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Chapter 3: Problem 15

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

Short Answer

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

Step by step solution

01

Identify the function

The function provided is \( f(x) = 3x^2 \). This is a quadratic function.
02

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 \( \frac{f(x_2) - f(x_1)}{x_2 - x_1} \).
03

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 \( x_2 = 2 + h \).
04

Calculate \( f(x_1) \)

Substitute \( x_1 = 2 \) into the function: \( f(2) = 3(2)^2 = 3 \times 4 = 12 \).
05

Calculate \( f(x_2) \)

Substitute \( x_2 = 2 + h \) into the function: \( f(2 + h) = 3(2 + h)^2 = 3(4 + 4h + h^2) = 12 + 12h + 3h^2 \).
06

Calculate the difference \( f(x_2) - f(x_1) \)

Calculate \( f(2 + h) - f(2) = (12 + 12h + 3h^2) - 12 = 12h + 3h^2 \).
07

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.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

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.

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Most popular questions from this chapter

Finding an Inverse "in Your Head" In the margin notes in this section we pointed out that the inverse of a function can be found by simply reversing the operations that make up the function. For instance, in Example 6 we saw that the inverse of $$ f(x)=3 x-2 \quad \text { is } \quad f^{-1}(x)=\frac{x+2}{3} $$ because the "reverse" of "Multiply by 3 and subtract \(2^{\prime \prime}\) is "Add 2 and divide by 3 ." Use the same procedure to find the inverse of the following functions. $$ \begin{array}{ll}{\text { (a) } f(x)=\frac{2 x+1}{5}} & {\text { (b) } f(x)=3-\frac{1}{x}} \\ {\text { (c) } f(x)=\sqrt{x^{3}+2}} & {\text { (d) } f(x)=(2 x-5)^{3}}\end{array} $$ Now consider another function: $$ f(x)=x^{3}+2 x+6 $$ Is it possible to use the same sort of simple reversal of operations to find the inverse of this function? If so, do it. If not, explain what is different about this function that makes this task difficult.

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