Problem 54 - Calculus: Early Transcendentals

Question

Use the definition of a derivative to find \( f'(x) \) and \( f''(x) \). Then graph \( f \), \( f' \), and \( f'' \) on a common screen and check to see if your answers are reasonable. \( f(x) = x^3 - 3x \)

Step-by-Step Solution

Answer

The derivatives are \( f'(x) = 3x^2 - 3 \) and \( f''(x) = 6x \). The graphs confirm these derivatives.

Step 1: Understand the Function

Given the function \( f(x) = x^3 - 3x \), we need to find the first and second derivatives. This function is a cubic polynomial.

Step 2: Use the Definition of the Derivative

The derivative of a function \( f(x) \) is given by the limit: \[ f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h} \] Apply this definition to \( f(x) = x^3 - 3x \).

Step 3: Calculate \( f'(x) \) Using the Definition

Substitute \( f(x+h) = (x+h)^3 - 3(x+h) \) into the definition: \[ f(x+h) = (x^3 + 3x^2h + 3xh^2 + h^3) - 3x - 3h \] \[ f(x+h) - f(x) = 3x^2h + 3xh^2 + h^3 - 3h \] Simplify the difference and divide by \( h \): \[ \frac{3x^2h + 3xh^2 + h^3 - 3h}{h} = 3x^2 + 3xh + h^2 - 3 \] Taking the limit as \( h \to 0 \), we get \( f'(x) = 3x^2 - 3 \).

Step 4: Calculate \( f''(x) \) Using the Derivative Definition

The second derivative is the derivative of \( f'(x) \). Calculate \( f''(x) \) from \( f'(x) = 3x^2 - 3 \). The derivative of \( 3x^2 \) is \( 6x \) and the derivative of \(-3\) is 0. Therefore, \( f''(x) = 6x \).

Step 5: Graphing the Functions

Graph \( f(x) = x^3 - 3x \), \( f'(x) = 3x^2 - 3 \), and \( f''(x) = 6x \) on a graphing tool. \( f(x) \) should appear as a cubic curve, \( f'(x) \) as a parabola, and \( f''(x) \) as a straight line.

Step 6: Check the Graph for Reasonability

Compare the graph of \( f(x) \) with \( f'(x) \) and \( f''(x) \). \( f'(x) = 0 \) at points where \( f(x) \) has horizontal tangents, and the sign of \( f'(x) \) indicates where \( f(x) \) is increasing or decreasing. \( f''(x) \) shows the concavity of \( f(x) \).

Key Concepts

First Derivative

The first derivative of a function, often denoted as \( f'(x) \), tells us how the function changes at any point. For the given cubic polynomial \( f(x) = x^3 - 3x \), finding the first derivative involves applying the limit definition:

• Substitute \( f(x+h) = (x+h)^3 - 3(x+h) \).
• Compute the expression \( (x+h)^3 - 3(x+h) - (x^3 - 3x) \).
• Simplify and divide each term by \( h \).
• Take the limit as \( h \to 0 \).

For this function, the first derivative \( f'(x) \) simplifies to \( 3x^2 - 3 \). This reveals the slope of the tangent line to \( f(x) \) at any given point. Where \( f'(x) = 0 \), the graph of \( f(x) \) has horizontal tangents.

Second Derivative

The second derivative, \( f''(x) \), gives insight into the curvature, or concavity, of the function \( f(x) \). It is the derivative of the first derivative, \( f'(x) \). By differentiating \( f'(x) = 3x^2 - 3 \), we find:

• Differentiate \( 3x^2 \) to get \( 6x \).
• \( -3 \) remains constant, so its derivative is 0.

Thus, \( f''(x) = 6x \). This function tells us at which rate \( f'(x) \) is changing and thus shows how \( f(x) \) bends: - If \( f''(x) > 0 \), \( f(x) \) is concave up.- If \( f''(x) < 0 \), \( f(x) \) is concave down.

Graph of Functions

Graphing functions is a powerful way to visualize derivatives and understand the behavior of the function. For this exercise:- Graph \( f(x) = x^3 - 3x \), which appears as a cubic curve.- Graph \( f'(x) = 3x^2 - 3 \), presenting as a parabola.- Graph \( f''(x) = 6x \), which is a straight line.These graphs verify our calculations. The points where \( f'(x) = 0 \) correspond to the peaks or valleys of \( f(x) \). The graph of \( f''(x) \) shows directly if these parts of the curve are bending upward or downward.

Cubic Polynomial

A cubic polynomial is an expression of degree three, such as \( x^3 - 3x \). These types of functions can have up to two turning points because they are defined by a third-degree term:

• The highest power of \( x \) determines the general shape of the graph.
• The linear and constant terms affect its direction and position.

Cubic polynomials are generally distinguished by their characteristic "S" like shape, exhibiting both increasing and decreasing spots due to their turning points, which can be revealed by analyzing the first derivative \( f'(x) \).

Concavity

Concavity in graph of a function tells us how it curves. The second derivative, \( f''(x) \), is key to determining concavity:

• When \( f''(x) > 0 \), the graph is concave up, resembling a cup.
• When \( f''(x) < 0 \), it is concave down, like a frown.

For \( f(x) = x^3 - 3x \), \( f''(x) = 6x \) implies the function is:- Concave up for \( x > 0 \), meaning the curve of \( f(x) \) bends upwards.- Concave down for \( x < 0 \), indicating the curve bends downwards.Identifying these changes in concavity can help predict the behavior of the function across different intervals.