Problem 52 - College Algebra and Calculus: An Applied Approach

Question

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 of the function \(f(x) = \frac{1}{\sqrt[3]{x^2}} - x\) at the point (-1,2) is \(y = -\frac{5}{3}x - \frac{1}{3}\).

Step 1: Compute the Derivative

We have to find the derivative of the given function. Using power rule for finding derivative,\(f'(x)=\frac{d(\frac{1}{\sqrt[3]{x^{2}}})}{dx} - \frac{d(x)}{dx}\).By derivating we get \(f'(x)=-\frac{2}{3}\frac{1}{{x^{\frac{5}{3}}}} - 1 \).

Step 2: Substitute Point into the Derivative

We need to evaluate the derivative at the given point \((-1,2)\). Substitute -1 into \(f'(x)\) to get \(f'(-1)\). After substituting, we obtained that \(f'(-1)=-\frac{2}{3} - 1 = - \frac{5}{3} \). So the slope of the tangent line at point \((-1,2)\) is \(-\frac{5}{3}\).

Step 3: Use Point Slope Form to Find Equation of Tangent Line

Use the point slope formula to determine the equation of the tangent line. The formula is \(y-y1=m(x-x1)\), where \(m\) is the slope, \((x1,y1)\) is the given point. Here, \(m=-\frac{5}{3}\) and \((x1,y1) = (-1,2) \). Thus the equation is \(y - 2 = -\frac{5}{3}(x + 1)\)

Step 4: Simplify the Equation

Simplify the equation to the form \(y = mx + b\): we get \(y = -\frac{5}{3}x - \frac{1}{3}\). This is the equation of the tangent line.

Step 5: Confirm Result Graphically

Use a graphing utility to graph the function and its tangent line. Also use the derivative feature of the graphing utility to verify our slope at the point (-1,2). Both the function and its tangent line at the point need to be graphed for visualization. It is not possible to demonstrate this step here, this must be done on your own using a graphing tool.

Key Concepts

Derivative of a Function

Understanding the derivative of a function is crucial when finding the equation of a tangent line in calculus. The derivative represents the rate at which a function's value changes as its input changes. In simple terms, it gives us the slope of the function at any given point, which is a measure of how steeply the graph rises or falls at that location.

Mathematically, the process of calculating the derivative is called differentiation. For the function in the exercise, \( f(x)=\frac{1}{\sqrt[3]{x^{2}}} - x \), the power rule is applied to find its derivative. The power rule states that if you have a function \( x^n \), its derivative is \( nx^{n-1} \). Applying this to both terms of the function yields the derivative, \( f'(x)=-\frac{2}{3}\frac{1}{{x^{\frac{5}{3}}}} - 1 \).

When the derivative is calculated at a specific point, it gives the slope of the tangent line to the function at that point. The ability to find this slope is a key skill for graphing the behavior of functions and predicting their future values.

Point-Slope Form

The point-slope form of a line is an essential concept in calculus for writing the equation of a line when you know its slope and a point through which it passes. The general formula is given by \( y - y_1 = m(x - x_1) \), where \( m \) is the slope and \( (x_1, y_1) \) is the point on the line.

In the given exercise, after finding the slope of the tangent line to be \( -\frac{5}{3} \) at the point \( (-1, 2) \), we can substitute these values into the point-slope form formula to find the equation of our tangent line: \( y - 2 = -\frac{5}{3}(x + 1) \). This equation can be further simplified to the slope-intercept form, \( y = mx + b \), for ease of graphing. Understanding how to manipulate and use the point-slope form allows students to seamlessly transition between different forms of linear equations and better grasp their geometric interpretations.

Graphing Utility

A graphing utility is an incredibly powerful tool in calculus and other branches of mathematics. It allows users to visually interpret functions and their derivatives, which can be especially helpful when dealing with complex equations where manual graphing is impractical.

For the function in our exercise, a graphing utility can be used to graph the function \( f(x) \) and its tangent line. By inputting the simplified equation of the tangent line, \( y = -\frac{5}{3}x - \frac{1}{3} \), into the utility, students can see how the line just touches the graph of the function at the point \( (-1, 2) \) without crossing through it anywhere else around that point. This visual confirmation strengthens the understanding of tangents. Additionally, most graphing utilities come with a derivative feature that allows students to confirm the derivative of the function at any point, reinforcing the connection between the algebraic derivative and the graphical representation. Overall, graphing utilities not only confirm our calculations but also enhance our comprehension of the behavior of functions.