Problem 51 - University Calculus: Early Transcendentals

Question

a. Find the derivative \(f^{\prime}(x)\) of the given function \(y=f(x)\) b. Graph \(y=f(x)\) and \(y=f^{\prime}(x)\) side by side using separate sets of coordinate axes, and answer the following questions. c. For what values of \(x\), if any, is \(f^{\prime}\) positive? Zero? Negative? d. Over what intervals of \(x\) -values, if any, does the function \(y=f(x)\) increase as \(x\) increases? Decrease as \(x\) increases? How is this related to what you found in part (c)? (We will say more about this relationship in Section \(4.3 .\) ) $$y=x^{3} / 3$$

Step-by-Step Solution

Answer

The derivative is \( f'(x) = x^2 \). \( f'(x) \) is zero at \( x = 0 \) and positive for all other \( x \), so \( y=f(x) \) increases everywhere except \( x = 0 \).

Step 1: Differentiate the Function

To find the derivative of the function \( y = \frac{x^3}{3} \), apply the power rule of differentiation. The power rule states that \( \frac{d}{dx}(x^n) = nx^{n-1} \). In this case, \( y = \frac{1}{3} x^3 \), so the derivative \( f'(x) \) is computed as follows:\[ f'(x) = \frac{d}{dx}\left(\frac{1}{3}x^3 \right) = \frac{1}{3} \cdot 3x^{3-1} = x^2. \] Thus, the derivative of the function is \( f'(x) = x^2 \).

Step 2: Graph the Functions

Graph the original function \( y = \frac{x^3}{3} \) and its derivative \( y = x^2 \) using separate sets of coordinate axes. The graph of \( y = \frac{x^3}{3} \) is a cubic function with an inflection point at the origin, while the graph of \( y = x^2 \) is a parabola that opens upwards.

Step 3: Determine Where \(f'(x)\) is Positive, Zero, or Negative

The derivative function \( f'(x) = x^2 \) is always non-negative because squaring any real number results in a non-negative number. - \( f'(x) = 0 \) only at \( x = 0 \).- \( f'(x) \) is positive for all \( x eq 0 \). Therefore, \( f'(x) \) is zero at \( x = 0 \) and positive for all other values of \( x \).

Step 4: Analyze the Increase/Decrease of \(y=f(x)\)

The function \( y = \frac{x^3}{3} \) increases when its derivative \( f'(x) = x^2 \) is positive.- Since \( f'(x) \) is positive for all \( x eq 0 \), the function \( y \) increases for all \( x otin \{0\} \).- At the point \( x = 0 \), \( f'(x) = 0 \), indicating a stationary point, but since the sign of \( f'(x) \) does not change, there is no interval of decrease.Hence, the function \( y \) increases for all \( x eq 0 \), which aligns with the intervals where \( f'(x) \) is more than zero.

Key Concepts

Power Rule

The power rule is a fundamental tool in calculus for finding the derivative of a polynomial function. It simplifies the process by providing a straightforward formula:

• For any expression of the form \( x^n \), the derivative is \( n \cdot x^{n-1} \).
  • For example, if you have \( x^3 \), applying the power rule gives \( 3x^{3-1} = 3x^2 \).

This rule makes it quick and easy to differentiate more complex expressions. In our exercise, we tackled \( y = \frac{x^3}{3} \). Applying the power rule here involves first rewriting the function as \( \frac{1}{3}x^3 \), then differentiating:

• Multiply the power (3) by the coefficient (\( \frac{1}{3} \)), resulting in \( x^2 \).

Understanding and mastering the power rule is essential for anyone studying differential calculus. This rule is just a first step but provides a solid foundation that is built upon in more advanced topics.

Cubic Function

A cubic function is a polynomial of degree three, and it takes the general form \( ax^3 + bx^2 + cx + d \). These functions have a wide range of applications due to their distinct shape and behavior.

• In our exercise, the original function given is \( y = \frac{x^3}{3} \), a simple cubic function.

Characteristic features of cubic functions include:

• An inflection point, which is where the function changes concavity, at \( x = 0 \) in our example.
• It can have up to two turning points.
• The graph will have an S-shaped curve.

These functions possess rich graphical and analytical properties, such as intersections with the axes and varying steepness. For students, understanding these character traits is vital in exploring how cubic functions can model real-world phenomenons.

Derivative Graph

Graphing the derivative of a function sheds light on the function's behavior concerning its increasing or decreasing nature. The derivative function, given by \( f'(x) \), displays the rate of change of the original function.

• For \( y = \frac{x^3}{3} \), the derivative is \( f'(x) = x^2 \).
• The graph of \( y = x^2 \) is a parabola that opens upwards.

From the graphs, important observations can be made:

• Where the derivative graph is above the x-axis, the original function is increasing.
• Where the derivative graph intersects the x-axis, there's usually a stationary point.

In this exercise, \( f'(x) = x^2 \) is non-negative, implying that the cubic function does not decline and thus continuously rises for \( x eq 0 \). Viewing these graphs side by side provides understanding of how the slope relates to the rate of change of the original function.

Function Increase and Decrease

Studying the behavior of a function as it increases or decreases helps us understand how it behaves across different intervals. This concept is closely tied to the derivative:

• When \( f'(x) \) is positive, the function increases.
• When \( f'(x) \) is zero, it indicates a potential stationary point (though not necessarily a maximum or minimum).
• If \( f'(x) \) were negative, it would indicate the function decreases, but that’s not the case here.

For the function \( y = \frac{x^3}{3} \), we determined that:

• It is monotonically increasing for all \( x eq 0 \).
• At \( x = 0 \), \( f'(x) = 0 \) indicating a flat tangent.

By linking the sign of the derivative to intervals of increase or decrease, we're able to predict the function's behavior without always needing to rely on its graph. This relationship is a cornerstone in calculus, making it easier to analyze function trends and predict future outcomes.