Problem 53 - Calculus with Concepts in Calculus

Question

Graph each pair of functions. Shade the region(s) the graphs enclose. \(f(x)=x^{3}+x\) and \(g(x)=3 x^{2}-x(\) Hint \(:\) The region has two parts.)

Step-by-Step Solution

Answer

Shade between \(x = 0\) and \(x = 2\), switching top functions at \(x = 1\).

Step 1: Sketch each function

First, identify the type and shape of each function. The function \(f(x) = x^3 + x\) represents a cubic curve, and \(g(x) = 3x^2 - x\) represents a quadratic parabola. Plot these functions separately on the same coordinate plane, accurately noting their behavior (cubic functions will pass through the origin and quadratics may open upwards or downwards).

Step 2: Find Intersection points

Set the equations equal to each other to find the points of intersection, \(x^3 + x = 3x^2 - x\). Rearrange the equation as \(x^3 - 3x^2 + 2x = 0\). Factor by taking out the common factor, \(x(x^2 - 3x + 2) = 0\), which gives \(x = 0\) and \(x^2 - 3x + 2 = 0\). Solve for \(x\) using the quadratic formula or factoring to get \(x = 1\) and \(x = 2\). Thus, the intersection points are \((0, 0)\), \((1, 0)\), and \((2, 2)\).

Step 3: Determine the regions to shade

Examine the graph to see the areas enclosed. Between \(x = 0\) and \(x = 1\), and from \(x = 1\) to \(x = 2\), look for areas bounded by \(f(x)\) and \(g(x)\). Check which function lies above the other by plugging in sample points within these intervals. Between \(x = 0\) and \(x = 1\), \(g(x)\) is above \(f(x)\). Between \(x = 1\) and \(x = 2\), \(f(x)\) is above \(g(x)\). Shade these regions.

Step 4: Check the correctness of the Shaded Region

Verify visually and algebraically that the shaded regions correspond to enclosed areas between the graphs in the provided intervals. Calculate the difference between the functions and ensure their signs match the expectations from earlier analysis: \(x^3 + x - (3x^2 - x)\) simplifies to \(x^3 - 3x^2 + 2x\), confirming the location and orientation of regions.

Key Concepts

Intersection Points

Finding the intersection points between two functions is crucial when determining enclosed areas on a graph. The intersection points tell us where the functions intersect or cross each other. Start by setting the two function equations equal to each other; in our case, that means solving the equation \(x^3 + x = 3x^2 - x\).

• First, rearrange the equation to \(x^3 - 3x^2 + 2x = 0\).
• This can be factored by taking out the common factor \(x\), giving us \(x(x^2 - 3x + 2) = 0\).

This results in potential solutions: \(x = 0\), \(x = 1\), and \(x = 2\). These x-values are where the cubic and quadratic functions meet.
To confirm these are intersection points, substitute these \(x\) values back into the original equations to find the corresponding \(y\) values. You’ll obtain points like \((0, 0)\), \((1, 0)\), and \((2, 2)\). Incorporating these points is essential for analyzing the overall behavior of the graphs.

Cubic and Quadratic Functions

Cubic functions, like \(f(x) = x^3 + x\), often appear as smooth, continuous curves that can rise or fall as they pass through the origin. In contrast, quadratic functions like \(g(x) = 3x^2 - x\) form parabolas, which are symmetrical and can open upwards or downwards depending on their coefficients.
Understanding the basic shapes and properties of these functions is key:

• The cubic function \(f(x)\) starts with a gentle slope through the origin and gradually steepens.
• Quadratic \(g(x)\) depicts a classic "U" shape.

Knowing this, you can identify how each function behaves across a given interval. For instance, near the origin, a cubic function’s leading term \(x^3\) plays a strong role in determining its initial direction.

Graphing Techniques

Graphing is an essential skill for visualizing the behavior of functions and identifying enclosed regions. When sketching functions like our cubic and quadratic, consider these graphing techniques:

• Plot key points and intersections first. This provides a framework for placing the curves accurately on the graph.
• Use the derived formula \(f(x) - g(x) = x^3 - 3x^2 + 2x\) to analyze which function dominates across intervals. This difference helps visualize where one graph is above or below another.

By accurately plotting these key points, you create reliable graphs. From there, observe trends like increasing or decreasing behavior, concavity, and symmetry. This deep understanding enhances your ability to pinpoint regions enclosed by multiple curves or functions.

Function Behavior Analysis

Analyzing function behavior involves determining how functions change over specific intervals. We'll look at regional behavior differences between \(f(x)\) and \(g(x)\):

• Over the interval \(0 \leq x \leq 1\), evaluate a few sample points. Here \(g(x)\) tends to lie above \(f(x)\).
• From \(x = 1\) to \(x = 2\), \(f(x)\) rises above \(g(x)\).

Function behavior analysis helps determine where shading occurs on the graph since it analyses which function is greater.
Through step-by-step analysis:

• Substitute \(x\) values into both function equations to see which yields a larger \(y\).
• Plot differences as mentioned to confirm region shading.

This process enables students to visually and numerically identify regions enclosed by specific function pairs across specified intervals.