Problem 41 - Brief Applied Calculus

Question

a. Find the equation of the tangent line to \(f(x)=x^{3}-3 x^{2}+2 x-2\) at \(x=2\) b. Graph the function and the tangent line on the window \([-1,4]\) by \([-7,5]\).

Step-by-Step Solution

Answer

Tangent line: \( y = 2x - 6 \). Graph both functions over \([-1,4]\) by \([-7,5]\).

Step 1: Find the Derivative of the Function

First, we need to find the derivative of the function \( f(x) = x^3 - 3x^2 + 2x - 2 \). The derivative, \( f'(x) \), will give us the slope of the tangent line at any point \( x \).Find \( f'(x) \):\[ f'(x) = 3x^2 - 6x + 2\]

Step 2: Evaluate the Derivative at x = 2

Now, evaluate the derivative at \( x = 2 \) to find the slope of the tangent line at that point.Calculate \( f'(2) \):\[ f'(2) = 3(2)^2 - 6(2) + 2 = 12 - 12 + 2 = 2\]So, the slope \( m \) of the tangent line is \( 2 \).

Step 3: Find the Function Value at x = 2

Next, we will determine the point \( (x, f(x)) \) on the function where the tangent line touches it by evaluating \( f(x) \) at \( x = 2 \).Calculate \( f(2) \):\[ f(2) = (2)^3 - 3(2)^2 + 2(2) - 2 = 8 - 12 + 4 - 2 = -2\]Thus, the point of tangency is \( (2, -2) \).

Step 4: Write the Equation of the Tangent Line

Using the point-slope form of the equation of a line \( y - y_1 = m(x - x_1) \), where \((x_1, y_1)\) is the point of tangency and \( m \) is the slope, we can write the equation of the tangent line.Substitute \( m = 2 \), \( x_1 = 2 \), and \( y_1 = -2 \) into the equation:\[ y - (-2) = 2(x - 2)\]Simplify to get:\[ y + 2 = 2x - 4\]So, the equation of the tangent line is:\[ y = 2x - 6\]

Step 5: Graph the Function and Tangent Line

Graph the function \( f(x) = x^3 - 3x^2 + 2x - 2 \) and its tangent line \( y = 2x - 6 \) on the given window \([-1,4]\) for the x-axis and \([-7,5]\) for the y-axis. The graph will show how the tangent line touches the curve at \( x = 2 \).

Key Concepts

Derivative

A derivative is a fundamental concept in calculus. It's a way to determine the rate at which a function is changing at any given point. Consider a function like our given example, \( f(x) = x^3 - 3x^2 + 2x - 2 \). To find derivatives, you use rules of differentiation, such as the power rule. The power rule states that for any term \( ax^n \), the derivative is \( nax^{n-1} \). Using this, the derivative \( f'(x) = 3x^2 - 6x + 2 \) was calculated. This derivative tells us the slope of the function's graph at any point \( x \). For example, evaluating the derivative at \( x = 2 \), we get \( f'(2) = 2 \), which tells us the slope of the tangent line to the curve is 2 at this point. Ultimately, the derivative gives us powerful insight into the behavior and slope of a function.

Tangent Line

A tangent line is a straight line that touches a curve at a single point without crossing over. This line shows the immediate rate of change of the function at that specific point. We use the derivative to find the slope of this tangent line. For the curve \( f(x) = x^3 - 3x^2 + 2x - 2 \) at \( x = 2 \), we found the slope (derivative) to be 2. Evaluating the function at the same point gives us the coordinates \((2, -2)\), the point of tangency. To find the equation of the tangent line, we use the point-slope form: \( y - y_1 = m(x - x_1) \). Plugging in \( m = 2 \), \( x_1 = 2 \), and \( y_1 = -2 \), we arrive at the equation \( y = 2x - 6 \). This equation is key as it allows us to draw and analyze how the graph behaves around the point \((2, -2)\).

Graphing Functions

Graphing functions is crucial for visualizing how they behave over their domains. It helps you see relationships and changes over specific intervals. In this exercise, we graph the function \( f(x) = x^3 - 3x^2 + 2x - 2 \) and its tangent line \( y = 2x - 6 \) over the window \([-1, 4]\) for the x-axis and \([-7, 5]\) for the y-axis. The graph shows how the tangent line intersects or touches the curve at the specific point \( x = 2 \), demonstrating how closely the tangent line approximates the curve at that point. Graphing can reveal important characteristics such as intercepts, slope, and curvature, giving a better understanding of the behavior and properties of the function. Using these visual tools helps deepen comprehension of the underlying mathematical principles.