step is crucial for establishing the point of tangency needed to write the equation of the tangent line.