Problem 68 - College Algebra and Calculus: An Applied Approach

Question

In Exercises, find the slope of the graph at the indicated point. Then write an equation of the tangent line to the graph of the function at the given point. $$ f(x)=2 \ln x^{3}, \quad(e, 6) $$

Step-by-Step Solution

Answer

The slope of the tangential line to the graph of the function \(f(x) = 2 \ln{x^3}\) at the point \((e, 6)\) is \(6e\), and the equation of the tangent line at this point is \(y-6 = 6e(x - e)\).

Step 1: Differentiation of the function

Differentiate the function \(f(x) = 2 \ln{x^3}\) using the chain rule: \(f'(x) = 2 \frac{1}{x} \cdot 3x^{3-1} = 2 \frac{1}{x} \cdot 3x^2 = 6x\). This derivative gives us the slope of the tangential line to the graph at any point \(x\).

Step 2: Calculate the slope at the point (e, 6)

Substitute \(x = e\) into \(f'(x)\) to calculate the slope of the tangent line at the required point: \(f'(e) = 6e\). So, the slope of the tangent line at the point \((e, 6)\) is \(6e\).

Step 3: Write an equation of the tangent line

Using the point-slope form (\(y-y_1 = m(x-x_1)\)) of the line, write down the equation of the tangent line to the graph at the point \((e, 6)\). Use the slope (m) of \(6e\) from steps 2 and coordinates \((e, 6)\) resulting in equation: \(y-6 = 6e(x-e)\).

Key Concepts

Differentiation

Differentiation is a key concept in calculus that enables us to determine how a function changes at any point. It's essentially the process of finding the derivative of a function. The derivative provides the rate at which a quantity changes with respect to change in another quantity. In simpler terms, it measures the sensitivity of one variable to changes in another.
A common application of differentiation is to find the slope of a function's graph at a particular point. The derivative at that point gives the slope of the tangent line, which approximates the function near that point.
The function given, \(f(x) = 2 \ln x^3\), can be differentiated using rules like the power rule, product rule, and notably here, the chain rule. Once differentiated, the function's derivative will reveal the slope of its tangent line at any chosen point.

Tangent Line

Tangent lines are straight lines that 'just touch' a curve at a given point, without crossing it. This notion is critical in understanding the behavior of functions locally. At the point of tangency, the tangent line perfectly represents the slope of the function.
When you find the equation of the tangent line to a function at a specific point, you're essentially stating the line's slope and its intersection at that point.

• The tangent captures how steeply the function is climbing or descending at that point.
• The equation often uses the point-slope form: \(y - y_1 = m(x - x_1)\).

In our exercise, the function \(f(x) = 2 \ln x^3\) is examined at the point \((e, 6)\), and using the previously determined slope, we find its tangent line equation.

Chain Rule

The chain rule is a fundamental technique in differentiation, particularly useful when dealing with composite functions. It provides a method for differentiating functions that are nested within one another.
The essence of the chain rule can be stated as: if you have a composite function \(y = f(g(x))\), the derivative \(y'\) is found as \(f'(g(x)) \cdot g'(x)\).

• This 'rule within a rule' approach allows us to break down complex functions into simpler parts.
• Each function's derivative is computed separately and then combined.

In our problem, to differentiate the function \(f(x) = 2 \ln x^3\), the chain rule is used to handle the composition of the logarithm and the power of \(x\). It helps in calculating the derivative efficiently, enabling us to find the slope at the desired point.

Slope

The slope of a function at a point is a measure of its steepness or incline at that specific location. It is determined by the derivative, which provides a single numerical value describing that steepness.
This concept is crucial for understanding how quickly or slowly a function is changing at any point along its graph.

• A positive slope indicates the function is increasing, while a negative slope shows it's decreasing.
• If the slope is zero, the function could be at a maximum, minimum, or just level at that point.

For the given exercise, substituting \(x = e\) into the derivative \(f'(x) = 6x\) gives us the slope \(6e\) at the point \((e, 6)\). This slope is then used to define the tangent line equation, helping us understand the behavior of the graph near that point.