Problem 32 - Calculus and its applications

Question

Draw a graph that is differentiable and has horizontal tangent lines at \(x=0, x=2,\) and \(x=4\).

Step-by-Step Solution

Answer

Use the function \(f(x) = x^3 - 3x^2\) to create a graph with horizontal tangents at \(x=0, 2, 4\).

Step 1: Understanding the Task

The goal is to draw a graph of a function that is differentiable and has horizontal tangent lines at three specific points: \(x = 0\), \(x = 2\), and \(x = 4\). A horizontal tangent line means that the derivative of the function is zero at those points.

Step 2: Choosing a Function

To satisfy the condition of having horizontal tangent lines at \(x = 0, 2, 4\), consider using a cubic polynomial, such as \(f(x) = ax^3 + bx^2 + cx + d\), because the flexibility of a cubic function allows us to control the slope at specific values of \(x\).

Step 3: Determining the Derivative

The derivative of the function \(f(x) = ax^3 + bx^2 + cx + d\) is \(f'(x) = 3ax^2 + 2bx + c\). We need \(f'(0) = f'(2) = f'(4) = 0\).

Step 4: Setting Up Equations

Substitute \(x=0, x=2,\) and \(x=4\) into the derivative equation: - For \(x=0\): \(f'(0) = c = 0\).- For \(x=2\): \(f'(2) = 12a + 4b = 0\).- For \(x=4\): \(f'(4) = 48a + 8b = 0\).

Step 5: Solving Equations

From \(f'(0) = 0\), we have \(c = 0\).Solving the two equations, \(12a + 4b = 0\) and \(48a + 8b = 0\), we find that they are dependent, meaning the same function can be obtained by scaling the coefficients. A simple solution is \(a = -rac{b}{3}\).

Step 6: Constructing the Function

Choosing \(a = 1\) and \(b = -3\), the function becomes \(f(x) = x^3 - 3x^2\). This satisfies \(f'(x) = 3x^2 - 6x\) with zeros at \(x = 0, 2,\) and \(x = 4\). Thus, the graph will have horizontal tangents at these points.

Step 7: Drawing the Graph

Plot the function \(f(x) = x^3 - 3x^2\). The graph will show where it flattens out at \(x = 0\), \(x = 2\), and \(x = 4\), creating horizontal tangent lines at these points.

Key Concepts

Horizontal Tangent Lines

When studying differentiable functions, the concept of horizontal tangent lines is crucial. A tangent line is a straight line that gently touches a curve at a particular point without intersecting it. If a tangent line is horizontal, it implies that its slope is zero.
This is significant because it provides insights into the behavior of the function at the point of tangency. For instance, if a graph has a horizontal tangent line at a certain point, the function might be experiencing a local maximum or minimum there. To find where the horizontal tangents occur, set the derivative of the function to zero, because the derivative represents the slope of the tangent line.

Cubic Polynomial

Cubic polynomials are a type of polynomial function where the highest power of the variable is three. They are generally represented as:

• \( f(x) = ax^3 + bx^2 + cx + d \)

These functions are highly versatile due to their ability to have turning points, making them useful in crafting graphs that meet specific requirements such as having horizontal tangent lines at particular points.
The flexibility of cubic polynomials is what makes them advantageous when needing to design a function to have specific properties. By adjusting the coefficients \(a, b, c,\) and \(d\), you can shape the curve to fit conditions like those seen in the exercise, where we needed the derivative, and consequently the slope, to be zero at certain points.
Thus, a cubic polynomial was selected for the task of having horizontal tangents at \(x=0, x=2,\) and \(x=4\), demonstrating its adaptability and usefulness.

Derivative of a Function

The derivative provides valuable information about a function's rate of change and is a cornerstone concept in calculus. It shows how the function value changes as the input changes, and is often seen symbolically as \(f'(x)\) for a function \(f(x)\).
To find the derivative, you apply differentiation rules, which aim to calculate the limit of the average rate of change as the interval approaches zero. For polynomial functions, this means applying the power rule which states:

• If \(f(x) = ax^n\), then \(f'(x) = nax^{n-1}\)

For example, with the cubic polynomial \(f(x) = ax^3 + bx^2 + cx + d\), the derivative is found by differentiating each term individually:

• \(f'(x) = 3ax^2 + 2bx + c\)

In the task, by setting the derivative equal to zero at specific points \(x = 0, 2, 4\), we determined where the function has horizontal tangent lines. Thus, the derivative not only gives the function's rate of change but also insights into the graph's behavior.