Problem 78 - Calculus Single Variable

Question

For each of the given functions, graph \(f\) and \(f^{\prime}\) in the given viewing rectangle \(R .\) Fill in the following table. $$ \begin{array}{|l|l|l|} \hline \begin{array}{l} \text { Interval } \\ \text { where } f \\ \text { increases } \end{array} & \begin{array}{l} \text { Interval } \\ \text { where } f \\ \text { decreases } \end{array} & \begin{array}{l} \text { Point at which } f \text { has } \\ \text { a horizontal tangent } \end{array} \\ \hline & & \\ \hline \begin{array}{l} \text { Interval } \\ \text { where } f^{\prime}>0 \end{array} & \begin{array}{l} \text { Interval } \\ \text { where } f^{\prime}<0 \end{array} & \begin{array}{l} \text { Point(s) at } \\ \text { which } f^{\prime}=0 \end{array} \\ \hline & & \\ \hline \end{array} $$ Use the table to draw inferences that relate the sign of \(f^{\prime}\) to the behavior of \(f\). (These relationships will be studied in Chapter 4.) \(f(x)=x^{2} \sin (x) /\left(1+x+x^{4}\right), R=[-3,1.2] \times[-0.9,2.1]\)

Step-by-Step Solution

Answer

Use the sign of \( f'(x) \) to determine intervals of increase and decrease of \( f(x) \). Horizontal tangents occur where \( f'(x) = 0 \).

Step 1: Understand the Function

We start by analyzing the given function: \( f(x) = \frac{x^2 \sin(x)}{1 + x + x^4} \). We need to investigate where this function increases, decreases, and where it has horizontal tangents.

Step 2: Find the Derivative of f

Next, calculate the derivative of the function \( f(x) \) using the quotient rule: \(\frac{d}{dx} \left[ \frac{u(x)}{v(x)} \right] = \frac{u'(x)v(x) - u(x)v'(x)}{[v(x)]^2}\). Choose \( u(x) = x^2 \sin(x) \) and \( v(x) = 1 + x + x^4 \).Calculate \( u'(x) = 2x\sin(x) + x^2\cos(x) \) and \( v'(x) = 1 + 4x^3 \).Substitute into the quotient rule:\[ f'(x) = \frac{(2x\sin(x) + x^2\cos(x))(1 + x + x^4) - (x^2\sin(x))(1 + 4x^3)}{(1 + x + x^4)^2} \]Simplify this expression to use it in the following steps.

Step 3: Determine Intervals of Increase and Decrease

Use the sign of \( f'(x) \) to determine where \( f(x) \) increases or decreases. When \( f'(x) > 0 \), \( f(x) \) is increasing; when \( f'(x) < 0 \), \( f(x) \) is decreasing. Solve \( f'(x) = 0 \) to find potential points where \( f(x) \) could change its behavior.

Step 4: Calculate and Analyze f'(x)

Solve \( f'(x) = 0 \) analytically or numerically using computational tools since the derivative is complex: set the numerator to zero and solve.Calculate \( f'(x) \) at critical points within the given interval \([-3, 1.2]\) and identify where \( f'(x) > 0 \), \( f'(x) < 0 \).

Step 5: Fill In the Table

With the critical points and the sign of \( f'(x) \), fill in:1. **Interval where \( f \) increases**: Where \( f'(x) > 0 \) within \( R \).2. **Interval where \( f \) decreases**: Where \( f'(x) < 0 \) within \( R \).3. **Point at which \( f \) has a horizontal tangent**: Calculated \( x \) points where \( f'(x) = 0 \).4. **Interval where \( f' > 0 \)**: Same as \( f \) increases.5. **Interval where \( f' < 0 \)**: Same as \( f \) decreases.6. **Point(s) where \( f' = 0 \)**: Point(s) of horizontal tangents.

Key Concepts

Derivative

The derivative is a fundamental tool in calculus that helps us understand how a function changes. It measures the rate at which a function's output value is changing with respect to its input value. In simple terms, the derivative gives us the slope of the tangent to the function's curve at any given point. This means it tells us how steep the curve is at that point. In the context of the function given, which is \( f(x) = \frac{x^2 \sin(x)}{1 + x + x^4} \), finding the derivative \( f'(x) \) allows us to determine how the function changes as \( x \) changes.
In this specific case, we used the quotient rule for differentiation, which is essential when you have a division of two functions. The quotient rule states: if you have a function \( \frac{u(x)}{v(x)} \), its derivative is \( \frac{u'(x)v(x) - u(x)v'(x)}{[v(x)]^2} \). By applying this rule, we isolate how each part contributes to the overall change of \( f(x) \). Calculating \( f'(x) \) for our given function involves differentiating the numerator and the denominator separately before combining them as the rule suggests.

Critical Points

Critical points are specific points on a graph where the derivative of a function is zero or undefined. At these points, the behavior of the function might change. They are important because they often correspond to local maxima or minima or to points where the function changes from increasing to decreasing, or vice versa. In this exercise, we identify the critical points by solving the equation \( f'(x) = 0 \).
When the derivative equals zero, the tangent to the curve is horizontal, indicating potential maxima or minima. Finding these points gives insight into the overall structure and behavior of the function. For complex functions like the one provided, finding these analytically may require setting complex calculations or numerical analysis using computational tools, due to the intricate nature of the derivative.

Function Behavior

Understanding the behavior of a function means knowing how the function behaves across different intervals of the domain. This includes identifying which sections of the function's curve are increasing, decreasing, and where any peaks or troughs occur. By analyzing the derivative, we can infer this behavior.
The derivative provides information on the function's slope: where \( f'(x) > 0 \), the function is increasing, and where \( f'(x) < 0 \), it is decreasing. Studying how the derivative transitions from positive to negative, or vice versa, helps identify turning points and understand the overall shape of the graph. In this case, using the intervals from the original function, we can draw a detailed image of what the function does at various sections across the domain \([-3, 1.2]\).

• Increasing behavior indicates that the values of \(f(x)\) are getting larger as \(x\) increases.
• Decreasing behavior, conversely, shows that \(f(x)\) values are getting smaller.

This information is essential for graph sketching and real-world applications where predicting a function's behavior is crucial.

Increasing and Decreasing Functions

An increasing function is one that ascends as you move from left to right along the x-axis, while a decreasing function descends. The derivative plays a crucial role in identifying these intervals. When the derivative \( f'(x) \) is positive over an interval, the function is considered increasing in that interval. Conversely, when \( f'(x) \) is negative, the function is decreasing.
To determine these intervals in the given problem, we calculate where the derivative is greater than or less than zero within the specified range \([-3, 1.2]\). Recognizing these points lets us fill out the table provided in the exercise, effectively charting where the function rises and falls. This understanding is pivotal in analyzing how outputs grow or shrink in response to changes in input, providing valuable insights into the nature of the function and predicting future behavior.