Question

In Exercises \(1-6,\) find the average rate of change of the function over each interval. \(f(x)=x^{3}+1\) (a) $$[2,3] \quad$$ \((b)\lceil- 1,1]\)

Short answer

The average rate of change of the function \(f(x) = x^3 + 1\) over the interval [2,3] is 19, and over the interval [-1, 1] is 1.

Step-by-step solution

Average rate of change for the interval [2,3]

We find the average rate of change of \(f(x) = x^3 + 1\) over the interval [2,3].We use the formula \(m_{avg}=\frac{f(b) - f(a)}{b-a}\).Here, \(a = 2\), \(b = 3\), \(f(a) = f(2) = 2^3 + 1 = 9\), and \(f(b) = f(3) = 3^3 + 1 = 28\).Substitute the values into the formula to get \(m_{avg} = \frac{28 - 9}{3 - 2} = 19\).

Average rate of change for the interval [-1, 1]

We find the average rate of change of \(f(x) = x^3 + 1\) over the interval [-1, 1].We use the formula \(m_{avg}=\frac{f(b) - f(a)}{b-a}\).Here, \(a = -1\), \(b = 1\), \(f(a) = f(-1) = (-1)^3 + 1 = 0\), and \(f(b) = f(1) = 1^3 + 1 = 2\).Substitute the values into the formula to get \(m_{avg} = \frac{2 - 0}{1 - (-1)} = 1\).

Key concepts

Calculus

Calculus is a branch of mathematics that deals with rates of change and the accumulation of quantities. It's built on two foundational concepts: differentiation and integration. Differentiation measures how a function changes as its input changes, giving rise to the concept of a derivative, which can be interpreted as an instantaneous rate of change. Integration, on the other hand, is about the accumulation of quantities, leading to the integral which measures the total accumulation of some quantity, such as area under a curve. Both elements are crucial for understanding how things change over time and space, making calculus essential in physics, engineering, economics, and more.

When students begin studying calculus, they often start with the concept of average rate of change before moving on to instantaneous rates of change, which are captured by the derivative. Understanding the average rate of change provides a solid foundation for the infinitesimal concepts that follow.

Average Rate of Change Formula

The average rate of change of a function is analogous to the slope of the secant line that intersects two points on the graph of the function. In essence, it measures how much the function's output (or 'y' value) changes per unit change in the input (or 'x' value) over a given interval. This is expressed mathematically by the formula:
\[ m_{avg} = \frac{f(b) - f(a)}{b - a} \]
where \(m_{avg}\) is the average rate of change, \(f(a)\) is the value of the function at the start of the interval, and \(f(b)\) is the value of the function at the end of the interval. This straightforward calculation is used widely to approximate the behavior of more complex functions over specified intervals. For students, mastering this formula is a key step in understanding variable change within different contexts.

Cubic Function

A cubic function is a type of polynomial that has the general form
\[ f(x) = ax^3 + bx^2 + cx + d \]
where the coefficients \(aeq0\), \(b\), \(c\), and \(d\) are real numbers, and \(x\) is the variable. The cubic term \(ax^3\) gives the function its name and its distinctive shape, including the possibility of having one, two, or three real roots, local maximum and minimum points, and points of inflection. These functions can model various real-world phenomena such as the volume of an expanding gas, the displacement of a particle over time, or the relationship between supply and demand in economics. Understanding cubic functions is important for analyzing complex relationships where variables are raised to higher powers.

Intervals

In mathematics, an interval represents a set of numbers lying between two endpoints. Intervals can be classified based on whether they include or exclude their endpoints. An interval including its endpoints is known as a closed interval, denoted as \([a, b]\), where 'a' and 'b' are the endpoints. Conversely, an open interval, denoted as \((a, b)\), excludes its endpoints. There are also half-open or half-closed intervals, like \([a, b)\) or \((a, b]\), which include one endpoint but exclude the other.

Intervals are fundamental in discussing domains and ranges of functions, continuity, and the behavior of functions within certain bounds. For instance, finding the average rate of change over an interval gives a snapshot of the function's performance between two specific input values, a concept frequently applied in various scientific and economic contexts.