Problem 52 - Algebra and Trigonometry

Question

Based on material learned earlier in the course. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for the final exam. Find the average rate of change of \(f(x)=x^{3}-5 x^{2}+27\) from -3 to 2 .

Step-by-Step Solution

Answer

The average rate of change is 12.

Step 1 - Understand the Problem

We need to find the average rate of change of the function \(f(x) = x^3 - 5x^2 + 27\) over the interval from \(x = -3\) to \(x = 2\). The average rate of change is calculated using the formula: \[\text{Average Rate of Change} = \frac{f(b) - f(a)}{b - a}\]where \(a\) and \(b\) are the endpoints of the interval.

Step 2 - Calculate \(f(a)\)

Substitute \(a = -3\) into the function \[f(-3) = (-3)^3 - 5(-3)^2 + 27 \] Simplify step-by-step: \[-27 - 45 + 27 = -45\]

Step 3 - Calculate \(f(b)\)

Substitute \(b = 2\) into the function \[f(2) = 2^3 - 5 \times 2^2 + 27 \] Simplify step-by-step: \[8 - 20 + 27 = 15\]

Step 4 - Apply the Average Rate of Change Formula

Using the values calculated:\[f(-3) = -45\] and \[f(2) = 15\]Substitute into the formula:\[\text{Average Rate of Change} = \frac{15 - (-45)}{2 - (-3)} = \frac{15 + 45}{2 + 3} = \frac{60}{5} = 12\]

Key Concepts

Functional Analysis

Functional analysis can be a challenging topic for many students. Essentially, it involves understanding how to evaluate and interpret functions. A function can be thought of as a rule that assigns a unique output to each input from a particular set, commonly referred to as the domain. In the step-by-step solution for finding the average rate of change, we are working with a polynomial function. Functional analysis helps us understand the behavior of this function over a specific interval. By evaluating the function at different points within the given interval, we can make sense of how the function changes and draw meaningful conclusions, such as the average rate of change in this exercise.

Polynomial Functions

Polynomial functions are an important subset of mathematical functions. These functions include terms that are non-negative integer powers of the variable—such as square, cubic, or even higher-order terms. In our exercise, the function is given by: f(x) = x^3 - 5x^2 + 27. This includes:

• cubic term: \(x^3\)
• quadratic term: \(-5x^2\)
• constant term: 27

Polynomial functions are fundamental in algebra and calculus, and understanding their characteristics helps solve a variety of mathematical problems.

Interval Calculation

Calculating values over an interval is crucial for understanding how functions behave over a range of inputs. In our given function, we're calculating the average rate of change from \(x = -3\) to \(x = 2\). This means we're interested in how the function value changes as we move from -3 to 2 on the x-axis.

We start by evaluating the function at the endpoints of the interval. Substituting \(x = -3\) and \(x = 2\) into the function gives us the values at these points. Then, using these values, we calculate the difference in function values as well as the difference in x-values. The formula for average rate of change is: \text{Average Rate of Change} = \frac{f(b) - f(a)}{b - a} Through this calculation, we find how much the function's value changes per unit change in x over the specified interval.

Algebraic Simplification

Algebraic simplification involves reducing expressions to more manageable forms. This makes calculations easier and results clearer. In our exercise, simplifying the function

after substitutions is key.

• For \(x = -3\), we simplified: \(f(-3) = (-3)^3 - 5(-3)^2 + 27 = -45\)
• For \(x = 2\), we simplified: \(f(2) = 2^3 - 5 \times 2^2 + 27 = 15\)

Each step involves operating on polynomial terms and combining like terms. The final step, applying the average rate of change formula, further involves simplifying a fraction. All calculations combined lead to an average rate of change of 12, emphasizing the value of algebraic techniques in simplifying solutions and obtaining clear results.