Problem 52 - Calculus an Applied Approach

Question

(a)Find an equation of the tangent line to the graph of the function at the given point, (b) use a graphing utility to graph the function and its tangent line at the point, and (c) use the derivative feature of a graphing utility to confirm your results. $$ f(x)=\frac{1}{\sqrt[3]{x^{2}}}-x \quad(-1,2) $$

Step-by-Step Solution

Answer

The equation of the tangent line to the graph at the point (-1,2) is \(y=x+3\). This is confirmed by graphing the function and the tangent line, and verifying using derivative feature of a graphing tool.

Step 1: Find the Derivative of the Function

Consider the function \(f(x)=\frac{1}{\sqrt[3]{x^{2}}}-x\). To find the slope of the tangent line, the derivative of the function must be found. The derivative of \(f(x)\) is \(f'(x)\). Using the power and chain rule of differentiation, \(f'(x)=\frac{-2}{3x\sqrt[3]{x^{4}}} - 1\).

Step 2: Find the Slope at (-1,2)

Substitute the x-coordinate of the point (-1,2) into the derivative to find the slope of the tangent line at this specific point. \(f'(-1)=\frac{-2}{3(-1)\sqrt[3]{(-1)^{4}}} - 1= 1\).

Step 3: Find the Equation of the Tangent Line

The equation of a line is given by \(y=mx+c\) where \(m\) is the slope and \(c\) is the y-intercept. We have \(m=1\) and the point (-1,2). Substituting these into the line equation gives \(2 = 1*(-1) + c\), solving this gives \(c=3\). So the equation of the tangent line is \(y=x+3\).

Step 4: Graph the Function and the Tangent Line

With a graphing tool, the original function \(f(x)=\frac{1}{\sqrt[3]{x^{2}}}-x\) and the tangent line \(y=x+3\), should be graphed.

Step 5: Confirm the Results

Use the derivative feature of the graphing tool to confirm the results. The derivative tool should provide the same slope at the point (-1,2) as calculated, confirming that the calculations and graph are correct.

Key Concepts

Derivative of a Function

Understanding the derivative of a function is crucial when dealing with calculus. It represents the rate at which a function's output value changes as its input value changes. Intuitively, when we look at the graph of a function, the derivative at any point gives us the slope of the tangent line to the graph at that specific point.

Let's use the function from the exercise, \(f(x)=\frac{1}{\sqrt[3]{x^{2}}}-x\), to demonstrate how to find the derivative. This involves the use of rules like the power rule, which tells us that the derivative of \(x^n\) is \(nx^{n-1}\), and the chain rule, which is applied when we have a function composed of other functions. Using these rules, we found that the derivative is \(f'(x)=\frac{-2}{3x\sqrt[3]{x^{4}}} - 1\).

The derivative not only guides us in graphing tangent lines but is also a central concept when we want to understand the behavior of functions, such as identifying maxima and minima, or solving problems related to rates of change in real-world applications.

Slope of a Tangent Line

The slope of the tangent line to a curve at a particular point is a numerical measure of how steep the tangent line is at that point. This slope is exactly the value of the derivative of the function at that point. For the given exercise, once we have calculated the derivative, we can find the slope by evaluating the derivative at the x-coordinate of the given point.

With our function \(f'(x)\), we substitute \(x = -1\) to obtain the slope at the point \((-1,2)\). After substitution, we discovered the slope is 1, indicating that at the point \((-1,2)\), the tangent line rises at the same rate as it runs; for every unit the x-coordinate increases, the y-coordinate also increases by precisely one unit.

This concept is tangible in many physical concepts such as velocity, where the slope of a position-time graph gives us an object's velocity, or in economics, where the slope of a cost function can represent the marginal cost.

Graphing Utility

A graphing utility is an indispensable tool in calculus that allows users to visualize functions and their properties, such as slopes of tangent lines and areas under curves. For exercises like the one given, it is used to plot the function as well as the tangent line at a particular point.

By inputting the function \(f(x)=\frac{1}{\sqrt[3]{x^{2}}}-x\) and the tangent line equation found in previous steps, \(y=x+3\), into the graphing utility, students can observe the tangency graphically and confirm that the tangent line only touches the curve at the single point (-1,2). The graphing utility also offers a derivative feature that can aid in confirming the slope found in previous calculations. It's crucial for students to learn how to effectively use this tool, not just for confirming calculations, but to gain a deeper visual comprehension of the abstract concepts involved in calculus.