Problem 8 - Thomas Calculus

Question

In Exercises 5–30, determine an appropriate viewing window for the given function and use it to display its graph. $$ f(x)=4 x^{3}-x^{4} $$

Step-by-Step Solution

Answer

The appropriate viewing window is \(-5 \leq x \leq 5\) and \(-50 \leq y \leq 30\).

Step 1: Analyze the Function

Let's start by identifying the characteristics of the function. The function given is \( f(x) = 4x^3 - x^4 \). It's a polynomial function of degree 4. The highest power term \(-x^4\) indicates the graph will extend downwards as \(x\) goes towards positive and negative infinity. The other term, \(4x^3\), will affect its shape.

Step 2: Find the Critical Points

To determine critical points, first find the derivative of the function: \[ f'(x) = 12x^2 - 4x^3. \]Set the derivative to zero to find critical points:\[ 12x^2 - 4x^3 = 0 \]Factor the equation:\[ 4x^2(3 - x) = 0 \]This gives critical points at \( x = 0 \) and \( x = 3 \).

Step 3: Determine the Function Behavior at Critical Points

Substitute the critical points into the original function to find \(f(x)\) values:\[ f(0) = 4(0)^3 - (0)^4 = 0, \]\[ f(3) = 4(3)^3 - (3)^4 = 108 - 81 = 27. \]Additionally, Observe the end behavior:As \( x \to \pm \infty \), \( f(x) \to -\infty \) due to the \(-x^4\) term.

Step 4: Propose a Viewing Window

Based on the critical points and behavior, choose a viewing window that covers these points and the function's movement towards infinity. A window from \(-5\) to \(5\) for \(x\) and \(-50\) to \(30\) for \(y\) captures the key features of the graph, including inflection points and intercepts with a good amount of space to understand the function's behavior.

Step 5: Display the Graph

Finally, using a graphing tool, enter the function and use the proposed viewing window: \((-5, 5)\) for \(x\) and \((-50, 30)\) for \(y\). Check if all critical features of the function like peaks, valleys, and asymptotic behavior are clearly displayed.

Key Concepts

Understanding Critical Points

In mathematics, critical points of a function are where its derivative equals zero or is undefined, giving us vital insight into the function's behavior. With the function \(f(x) = 4x^3 - x^4\), we calculated its derivative and set it to zero, resulting in critical points at \(x = 0\) and \(x = 3\). These points indicate where the rate of change of the function temporarily pauses, signaling potential maximums, minimums, or points of inflection. To delve deeper:

• Local maximum: A point where the function reaches a peak within a nearby neighborhood.
• Local minimum: A point where the function reaches its lowest value within a nearby neighborhood.
• Inflection point: Where the function changes concavity, indicating a shift from a peak to trough, or vice versa.

Substituting these critical points into the original function reveals the output values, giving you a clearer understanding of the function's hills and valleys at these points.

The Role of Derivatives

Derivatives are a fundamental tool in calculus, crucial for assessing the rate at which functions change. In the context of graphing polynomial functions like \(f(x) = 4x^3 - x^4\), derivatives help pinpoint critical points and indicate where a function increases or decreases. Calculating the derivative, \(f'(x) = 12x^2 - 4x^3\), involves applying basic rules of differentiation to each term, offering insight into the behavior of \(f(x)\).
Particularly:

• First derivative test: By evaluating \(f'(x)\), we can determine intervals where the function is increasing or decreasing. Positive values suggest an increasing function, while negative values indicate a decrease.
• Interpreting critical points: When \(f'(x) = 0\), check surrounding values to establish if the points correspond to local maxima or minima.

Derivatives offer a microscopic view of graph behavior, making it possible to predict how polynomial graphs may shift, bend, or even oscillate.

Understanding End Behavior

End behavior in polynomial functions describes how the graph acts as the input \(x\) approaches positive or negative infinity. With the polynomial \(f(x) = 4x^3 - x^4\), the "lead term" \(-x^4\) governs the overall direction as \(x\) expands or contracts to extremes.
For polynomial functions, this term will typically determine:

• Degree of polynomial: The highest power of \(x\) indicates the number of possible tails on the graph as \(x\) goes to infinity.
• Coefficient sign of the lead term: If it's negative, like in \(-x^4\), then the ends head downwards.

Thus, as \(x \to \infty\) and \(x \to -\infty\), both ends of our specific function graph trend downward, making understanding this behavior essential for setting a comprehensive viewing window.