Problem 52 - 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 \(4.3 .\) ) $$y=x^{4} / 4$$

Step-by-Step Solution

Answer

\(f'(x)=x^3\); \(f'(x)\) is positive for \(x>0\), zero at \(x=0\), negative for \(x<0\); \(f(x)\) increases for \(x>0\) and decreases for \(x<0\).

Step 1: Differentiate the Function

To find the derivative of the function \( y = f(x) = \frac{x^4}{4} \), we apply the power rule, which states that \( \frac{d}{dx} x^n = n x^{n-1} \). The coefficient \( \frac{1}{4} \) is constant, so it remains as it is.Differentiating, we have:\[ f'(x) = \frac{d}{dx} \left( \frac{x^4}{4} \right) = \frac{1}{4} \cdot 4x^{3} = x^{3} \]

Step 2: Graph the Functions

Graph \( y = f(x) = \frac{x^4}{4} \) and \( y = f'(x) = x^3 \) separately. - For \( y = \frac{x^4}{4} \), the graph is a quartic curve that opens upwards, symmetric about the y-axis with a minimum at the origin.- For \( y = x^3 \), the graph is a cubic curve that passes through the origin, increasing on both sides.

Step 3: Analyze Sign of the Derivative

Examine the sign of \( f'(x) = x^3 \) to determine where it is positive, zero, or negative.- \( f'(x) = 0 \) at \( x = 0 \).- \( f'(x) > 0 \) when \( x > 0 \).- \( f'(x) < 0 \) when \( x < 0 \).

Step 4: Determine Intervals of Increase and Decrease

The function \( y = f(x) = \frac{x^4}{4} \) increases where \( f'(x) > 0 \) and decreases where \( f'(x) < 0 \).- The function increases for \( x > 0 \) because \( f'(x) > 0 \) (positive slope).- The function decreases for \( x < 0 \) because \( f'(x) < 0 \) (negative slope).This behavior matches the sign of \( f'(x) \) from Step 3.

Key Concepts

Power Rule

The power rule is a core concept in calculus for finding the derivative of a function. It's straightforward and extremely handy, especially when dealing with polynomial functions. The power rule states that if you have a function in the form of a monomial, like \( x^n \), then the derivative of that function, with respect to \( x \), is \( nx^{n-1} \).
In our original exercise, the function given was \( y = \frac{x^4}{4} \). By applying the power rule, we start by acknowledging the exponent \( n = 4 \). The rule dictates that we multiply the original exponent by the coefficient and then decrease the exponent by one. Since the coefficient in this problem is \( \frac{1}{4} \), the differentiation proceeds as:

• Multiply \( 4 \) by \( \frac{1}{4} \) which yields \( 1 \).
• Subtract one from the exponent to get \( x^{4-1} = x^3 \).

So, the derivative of our function \( y = \frac{x^4}{4} \) is simply \( x^3 \). This rule is powerful as it simplifies processes that might otherwise involve more complicated algebraic manipulation.

Graphing Functions

Graphing functions provides a visual way to understand and interpret the behavior of functions and their derivatives. For the given exercise, we deal with two functions: the original function \( y = \frac{x^4}{4} \) and its derivative \( y' = x^3 \).
Graphing \( y = \frac{x^4}{4} \), we find a quartic curve, which is symmetric about the y-axis. This graph's key feature includes a minimum at the origin, which signifies that there are no dips or turns outside the origin.
On the other hand, the graph of \( y = x^3 \) showcases a cubic curve that passes through the origin. Unlike the quartic curve, this cubic curve is purely increasing on the entire real number line, but with differing signs on either side of the origin. The curve's gradient changes its sign at \( x = 0 \), marking the transition from increasing to decreasing as you move along the x-axis.

Intervals of Increase and Decrease

The intervals of increase and decrease of a function are determined by the sign of its derivative. For our function \( y = \frac{x^4}{4} \), the derivative \( f'(x) = x^3 \) helps us identify these intervals.
To find these intervals, look at where \( f'(x) \) is positive or negative:

• The function \( y = \frac{x^4}{4} \) increases on intervals where \( f'(x) > 0 \). In this case, it's for \( x > 0 \).
• It decreases where \( f'(x) < 0 \). Here, it's for \( x < 0 \).

The derivative is zero at \( x = 0 \), highlighting a critical point and marking the boundary between increasing and decreasing intervals.
Understanding these intervals is crucial because they give insight into the behavior and nature of the function. For instance, knowing where the function increases or decreases helps in sketching accurate graphs and predicting the function's behavior.

Sign of Derivative

The sign of a derivative at different intervals helps elucidate the nature of a function's slope over various sections of its domain. For the exercise, analyze \( f'(x) = x^3 \):
- At \( x = 0 \), the derivative \( f'(x) = 0 \). This indicates a horizontal tangent at the origin of the original function \( y = \frac{x^4}{4} \).- When \( x > 0 \), the derivative \( f'(x) > 0 \). This implies that the function \( y = \frac{x^4}{4} \) has a positive slope in this region, thus increasing.- Conversely, at \( x < 0 \), the derivative \( f'(x) < 0 \). Here, the function has a negative slope, indicating a decrease in value as \( x \) decreases.
The sign analysis of the derivative gives a clear understanding of the function's behavior over different intervals and helps in visualizing the direction in which the function is headed across its domain.