Problem 34 - College Algebra

Question

For the following exercises, find the average rate of change of each function on the interval specified. $$k(t)=6 t^{2}+\frac{4}{t^{3}} \text { on }[-1,3]$$

Step-by-Step Solution

Answer

The average rate of change is \(\frac{352}{27}\).

Step 1: Identify the formula for the average rate of change

The average rate of change of a function over an interval \([a, b]\) is found using the formula: \[\text{Average Rate of Change} = \frac{k(b) - k(a)}{b - a}\]. In this problem, \(a = -1\) and \(b = 3\).

Step 2: Calculate \(k(-1)\)

Substitute \(t = -1\) into the function \(k(t) = 6t^2 + \frac{4}{t^3}\). This gives: \[k(-1) = 6(-1)^2 + \frac{4}{(-1)^3} = 6(1) - 4 = 6 - 4 = 2.\]

Step 3: Calculate \(k(3)\)

Substitute \(t = 3\) into the function \(k(t) = 6t^2 + \frac{4}{t^3}\). This gives: \[k(3) = 6(3)^2 + \frac{4}{(3)^3} = 6(9) + \frac{4}{27} = 54 + \frac{4}{27}.\] Simplify \(\frac{4}{27}\) to get a precise value or leave it as a fraction if preferred.

Step 4: Calculate the Average Rate of Change

Use the formula from Step 1: \[\text{Average Rate of Change} = \frac{k(3) - k(-1)}{3 - (-1)}\]. Substituting the calculated values: \[\text{Average Rate of Change} = \frac{\left(54 + \frac{4}{27}\right) - 2}{3 + 1} = \frac{54 + \frac{4}{27} - 2}{4}.\]First, simplify the numerator: \[54 - 2 + \frac{4}{27} = 52 + \frac{4}{27}.\]Convert \(52\) to fraction form \( \frac{1404}{27} + \frac{4}{27} = \frac{1408}{27}\).Divide by 4: \[\frac{1408}{27} \div 4 = \frac{1408}{108} = \frac{352}{27}.\]

Step 5: Simplify the Fraction

Finally, simplify \(\frac{352}{27}\) if possible. The fraction is already in its simplest form given that there are no common factors to simplify further.

Key Concepts

Function Evaluation

Function evaluation is simply about finding the value of a function at a specific point. To evaluate a function, replace the variable in the function with a specific number. This can give a quick understanding of how the function behaves. In our problem, we evaluated the function \(k(t) = 6t^2 + \frac{4}{t^3}\) at points \(t = -1\) and \(t = 3\).
Here's how you do it:

• Substitute the given value into the function
• Perform the arithmetic operations according to mathematics rules
• The result is the function’s value at that point

Understanding function evaluation is crucial because it forms the foundation for more complex mathematical processes like finding the average rate of change.

Interval Notation

Interval notation provides a concise way to describe a range of numbers, or an interval, on the real number line. It's often used in calculus and algebra to specify the domain of a function or the points over which you want to analyze the function's behavior. In this exercise, the interval is written as \([-1, 3]\). Here's what the notation means:

• The square bracket \(\left[ \right]\) signifies that the endpoints are included in the interval. This is called a closed interval.
• The smallest number \(-1\) is the starting point, and the largest number \(3\) is the endpoint.

Understanding intervals is important as it outlines the specific part of a function you are focusing on. This focus can be crucial when discussing limits, continuity, and the concept of derivatives.

Polynomial Functions

Polynomial functions are algebraic expressions that involve variables raised to whole number exponents. They are summed together and typically appear in the form \(a_nx^n + a_{n-1}x^{n-1} + ... + a_1x + a_0\), where each \(a_n\) is a coefficient. In our function \(k(t) = 6t^2\), the part \(6t^2\) is a polynomial term.
Key features of polynomial functions include:

• They are continuous and smooth, meaning the graph can be drawn without lifting the pen.
• They have a finite number of "turns" or changes in direction, known as critical points.
• The degree of the polynomial is the highest power of the variable, influencing the shape of the graph.

This understanding is crucial, especially when dealing with problems on the average rate of change, as it helps predict the function’s behavior within the specified interval.

Rational Functions

Rational functions are ratios of two polynomials. In the function \(k(t) = 6t^2 + \frac{4}{t^3}\), the term \(\frac{4}{t^3}\) represents a rational component.
Main characteristics of rational functions are:

• They can have discontinuities, like vertical asymptotes, where the function is undefined.
• The behavior of the function can significantly change around points where the denominator is zero.
• They often have horizontal or oblique asymptotes, which describe the end behavior of the function.

Understanding rational functions is essential, as they are common in many calculus applications. The presence of rational terms in a function means you need to be cautious about restrictions due to undefined parts of the domain.