Problem 54 - Calculus with Concepts in Calculus

Question

Graph each pair of functions. Shade the region(s) the graphs enclose. $$ f(x)=x^{3}+x^{2}+1 \text { and } g(x)=x^{3}+x+1 $$

Step-by-Step Solution

Answer

Shade the region between the curves from \( x = 0 \) to \( x = 1 \), where \( f(x) > g(x) \).

Step 1: Understand the Functions

We are given two functions, \( f(x) = x^3 + x^2 + 1 \) and \( g(x) = x^3 + x + 1 \). Both are cubic functions but differ in the middle term.

Step 2: Determine Intersection Points

Set the functions equal to each other to find their intersection: \( x^3 + x^2 + 1 = x^3 + x + 1 \). Simplifying, we get \( x^2 = x \), which gives \( x(x - 1) = 0 \). Thus, the intersection points are at \( x = 0 \) and \( x = 1 \).

Step 3: Evaluate Functions at Intersection Points

Calculate \( f(x) \) and \( g(x) \) at the intersection points. For \( x = 0 \), both functions equal \( 1 \). At \( x = 1 \), both functions equal \( 3 \). Hence, the coordinates of the intersection points are \((0, 1)\) and \((1, 3)\).

Step 4: Graph the Functions

Sketch the graphs of \( f(x) = x^3 + x^2 + 1 \) and \( g(x) = x^3 + x + 1 \) on the same coordinate plane. Notably, observe that the graphs intersect at the points \((0, 1)\) and \((1, 3)\). Function \( f(x) \) is above \( g(x) \) between these points.

Step 5: Shade the Enclosed Region

Identify the region between the graphs from \( x = 0 \) to \( x = 1 \). Since \( f(x) > g(x) \) in this interval, shade the area between the two graphs, which represents the region enclosed by the two curves.

Key Concepts

Intersection Points

In mathematics, intersection points are the locations where two curves or lines meet on a graph. Finding these points is crucial for understanding where two functions coincide.

• For this exercise, we set the functions equal: \( f(x) = x^3 + x^2 + 1 \) and \( g(x) = x^3 + x + 1 \).
• Simplifying gives \( x^3 + x^2 + 1 = x^3 + x + 1 \), which can further be simplified to \( x^2 = x \).
• Solving \( x^2 = x \) leads to \( x(x - 1) = 0 \), resulting in intersection points at \( x = 0 \) and \( x = 1 \).

Evaluating the functions at these points confirms our intersection locations: both functions output \( 1 \) at \( x = 0 \) and \( 3 \) at \( x = 1 \), leading to the coordinates \((0, 1)\) and \((1, 3)\). Determining these points allows us to begin exploring how the functions interact, which is key in visualizing where they overlap and how to approach graphing them.

Cubic Functions

Cubic functions are polynomial expressions where the highest power of the variable is three. These functions can create various graph shapes, often with curves reminiscent of an "S" shape, which can be more or less pronounced depending on the coefficients of the terms.

• The general form of a cubic function is \( ax^3 + bx^2 + cx + d \). For the given functions, \( f(x) = x^3 + x^2 + 1 \) and \( g(x) = x^3 + x + 1 \), the highest power of the variable is three.
• As these are both cubic functions, they may intersect at one or more points, which can involve solving a quadratic equation, as seen in the intersection solution.

Despite having the same cubic profile \( (x^3) \), these functions differ by their quadratic and linear terms, affecting how they navigate the graph. Consequently, understanding the behavior and intersection of cubic functions is essential, as it impacts the potential enclosed regions and how effectively we can visualize their interaction on a graph.

Enclosed Regions

Enclosed regions refer to the area between two curves on a graph where one function lies above the other. Identifying these regions is important for tasks like calculating area or understanding function behavior within specific domains.

• To find enclosed regions, examine where one function overtakes the other between determined points of intersection. Here, \( f(x) = x^3 + x^2 + 1 \) is above \( g(x) = x^3 + x + 1 \) from \( x = 0 \) to \( x = 1 \).
• Shading the region helps to visually represent the difference between the two functions over this domain.

This shaded region from \( x = 0 \) to \( x = 1 \) shows how the functions encase a part of the coordinate plane, delineating a space that is influenced by the interaction of two cubic functions. Understanding these enclosed areas is vital, especially in calculus, where they might be used to find integrals that determine the specific area between the curves.