Question

Find an equation of the tangent line to the function at the given point. Then find the function values and the tangent line values at \(f(x+\Delta x)\) and \(y(x+\Delta x)\) for \(\Delta x=-0.01\) and \(0.01\). \(f(x)=\frac{x}{x^{2}+1}\) \((0,0)\)

Short answer

The equation of the tangent line is \(y=x\). The function values at \(f(-0.01)\) and \(f(0.01)\) are approximately -0.0099 and 0.0099 respectively. The tangent line values at \(y(-0.01)\) and \(y(0.01)\) are -0.01 and 0.01 respectively.

Step-by-step solution

Find the derivative of the function

The derivative \(f'(x)\) of the function \(f(x)=\frac{x}{x^{2}+1}\) can be calculated using the quotient rule which states that \((\frac{u}{v})' = \frac{u'v-uv'}{v^2}\). In this case, u = x and v = \(x^2 + 1\). So \(f'(x)=\frac{(x^2+1)-2x^2}{(x^2+1)^2} = \frac{1-x^2}{(x^2+1)^2}\).

Find the slope at the given point

By substituting the x-coordinate of the point into the derivative, we get the slope of the tangent line at the point. So the slope \(m = f'(0) = \frac{1-0^2}{(0^2+1)^2} = 1.\)

Find the equation of the tangent line

Using the equation of a line \(y - y_1 = m(x - x_1)\) where m is the slope and \((x_1, y_1)\) is the given point, we find that the equation of the tangent line is \(y = x\).

Compute the function values and tangent line values

For \(\Delta x=-0.01\) and \(0.01\), calculate the function values as \(f(x+\Delta x)\) and tangent line values as \(y(x+\Delta x)\). Therefore, \(f(-0.01)= \frac{-0.01}{(-0.01)^{2}+1}\) and \(f(0.01)=\frac{0.01}{0.01^{2}+1}\). Similarly, find the tangent line values at these points as \(y(-0.01) = -0.01\) and \(y(0.01) = 0.01\).

Key concepts

Derivative Calculation

Understanding the derivative of a function is at the heart of calculus. It's essential for students to grasp this concept because it measures how a function's output value changes concerning a change in its input value. In simpler terms, the derivative represents the rate of change of a function at a particular point, which can be thought of as the function's slope at that point.

To calculate the derivative of a function, such as the example given, we often use differentiation rules like the power rule, product rule, or quotient rule. The derivative gives us the power to find the precise slope of a tangent line to a curve at any given point, which is critical for many applications in science and engineering. By mastering derivative calculations, students can solve a wide array of mathematical problems ranging from finding instantaneous rates of change to optimizing functions.

Quotient Rule

When a function is a ratio of two other functions, like in the provided exercise where we have \(f(x)=\frac{x}{x^{2}+1}\), the quotient rule is a handy tool for finding its derivative. The quotient rule states that the derivative of \(\frac{u}{v}\) is \(\frac{u'v-uv'}{v^2}\), with \('\) denoting the derivative of a function.

This rule allows us to differentiate complex rational functions systematically. To apply the quotient rule, we identify the numerator (\(u\)) and the denominator (\(v\)) of our function, then differentiate them separately to get \('u\) and \('v\) before plugging these values into the formula. While this rule may seem daunting at first, with practice, it becomes a fundamental technique that enables the solving of problems involving rates of change in various fields, such as physics, economics, and biology.

Slope of Tangent Line

The slope of a tangent line to a curve at a given point is a critical concept that sheds light on the behavior of the curve at that specific spot. Essentially, it's a measure of the steepness of the line that just grazes the curve without cutting through it. In the context of the function under study, \(f(x)=\frac{x}{x^{2}+1}\), we needed to find the slope of the tangent line at the point \(0,0\).

By first finding the derivative, we then evaluated it at the desired x-coordinate to determine the slope. This slope tells us the direction of the tangent line and, visually, how the function is increasing or decreasing at that point. Knowing how to find this slope is invaluable not just for graphing purposes, but also for solving more complex problems involving motion, optimization, and other applications where change is involved.