Question

Famous Curves In Exercises \(69-72\) , find an equation of the tangent line to the graph at the given point. (The graphs in Exercises 69 and 70 are called Witches of Agnesi. The graphs in Exercises 71 and 72 are called serpentines.) Graph cannot copy $$ f(x)=\frac{4 x}{x^{2}+6} $$

Short answer

The process to find a tangent line to the function involves finding its derivative to get the slope and then using point-slope form. The generic slope of the tangent line to the function at any point x is given by \(f'(x) = \frac{-4x^2 + 24}{(x^2+6)^2}\). The equation of the tangent line, in general, will be dependent on the x-value.

Step-by-step solution

Find the derivative of the function

The derivative of a function gives us the slope of the tangent line to the function at any point. To find the derivative of the function \(f(x) = \frac{4x}{x^{2}+6}\), use the quotient rule, which states that the derivative of \(\frac{u}{v}\) is \( \frac{vu' - uv'}{v^2}\), where \(u = 4x\), u' = 4 and \(v = x^2 + 6\), v' = 2x. So, \[f'(x) = \frac{(x^2 + 6)4 - 4x*2x}{(x^2+6)^2} = \frac{4x^2 + 24 - 8x^2}{(x^2+6)^2} = \frac{-4x^2 + 24}{(x^2+6)^2}\]

Substitute the given point into the derivative

The slope of the tangent line at a given point is the value of the derivative at that point. However, since no specific point is given for this exercise, a generic form is provided: \( m = f'(x) = \frac{-4x^2 + 24}{(x^2+6)^2}. This is the slope of the tangent line to the function at any point x.

Use the point-slope form to write the equation of the line

The point-slope form of a line is \( y - y_1 = m(x - x_1) \) where \((x_1, y_1)\) is the point on the line and \(m\) is the slope. However, with no specific given point in this task, an equation of the line in general is provided. It should be noted that this is an equation in terms of \(x\), and \(y = f(x)\), so the actual line will be dependent on the x-value.

Key concepts

Derivative

The concept of a derivative is crucial for understanding how functions change. When you find a derivative, you're essentially measuring how steep a graph is at any given point. This steepness is referred to as the "slope". Using derivatives helps us find the slope of tangent lines to curves.

To determine the derivative of a function such as \(f(x) = \frac{4x}{x^{2}+6}\), we can utilize rules like the power rule, product rule, or in this case, the quotient rule, which is used for functions expressed as a division of two other functions.

Quotient Rule

When dealing with functions that are divisions, like \( \frac{u}{v} \), the quotient rule becomes a useful tool. It tells us how to take the derivative of such functions. The general formula for the quotient rule is:

• \( \frac{d}{dx} \left( \frac{u}{v} \right) = \frac{v \cdot u' - u \cdot v'}{v^2} \)

In this exercise, we have \( u = 4x \) and \( v = x^2 + 6 \), leading to their respective derivatives, \( u' = 4 \) and \( v' = 2x \).

Plugging these into the quotient rule, we get:
\[ f'(x) = \frac{(x^2 + 6)4 - 4x \cdot 2x}{(x^2+6)^2} \]
Which simplifies further to \( \frac{-4x^2 + 24}{(x^2+6)^2} \). This result represents the slope of the tangent line at any point \( x \) on the curve.

Slope

In mathematics, finding the slope of a line is fundamental. The slope measures the steepness or incline of the line, indicating how one variable changes in response to another. When we talk about the slope in terms of calculus, we often refer to the slope of tangent lines, which are lines that just "touch" the curve at a point. The derivative evaluated at a point gives us this specific value.

For the function given and using its derivative \( f'(x) = \frac{-4x^2 + 24}{(x^2+6)^2} \), the slope of the tangent line at a point \( x \) tells us how the function behaves at that exact position.

Point-Slope Form

Once you have determined the slope from the derivative, the next step is to find the equation of the tangent line. This is done using the point-slope form of a linear equation:

• \( y - y_1 = m(x - x_1) \)

Here, \( (x_1, y_1) \) is the point at which the tangent line touches the curve, and \( m \) is the slope obtained from the derivative. This form is especially useful because it directly utilizes the slope and a specific point.

In scenarios where no specific point is provided, your tangent line's equation will depend on variables of \( x \). Therefore every line must be considered in terms of relating \( y = f(x) \). This conceptual approach is key to constructing equations of tangent lines accurately.