Problem 2 - Thomas Calculus

Question

Use graphing software to determine which of the given viewing windows displays the most appropriate graph of the specified function. \(\begin{equation} f(x)=x^{3}-4 x^{2}-4 x+16 \end{equation} \begin{equation} \begin{array}{ll}{\text { a. }[-1,1] \text { by }[-5,5]} & {\text { b. }[-3,3] \text { by }[-10,10]} \\ {\text { c. }[-5,5] \text { by }[-10,20]} & {\text { d. }[-20,20] \text { by }[-100,100]}\end{array} \end{equation}\)

Step-by-Step Solution

Answer

Option b \([-3,3]\) by \([-10,10]\) is the most appropriate viewing window.

Step 1: Identify Turning Points and Asymptotes

Before plotting, analyze the function \(f(x) = x^3 - 4x^2 - 4x + 16\) to find critical points, which might indicate regions of interest. First, find the first derivative \(f'(x)\). Solve \(f'(x) = 0\) to find critical points and examine changes in intervals to understand the behavior of the graph within the window options.

Step 2: First Derivative Calculation

Compute the derivative of the function: \(f'(x) = 3x^2 - 8x - 4\). This derivative comes from applying standard differentiation rules to \(f(x) = x^3 - 4x^2 - 4x + 16\).

Step 3: Solve for Critical Points

Set \(f'(x) = 0\): \[3x^2 - 8x - 4 = 0\]Use the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\) to solve for \(x\). Find the discriminant \(b^2 - 4ac = (-8)^2 - 4(3)(-4) = 64 + 48 = 112\). This discriminant is positive, indicating two real roots.

Step 4: Calculate Real Roots

Using \(a = 3\), \(b = -8\), \(c = -4\), find\[x = \frac{-(-8) \pm \sqrt{112}}{6}\]\[x = \frac{8 \pm 10.583}{6}\]This calculates to approximate roots, \(x \approx 3.097\) and \(x \approx -0.431\). These are critical points which suggest where curve changes concavity.

Step 5: Determine Appropriate Viewing Window

Considering roots \(x \approx 3.097\) and \(x \approx -0.431\), and since \(f(x)\) is polynomial, it will extend towards infinity for extreme values of \(x\). Analyze window coverage: - Option a only partially shows the critical points.- Option b covers both roots well and maintains detail in the significant areas.- Option c includes possible relevant points, especially vertical diversity.- Option d covers broadly but can obscure detail near roots. Option b \([-3,3]\) includes the relevant curve without losing detail to excessive scale.

Key Concepts

Understanding Critical Points

Critical points in a polynomial function like \(f(x) = x^3 - 4x^2 - 4x + 16\) are points on the graph where the function reaches a local maximum or minimum, or may even plateau. These points occur when the first derivative \(f'(x)\) is equal to zero.
To find these critical points, we solve \(f'(x) = 0\). Critical points help in dividing the graph of the function into different sections where the function might increase or decrease. By identifying these points, we gain insight into the behaviour of the polynomial and decide the best window of observation that makes it easy to draw key insights.

• Local Maximum: The function transitions from increasing to decreasing.
• Local Minimum: The function changes from decreasing to increasing.
• Plateaus: A flat segment where the slope is zero but no change in direction occurs.

For the given polynomial, solving the derivative helps identify these key areas, which often serve as landmarks where an important behaviour of the function changes.

First Derivative Analysis

The first derivative analysis involves computing the derivative of a function and investigating its sign changes to infer the behaviour of the function over an interval.
For the function \(f(x) = x^3 - 4x^2 - 4x + 16\), the derivative is \(f'(x) = 3x^2 - 8x - 4\).
By setting \(f'(x) = 0\) and solving, we can pinpoint critical points. The roots found, approximately \(x \approx 3.097\) and \(x \approx -0.431\), mark where the function shifts from increasing to decreasing, or vice versa.

• Positive Derivative: Implies the original function is increasing over that interval.
• Negative Derivative: Indicates the original function is decreasing.

This analysis gives a detailed overview of where the function's tendencies to rise or fall occur, thus enabling one to appreciate the function’s global shape and aiding in selecting an optimal viewing window.

Solving with the Quadratic Formula

The quadratic formula is a crucial tool for finding the roots of a quadratic equation. In our context, it aids in solving \(3x^2 - 8x - 4 = 0\), derived from the first derivative of our polynomial function. The quadratic formula is given by:\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]To solve our equation, we identify:

• \(a = 3\)
• \(b = -8\)
• \(c = -4\)

With these, we calculate the discriminant: \[b^2 - 4ac = (-8)^2 - 4(3)(-4) = 112\]A positive discriminant indicates two distinct real solutions. Solving gives us:

• \(x = \frac{8 \pm 10.583}{6}\)
• Approximate roots: \(x \approx 3.097\) and \(x \approx -0.431\)

These solutions highlight points where the function's slope is zero, signifying crucial turning points critical for efficiently graphing the function and choosing an appropriate viewing window.