Problem 22 - Algebra and Trigonometry

Question

\(17-28\) A function is given. Determine the average rate of change of the function between the given values of the variable. $$ f(x)=x+x^{4} ; \quad x=-1, x=3 $$

Step-by-Step Solution

Answer

The average rate of change of the function is 21.

Step 1: Understand the Concept

The average rate of change of a function between two points is similar to finding the slope of the line that connects the two points on the function's graph. This is calculated using the formula \( \frac{f(b) - f(a)}{b - a} \) where \( a \) and \( b \) are the given values of \( x \).

Step 2: Substitute the Values into the Formula

We have the function \( f(x) = x + x^4 \), and we need to find \( f(-1) \) and \( f(3) \). Then apply the average rate of change formula with \( a = -1 \) and \( b = 3 \).

Step 3: Calculate \( f(-1) \)

Substitute \( x = -1 \) into the function: \( f(-1) = (-1) + (-1)^4 = -1 + 1 = 0 \).

Step 4: Calculate \( f(3) \)

Substitute \( x = 3 \) into the function: \( f(3) = 3 + 3^4 = 3 + 81 = 84 \).

Step 5: Compute Average Rate of Change

Use the formula: \( \frac{f(3) - f(-1)}{3 - (-1)} = \frac{84 - 0}{3 + 1} = \frac{84}{4} = 21 \).

Key Concepts

Function Evaluation

Function evaluation is a fundamental concept in calculus used for understanding how functions behave at specific points. To evaluate a function, you substitute the input value (usually represented by the variable \( x \)) into the function and calculate the result.

Here's how you would evaluate the function given in the exercise:

• The function is \( f(x) = x + x^4 \).
• To find \( f(-1) \), replace \( x \) with \(-1\) to get: \( f(-1) = (-1) + (-1)^4 = -1 + 1 = 0 \).
• Similarly, for \( f(3) \), replace \( x \) with \( 3 \) and solve: \( f(3) = 3 + 3^4 = 3 + 81 = 84 \).

Evaluating a function in this manner is crucial for determining other properties, such as the average rate of change.

Slope of Secant Line

The slope of a secant line is an average measure of how a function changes between two points. Imagine a straight line connecting two points on a curved graph; the slope tells us how steep this line is. This concept links directly to the average rate of change.

To compute the slope of the secant line between two points \((a, f(a))\) and \((b, f(b))\), use the formula:\[ \frac{f(b) - f(a)}{b - a} \]
In the exercise example:

• We calculated \( f(-1) = 0 \) and \( f(3) = 84 \).
• Using the points \((-1,0)\) and \((3,84)\), the slope is \( \frac{84 - 0}{3 - (-1)} = \frac{84}{4} = 21 \).

This slope tells us the average rate at which the function increases from \( x = -1 \) to \( x = 3 \).

Polynomial Functions

Polynomial functions are mathematical expressions involving a sum of powers of a variable, like \( x + x^4 \). They are widely used in mathematics because of their versatility and straightforward nature.

• They consist of one or more terms, each involving a variable raised to a non-negative integer power and a coefficient.
• The degree of a polynomial is the highest power of the variable in the expression.

In the given exercise, the polynomial \( f(x) = x + x^4 \) has terms up to the fourth power, making it a degree 4 polynomial.

Polynomial functions can display a variety of behaviors depending on their degree and coefficients:

• They may cross the x-axis at one or more points.
• Their graphs can have several turning points where they change direction.

Understanding these aspects helps in predicting how the polynomial behaves over a range of input values.