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Chapter 3: Problem 73

The graph of \(y=\frac{a^{3}}{x^{2}+a^{2}},\) where \(a\) is a constant, is called the witch of Agnesi (named after the 18th-century Italian mathematician Maria Agnesi). a. Let \(a=3\) and find an equation of the line tangent to \(y=\frac{27}{x^{2}+9}\) at \(x=2\) b. Plot the function and the tangent line found in part (a).

Short Answer

Expert verified
Question: Find the equation of the tangent line to the function \(y = \frac{27}{x^2 + 9}\) at \(x = 2\), and plot the function and the tangent line. Answer: The equation of the tangent line at \(x = 2\) is \(y = -\frac{108}{169}x + \frac{567}{169}\). To plot the function and the tangent line, use a graphing tool like Desmos, Geogebra, or any other graphing calculator.

Step by step solution

01

Abalysis and Prequisites

Before we begin, we need to understand a few things. The equation of a line in the slope-intercept form is given by \(y = mx + c\), where m is the slope and c is the y-intercept. To find the tangent of a function at a given point, we need the slope (m) and the coordinates (x, y) at that point.
02

Find the value of y when x equals 2

Substitute x=2 in the function \(y=\frac{27}{x^2+9}\): \(y = \frac{27}{(2)^2 + 9} = \frac{27}{13}\) So, the coordinates of the point are \((2, \frac{27}{13})\).
03

Find the derivative

To find the slope of the tangent, we need to derive function with respect to x. \(y = \frac{27}{x^2+9}\) \(y'= \frac{-54x}{(x^2+9)^2}\)
04

Find the slope at x equals 2

Substitute x=2 in the derivative: \(y' = \frac{-54(2)}{((2)^2+9)^2} = -\frac{108}{169}\) So, the slope of the tangent at the point (2, \(\frac{27}{13}\)) is \(-\frac{108}{169}\).
05

Find the equation of the tangent line

Now that we have the point (2, \(\frac{27}{13}\)) and the slope, \(-\frac{108}{169}\), we can write the equation of the tangent line in point-slope form: \(y - \frac{27}{13} = -\frac{108}{169}(x - 2)\)
06

Put the equation in slope-intercept form

If we multiply both sides by \(169\) and simplify, we have: \(169(y - \frac{27}{13}) = -108(x - 2)\) \(169y - 351 = -108x + 216\) \(169y = -108x + 216 + 351\) So, the equation of the tangent in slope-intercept form is: \(y = -\frac{108}{169}x + \frac{567}{169}\)
07

Plot the function and the tangent line

Now we can plot the witch of Agnesi function (\(y = \frac{27}{x^2 + 9}\)) and the tangent line found in part (a) (\(y = -\frac{108}{169}x + \frac{567}{169}\)) using a tool of your choosing like Desmos, Geogebra or any other graphing calculator.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Tangent Line
The tangent line is a straight line that just "touches" a curve at a single point. This point is called the point of tangency. The tangent line represents the instantaneous rate of change of the function at a particular point. It can be thought of as the direction in which the curve is "going" at that specific spot.
  • To find this line, we need the slope of the function at the point of tangency.
  • This requires us to calculate the derivative of the function (more on this next!), which tells us how steep the curve is at any given point.
  • The equation of the tangent line uses this slope and the coordinates of the tangency point.
When calculating the tangent line equation, we can use the point-slope form, which is: \(y - y_1 = m(x - x_1)\), where \(m\) is the slope at \((x_1, y_1)\). Simplifying this equation helps us better analyze how the tangent line behaves when plotted with the curve.
Derivative
In calculus, the derivative of a function helps us find how the function changes. Derivatives are central to understanding how a curve behaves, as they provide the slope of the function at any given point. This is exactly what we need when finding the tangent line.
  • The derivative can be understood as the rate of change of the function with respect to a variable, often time or space.
  • By computing the derivative, you can determine where the function is increasing or decreasing, and how steeply.
  • It guides us in plotting the tangent line by providing the exact slope needed.
For the function \(y = \frac{27}{x^2+9}\), the derivative \(y'\) was calculated as \(\frac{-54x}{(x^2+9)^2}\). By substituting \(x = 2\), we found the slope of the tangent line to be \(-\frac{108}{169}\). This slope was crucial for determining the final equation of the tangent line.
Function Graph
Graphing a function is a visual way to understand how a function behaves over a range of inputs. The function we are working with here, known as the "witch of Agnesi," has a distinctive shape that can be better understood through graphing.
  • The function graph shows you where the function increases, decreases, and where it might level out.
  • It visually illustrates the point of tangency, where the tangent line just "kisses" the curve.
  • Having both the function and its tangent line on the same graph can help you see how accurately the tangent line reflects the 'slope' of the curve at a particular point.
Using tools like Desmos or GeoGebra, one can observe how the calculations come to life. The graph of \(y = \frac{27}{x^2+9}\), paired with the tangent line \(y = -\frac{108}{169}x + \frac{567}{169}\), clearly shows the relationship between the tangent and the curve, enhancing comprehension of both the curve's behavior and the significance of the tangent line.

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