Problem 15 - Algebra and Trigonometry

Question

A function is given. Determine the average rate of change of the function between the given values of the variable. $$ f(x)=3 x^{2} ; \quad x=2, x=2+h $$

Step-by-Step Solution

Answer

The average rate of change is \( 12 + 3h \).

Step 1: Identify the function

The function provided is \( f(x) = 3x^2 \). This is a quadratic function.

Step 2: Understand the formula for average rate of change

The average rate of change of a function \( f(x) \) over an interval \([x_1, x_2]\) is given by the formula \( \frac{f(x_2) - f(x_1)}{x_2 - x_1} \).

Step 3: Substitute given points into the formula

We want to find the average rate of change between \( x = 2 \) and \( x = 2 + h \). So \( x_1 = 2 \) and \( x_2 = 2 + h \).

Step 4: Calculate \( f(x_1) \)

Substitute \( x_1 = 2 \) into the function: \( f(2) = 3(2)^2 = 3 \times 4 = 12 \).

Step 5: Calculate \( f(x_2) \)

Substitute \( x_2 = 2 + h \) into the function: \( f(2 + h) = 3(2 + h)^2 = 3(4 + 4h + h^2) = 12 + 12h + 3h^2 \).

Step 6: Calculate the difference \( f(x_2) - f(x_1) \)

Calculate \( f(2 + h) - f(2) = (12 + 12h + 3h^2) - 12 = 12h + 3h^2 \).

Step 7: Divide the difference by \( x_2 - x_1 \)

The difference in the x-values is \( (2 + h) - 2 = h \). Thus, the average rate of change is \( \frac{12h + 3h^2}{h} = 12 + 3h \), after cancelling \( h \) in the numerator and denominator.

Key Concepts

Quadratic Function

A quadratic function is a type of function represented by the expression \( ax^2 + bx + c \). In this expression, the highest power of \( x \) is 2, which makes it a quadratic function.
Quadratic functions have a U-shaped graph known as a parabola. The standard form of a quadratic function helps to determine the direction of the parabola.

• If \( a > 0 \), the parabola opens upwards.
• If \( a < 0 \), the parabola opens downwards.

In our exercise, the function given is \( f(x) = 3x^2 \). Here, \( a = 3 \), meaning the parabola opens upwards. This characteristic shape significantly influences the behavior of the function, particularly in how it grows or shrinks as \( x \) changes.

Rate of Change Formula

The rate of change formula is crucial in understanding how functions behave between two points. The average rate of change of a function between two points \( x_1 \) and \( x_2 \) is calculated using the formula:\[\frac{f(x_2) - f(x_1)}{x_2 - x_1}\]This formula helps us measure how much the function's output, or \( y \)-value, changes per unit increase in the \( x \)-value.
This is particularly useful when dealing with non-linear functions such as quadratic functions, where the rate of change is not constant. By comparing outputs across two points, we can get a snapshot of how quickly the function is increasing or decreasing over that interval.

Function Substitution

Function substitution involves finding the output of a function by replacing the input variable with specific values.
In our scenario, you substitute particular values of \( x \) into \( f(x) = 3x^2 \) to compute \( f(x) \) at those points.
Let's break it down:

• For \( x_1 = 2 \), substitute into the function: \( f(2) = 3 \cdot (2)^2 = 12 \).
• For \( x_2 = 2 + h \), substitute: \( f(2 + h) = 3 \cdot (2 + h)^2 = 12 + 12h + 3h^2 \).

These substitutions enable you to find \( f(x_1) \) and \( f(x_2) \), which are crucial for calculating the average rate of change.

Difference Quotient

The difference quotient is a term that frequently comes up when calculating the average rate of change.
In our example, the difference quotient is formed by the expression \( \frac{f(x_2) - f(x_1)}{x_2 - x_1} \), which simplifies to \( \frac{12h + 3h^2}{h} \).
This expression shows the change in the function's output relative to the change in input, a fundamental idea in calculus.

• First, find the difference \( f(x_2) - f(x_1) \), resulting in \( 12h + 3h^2 \).
• Then, divide this difference by \( x_2 - x_1 \), which is \( h \).
• Simplify the quotient to \( 12 + 3h \).

The difference quotient thus gives a simplified expression representing the average rate of change over the interval, which is helpful in understanding the behavior of the function on that interval.