Problem 13 - College Algebra

Question

For the following exercises, find the average rate of change of each function on the interval specified for real numbers \(b\) or \(h\) in simplest form. $$ j(x)=3 x^{3} \text { on }[1,1+h] $$

Step-by-Step Solution

Answer

The average rate of change is \(9 + 9h + 3h^2\).

Step 1: Understand the formula for average rate of change

The average rate of change of a function \(j(x)\) from \(x=a\) to \(x=b\) is given by the formula: \[ \frac{j(b) - j(a)}{b-a} \] . We are asked to find the average rate of change of the function \(j(x) = 3x^3\) from \(x=1\) to \(x=1+h\).

Step 2: Evaluate the function at the endpoints of the interval

Compute \(j(1)\) and \(j(1+h)\) using the function definition. - \(j(1) = 3(1)^3 = 3\) - \(j(1+h) = 3(1+h)^3 = 3(1 + 3h + 3h^2 + h^3) = 3 + 9h + 9h^2 + 3h^3\).

Step 3: Substitute into the average rate of change formula

Substitute the values from Step 2 into the average rate of change formula and perform the subtraction:\[\frac{3 + 9h + 9h^2 + 3h^3 - 3}{(1+h) - 1} = \frac{9h + 9h^2 + 3h^3}{h}\].

Step 4: Simplify the expression

Cancel \(h\) in the numerator and denominator in the expression:\[9 + 9h + 3h^2 \]. So, the average rate of change is \(9 + 9h + 3h^2\).

Key Concepts

Functions

Functions are critical building blocks in mathematics. A function is like a machine where you input a number, it processes it, and outputs another number. Essentially, a function describes a relationship between two sets of numbers. In our example, the function is expressed as \( j(x) = 3x^3 \). Here, \( x \) is the input, and \( j(x) \) - or occasionally denoted as the output - is the result after applying the function on \( x \). Functions behave consistently based on their definitions. For instance, at \( x = 1 \), we simply replace \( x \) with \( 1 \) in \( 3x^3 \) and get 3. This same substitution principle applies at any point along the interval.When dealing with functions, it is important to understand:

• How the inputs and outputs are related.
• What happens to the function when the input changes slightly, which directly ties into the average rate of change concept.

Understanding these relationships helps us make predictions and analyze different scenarios in both practical and theoretical frameworks.

Calculus

Calculus is a branch of mathematics focused on change. One of its core ideas is understanding how things change and accumulate over time or space. The average rate of change is a fundamental concept in calculus. Imagine tracking how a plant grows over a week. The average rate of change calculates how much the plant's height increased divided by the duration it took. For functions like \( j(x) = 3x^3 \), we look at how the output value \( j(x) \) changes as we adjust \( x \) from \( 1 \) to \( 1 + h \).With calculus, you learn to:

• See how functions behave as inputs change, emphasizing growth and decay.
• Determine how these changes accumulate over particular intervals.

Average rate of change helps set the stage for more advanced concepts like derivatives, where calculus explores change on infinitely small scales. In our exercise, we learn to express this change on the interval \([1, 1 + h]\). By finding the difference: \( \frac{j(1+h) - j(1)}{h} \), we see an example of how calculus starts to quantify and picture change.

Polynomials

Polynomials are a type of mathematical expression that are made up of variables raised to whole numbers (like \(x^3\)), coefficients (like \(3\)), and the operation of addition. They can appear simple, like \( j(x) = 3x^3 \), or far more complex with multiple terms and higher degrees.Working with polynomials involves understanding a few basics:

• Each term in a polynomial consists of a coefficient and a degree, which is the power to which the variable is raised.
• They are heavily utilized in calculus since they are easy to manipulate and differentiate.

Understanding a polynomial function involves calculating its values for different inputs and observing its behavior over intervals, much like our task to find the average rate of change over \([1, 1 + h]\).In the exercise, we observe how changes in \(x\) affect \( j(x) \), unveiling the multiplying effect of \( h \) on \( j(x) = 3\). This action reveals how polynomials evolve and fluctuate, serving as a stepping stone to learning more about curve sketching and polynomial roots in calculus.