Problem 72 - Calculus an Applied Approach

Question

Find an equation of the tangent line to the graph of the function at the given point. Then use a graphing utility to graph the function and the tangent line in the same viewing window. $$\begin{array}{ll}{\text { Function }} & {\text { Point }} \\\y=\frac{x}{\sqrt{25+x^{2}}} & {(0,0)}\end{array}$$

Step-by-Step Solution

Answer

The equation of the tangent line to the function \(y = \frac{x}{\sqrt{25+x^{2}}}\) at the point \((0,0)\) is \(y = 0\). Plotting the graph confirms this.

Step 1: Find Derivative of the Function

First, find the derivative \(y'\) of the function \(y = \frac{x}{\sqrt{25+x^{2}}}\). You can do this by applying the quotient rule which states that if we have a function in the form \(f(x) = \frac{g(x)}{h(x)}\), then the derivative of this function is given by \(f'(x) = \frac{g'(x)h(x) - g(x)h'(x)}{[h(x)]^2}\). Apply this to our current function to compute its derivative.

Step 2: Evaluate the Derived Function at the Given Point

Once you have found the derivative, the next step is to substitute the x-coordinate (which is 0 in this case) of the given point into the derivative to find the slope of the tangent line. The slope of the tangent line is therefore given by \(y'\) evaluated at \(x = 0\).

Step 3: Find the Equation of the Tangent Line

Having found the slope of the tangent line, we can now find its equation by using the point-slope form of a line which is \(y - y_1 = m(x - x_1)\), where \((x_1, y_1)\) is the given point on the line and \(m\) is the slope of the line. Substitute the given point \((0,0)\) and the slope into this formula to get the equation of the tangent line.

Step 4: Graph the Function and the Tangent Line

Finally, use a graphing utility to plot both the original function and its tangent at the given point in the same window. This is to verify the work already done.

Key Concepts

Derivative

In calculus, the derivative is a fundamental concept that measures how a function changes as its input changes. It tells us the rate at which a function is changing at any given point. Consider our original function \[y = \frac{x}{\sqrt{25+x^2}}\]The derivative of this function, denoted as \(y'\), helps us to understand how \(y\) changes with respect to changes in \(x\) around any point. Derivatives can be computed using a variety of rules, one of which is the Quotient Rule.

• Providing insight into the behavior of functions: As the slope, it indicates if and how fast a function is increasing or decreasing.
• Computational foundation: Essential for finding tangent lines, which provide linear approximations of functions around a point.

The process of differentiation provides the slope necessary for determining the equation of the line that best touches or approximates the function's curve at a specific point.

Tangent Line

The tangent line to a graph of a function at a point is a straight line that just "touches" the curve at that point. It provides the best linear approximation of the function near that point. When dealing with the function \[y=\frac{x}{\sqrt{25+x^2}}\] and a point of interest, say (0, 0), the tangent line captures the behavior of the function closely around (0, 0).

• Slope: The derivative at a given point gives us the slope of the tangent line.
• Equation of Tangent Line: By using the point-slope form \(y - y_1 = m(x - x_1)\), where \(m\) is the slope and \((x_1, y_1)\) is a point on the line.

To solve for the tangent line, calculate the derivative and evaluate it at \(x = 0\). Use this slope in the point-slope formula with point \((0,0)\) to obtain the tangent line's equation. This line visually approximates how the function behaves just around that specific point.

Quotient Rule

The Quotient Rule is a technique in calculus for finding the derivative of a quotient of two functions. If you have a function \[f(x) = \frac{g(x)}{h(x)}\] the quotient rule states:\[f'(x) = \frac{g'(x)h(x) - g(x)h'(x)}{[h(x)]^2}\]When applying it to our given functionyou identify:

• \(g(x) = x\) and \(h(x) = \sqrt{25 + x^2}\).
• Differentiate each part to find \(g'(x) = 1\) and \(h'(x) = \frac{x}{\sqrt{25+x^2}}\), using chain rule for \(h(x)\).

Plug these into the quotient rule formula to find the derivative. This calculated derivative will be used to find the slope of the tangent line at (0,0). The quotient rule is vital to dealing with functions expressed as a ratio, especially when differentiating products could be challenging otherwise.

Graphing Utility

Graphing utilities, like graphing calculators or software, are essential tools in calculus for visualizing functions and their properties. These utilities allow students to explore the behavior of both the function and its tangent line in a visual format. For our function and its tangent line, these utilities help:

• Visual Confirmation: Plotting both the function \(y = \frac{x}{\sqrt{25+x^2}}\) and the derived tangent line in the same frame to verify points of tangency.
• Exploration: Students can adjust the view window to observe the function's behavior across different intervals of \(x\) and see how the tangent line aligns with the curve at (0,0).

Using a graphing utility not only helps validate analytical solutions but also enhances understanding by allowing learners to visualize complex calculus concepts dynamically.