Problem 54 - Calculus Early Transcendentals

Question

Give a graph of the polynomial and label the coordinates of the intercepts, stationary points, and inflection points. Check your work with a graphing utility. $$ p(x)=2-x+2 x^{2}-x^{3} $$

Step-by-Step Solution

Answer

Intercepts: (0,2); Stationary: \((\frac{1}{3}, \frac{56}{27})\), (1,2); Inflection: \((\frac{2}{3}, \frac{52}{27})\).

Step 1: Find the Intercepts

First, determine where the graph intersects the axes. To find x-intercepts, solve for \( p(x) = 0 \): \[ 2 - x + 2x^2 - x^3 = 0 \]. This polynomial may be solved using numerical methods or graphing tools as it is non-trivial to factor analytically. For y-intercept, substitute \( x = 0 \): \[ p(0) = 2 \]Hence, the y-intercept is at \((0, 2)\).

Step 2: Find Stationary Points

To find stationary points (where the derivative equals zero), find the derivative: \[ p'(x) = -1 + 4x - 3x^2 \].Set the derivative to zero and solve: \[ -1 + 4x - 3x^2 = 0 \equiv 3x^2 - 4x + 1 = 0 \].Solve using the quadratic formula: \( x = \frac{-b \pm \sqrt{b^2-4ac}}{2a} = \frac{4 \pm \sqrt{4}}{6} \), resulting in \( x = \frac{1}{3} \) and \( x = 1 \).Calculate \( p\left(\frac{1}{3}\right) \) and \( p(1) \) to get the y-coordinates of these points.

Step 3: Determine Inflection Points

To find inflection points (where the second derivative changes sign), find the second derivative:\[ p''(x) = 4 - 6x \].Set the second derivative to zero and solve: \[ 4 - 6x = 0 \equiv x = \frac{2}{3} \].Substitute back into the original function \( p(x) \) to get the corresponding \( y \)-value, \( p\left(\frac{2}{3}\right) \).

Step 4: Graphing Utility Verification

Graph the polynomial using a graphing utility. Plot the intercepts, stationary points, and the inflection point. Verify that these points correctly match the locations on the graph. Adjust for any calculations if discrepancies arise.

Key Concepts

Intercepts in Polynomial Graphs

Intercepts are the points where a graph intersects the axes. These are crucial in understanding the behavior of polynomials. For the polynomial function \( p(x) = 2 - x + 2x^2 - x^3 \), finding these intercepts is the first step to graphing.

• **X-Intercepts:** These occur when \( p(x) = 0 \). For complex polynomials, numerical methods or graphing tools may be necessary as they can be challenging to factor by hand.
• **Y-Intercept:** This is found by evaluating the polynomial at \( x = 0 \). For our polynomial, it simplifies to \( p(0) = 2 \), resulting in a y-intercept at \((0, 2)\).

Intercepts give a basic framework for sketching the graph. While y-intercepts are straightforward, x-intercepts may need more advanced techniques when they do not simplify easily.

Stationary Points

Stationary points occur where the slope of the graph changes direction, i.e., when the first derivative of the function equals zero. These points can either be maxima, minima, or points of horizontal inflection.

• **Finding Derivative:** For the polynomial \( p(x) = 2 - x + 2x^2 - x^3 \), the first derivative is \( p'(x) = -1 + 4x - 3x^2 \).
• **Solve for Zero:** Set the derivative equal to zero: \(-1 + 4x - 3x^2 = 0\). Solving gives the x-values \( x = \frac{1}{3} \) and \( x = 1 \).
• **Determine Y-Coordinates:** Calculate the y-coordinates by substituting back into the original polynomial, finding \( p\left(\frac{1}{3}\right) \) and \( p(1) \).

Stationary points provide insight into the behavior of the graph at critical values, letting you identify potential peaks and troughs.

Inflection Points

Inflection points are where the graph changes its curvature. This is identified by taking the second derivative and noting where it changes sign.

• **Calculate Second Derivative:** For our function, the second derivative is \( p''(x) = 4 - 6x \).
• **Set Second Derivative to Zero:** Finding \( 4 - 6x = 0 \) gives \( x = \frac{2}{3} \).
• **Find Corresponding Y-Value:** Substitute this back into the original polynomial to obtain the y-coordinate, \( p\left(\frac{2}{3}\right) \).

Identifying inflection points helps understand how the graph curves, showing transitions between concave upwards and downwards.