Problem 21 - Calculus for Biology and Medicine

Question

Use a graphing calculator to graph \(f(x)=x^{2}, x \geq 0\), and \(g(x)=x^{3}, x \geq 0\), together. For which values of \(x\) is \(f(x)>g(x)\) and for which is \(f(x)

Step-by-Step Solution

Answer

For \(0 < x < 1\), \(f(x) > g(x)\). For \(x > 1\), \(f(x) < g(x)\).

Step 1: Graph the Functions

Use a graphing calculator to input the functions \(f(x) = x^2\) and \(g(x) = x^3\). Make sure to set the domain for both functions as \(x \geq 0\). Observe where these graphs intersect and their behavior for different \(x\) values.

Step 2: Find Intersection Points

Look at the graph to find the intersection point(s) of \(f(x) = x^2\) and \(g(x) = x^3\). Notice that they intersect at \(x = 0\) and \(x = 1\) because that is where the values of the two functions are equal.

Step 3: Analyze Intervals between Intersection Points

Analyze the intervals around the intersection points. For \(0 < x < 1\), observe that \(f(x) > g(x)\) because the parabola \(f(x) = x^2\) is above the cubic graph \(g(x) = x^3\). For \(x > 1\), observe that \(f(x) < g(x)\), since the cubic function starts increasing more rapidly than the quadratic function.

Key Concepts

Function Intersection

A function intersection is the point or points where two or more functions meet on a graph. In this scenario, we are interested in the intersections of the quadratic function, \(f(x) = x^2\), and the cubic function, \(g(x) = x^3\), but constrained to the domain \(x \geq 0\). For these functions, the intersections occur where their values are equal, which is when \(x^2 = x^3\). By solving the equation, we find the intersections at \(x = 0\) and \(x = 1\). This means that at these points, both functions share the same value on the graph. These intersections are foundational when analyzing how the functions behave relative to each other in other intervals.

Function intersections are crucial because they highlight points where the output from two functions is identical. Understanding these points helps in determining which function produces greater values over specific intervals.

Quadratic Function

A quadratic function is a type of polynomial function that can be represented as \(f(x) = ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants, and \(a eq 0\). In our case, \(f(x) = x^2\) is a simple quadratic function where \(a = 1\), \(b = 0\), and \(c = 0\).

Quadratic functions graph into a U-shaped curve known as a parabola. This curve opens upwards when \(a\) is positive, as in \(f(x) = x^2\). Working with quadratics, especially when graphed, requires attention to their parabolic shape, vertex, and symmetry. When we set the domain to \(x \geq 0\), we only look at the right half of the parabola.

The nature of quadratic functions is that they grow at a rate that’s proportional to the square of the increase in \(x\). Thus, between the intersection points \(x = 0\) and \(x = 1\), the quadratic function is above the cubic function, making \(f(x) > g(x)\).

Cubic Function

A cubic function can be expressed as \(g(x) = ax^3 + bx^2 + cx + d\), with \(a eq 0\). Our function, \(g(x) = x^3\), simplifies to a monomial, where \(a = 1\), \(b = 0\), \(c = 0\), and \(d = 0\).

This type of function is graphed as a curve with an S-shape, characterized by its inflection point, which in our example is at the origin \((0,0)\). The behavior of cubic functions is distinct; they tend to increase more rapidly than quadratic functions as \(x\) becomes large, especially for \(x > 1\).

For \(x > 1\), \(g(x) = x^3\) becomes larger than \(f(x) = x^2\), making \(f(x) < g(x)\). Cubic functions, as they extend further out along the x-axis, surpass quadratic functions due to their growing rate based on the cube of \(x\), showing a more pronounced and steeper climb.