Question

(a) find an equation of the tangent line to the graph of \(f\) 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. $$ f(x)=\left(x^{3}-3 x+1\right)(x+2) $$

Short answer

The equation of the tangent line to the graph of the function at an arbitrary point 'p' is \(y = f\'(p)*(x - p) + f(p)\) where \(f\'(p) = ((3*p^{2}-3) * (p+2)) + ((p^{3}-3*p+1)*1)\). The function and its tangent at point 'p' can be visualized by plotting on a graphing utility and the result can be confirmed using the derivative feature of the graphing utility.

Step-by-step solution

Find the derivative of the function

First, use the product rule, which states that the derivative of two multiplied functions is the first function times the derivative of the second function plus the second function times the derivative of the first function. The derivative of the function given in the exercise: \(f(x) = (x^{3}-3x+1) * (x+2)\) can be calculated as: \(f\'(x) = ((3x^{2}-3) * (x+2)) + ((x^{3}-3x+1)*1)\)

Substitute x-value in the derivative function to find the slope

The slope of the tangent line at a given point (a, f(a)) is equal to the value of the derivative at that point. Let's consider an arbitrary point 'p' for demonstration. Then we can plug the x coordinate of the point 'p' into the derivative function to get the slope of the tangent line \(f\'(p) = ((3p^{2}-3) * (p+2)) + ((p^{3}-3p+1)*1) . This got us the slope of tangent at 'p'.

Find equation of the tangent line

The equation of the tangent line in the slope-intercept form (y = mx + b) can be found once we have the slope and a point on the line (p, f(p)). M is the slope from the previous step and 'b' is the y-intercept that can be found by solving the equation \(y - f(p) = m*(x - p)\) for \(y\), then the equation of the tangent line is \(y = f\'(p)*(x - p) + f(p)\)

Plot the function and its tangent line

Use a graphing utility to plot the function \(f(x) = (x^{3}-3x+1) * (x+2)\) and its tangent at the point (p, f(p)). The tangent line should touch the graph of the function only at the point (p, f(p)) and nowhere else.

Confirmation using the derivative feature of the graphing utility

Now, use the derivative feature of the graphing utility to confirm your result. This tool will automatically calculate the derivative of the function and display the tangent line on the graph. This tangent must match the one you drew.

Key concepts

Product Rule

Understanding the product rule is essential when tackling calculus problems involving derivatives of functions that are products of two or more functions. It's a fundamental tool that enables us to differentiate expressions where two functions are multiplied together.

The rule states that if you have two functions, let's call them u(x) and v(x), the derivative of their product, u(x) * v(x), is given by:
\[u'(x)v(x) + u(x)v'(x)\]
To clarify, you take the derivative of the first function (u'(x)) and multiply it by the second function (v(x)) as is. Then, you add the product of the first function as is (u(x)) with the derivative of the second function (v'(x)). This results in the derivative of the entire expression.

When we apply the product rule to our example function f(x), we first identify each part of the product (x^3 - 3x + 1) and (x + 2). Then, we find their derivatives separately. Finally, we use the product rule to combine these results and find the derivative of the entire expression f(x)

Derivative of a Function

A derivative represents the rate of change of a function with respect to a variable. It's a crucial concept in calculus, especially when analyzing the motion of objects or changes in quantities. For any given function, the derivative is found using rules of differentiation such as the power rule, chain rule, product rule, or quotient rule.

The derivative has many applications, including finding the slope of a tangent line to a graph at a particular point. This slope is simply the value of the derivative at that specific point. In our exercise, finding the derivative of f(x) at a point p gives us the slope of the tangent line at (p, f(p)). This information is what allows us to construct the tangent line equation and understand how the function behaves locally around the point p.

Graphing Utility Usage

Graphing utilities are powerful tools that can greatly assist in visualizing functions and their characteristics, including tangent lines. They allow users to see the shape of the curve produced by the function and understand where it increases, decreases, and where the function's critical points are.

In our exercise scenario, we can input the function f(x) into a graphing utility to produce its graph. Additionally, we can use the utility's features to draw the tangent line at a specified point, providing a clear visual confirmation that the tangent line indeed touches the curve at just one point. Moreover, graphing utilities often have features that can calculate derivatives directly, which enables students to check their manual calculations with the result provided by the utility.

Slope-Intercept Form

The slope-intercept form is a way of writing the equation of a line. By using this form, y = mx + b, where m is the slope of the line, and b is the y-intercept, we create a straightforward method for plotting linear functions. This form is especially useful when we have the slope of a line (from the derivative at a specific point, for instance) and a point through which the line passes.

For our exercise's tangent line equation, we begin with the slope discovered from the derivative and the coordinates of the given point. Substituting these into the slope-intercept form allows us to create the equation of the tangent line with ease. Understanding this form enhances one's ability to analyze, graph, and interpret linear relationships quickly.