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Chapter 1: Problem 167

Find the average rate of change of each function on the interval specified for real numbers \(b\) or \(h.\) \(r(t)=4 t^{3}\) on \([2,2+h]\)

Short Answer

Expert verified
The average rate of change is \(48 + 24h + 4h^2\).

Step by step solution

01

Understand the Average Rate of Change Formula

The average rate of change of a function between two points is given by the formula \( \frac{f(b) - f(a)}{b - a} \). Here, \( b \) and \( a \) are the two points over an interval. For function \( r(t) = 4t^3 \), we need to calculate it over the interval \([2, 2+h]\). So, we identify \( a = 2 \) and \( b = 2+h \).
02

Calculate Function Values

Evaluate \( r(t) = 4t^3 \) at the endpoints of the interval:- At \( t = 2 \), \( r(2) = 4(2)^3 = 4 \times 8 = 32 \).- At \( t = 2+h \), \( r(2+h) = 4(2+h)^3 \).
03

Expand the Expression for \(r(2+h)\)

First, expand \( (2+h)^3 \):\( (2+h)^3 = (2+h)(2+h)(2+h) = 8 + 12h + 6h^2 + h^3 \).Thus, \( r(2+h) = 4(8 + 12h + 6h^2 + h^3) = 32 + 48h + 24h^2 + 4h^3 \).
04

Compute the Difference \(r(b) - r(a)\)

We subtract \( r(2) \) from \( r(2+h) \):\( r(2+h) - r(2) = (32 + 48h + 24h^2 + 4h^3) - 32 = 48h + 24h^2 + 4h^3 \).
05

Apply the Average Rate of Change Formula

Use the average rate of change formula:\[ \text{Average rate of change} = \frac{r(2+h) - r(2)}{(2+h) - 2} = \frac{48h + 24h^2 + 4h^3}{h} \].Simplifying, divide each term by \( h \):\[ = 48 + 24h + 4h^2 \].
06

Conclusion

The average rate of change of the function \( r(t)=4t^3 \) on the interval \([2, 2+h]\) is expressed as a function of \( h \):\( 48 + 24h + 4h^2 \).

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Key Concepts

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

Understanding Polynomials
A polynomial is a mathematical expression consisting of variables and coefficients, constructed using only addition, subtraction, multiplication, and non-negative integer exponents of variables. In this example, we deal with a function given by a polynomial:
  • The function is given as: \( r(t) = 4t^3 \).
  • This is a cubic polynomial because the highest power of the variable \( t \) is 3.
  • Polynomials are fundamental in algebra and serve as a base for more complex calculations.
Identifying the type of polynomial helps in understanding its behavior, like how it will change over an interval or react to different operations.
Explaining Interval Notation
Interval notation is a way of describing the range of values on which a function is evaluated. In our exercise, the interval is given as
  • \([2, 2+h]\), where 2 is a prescriptive start value, and \(2 + h\) suggests a flexible endpoint.
  • The square brackets \([ ]\) indicate that both endpoints are included in the interval.
This is a closed interval, and it's crucial for deciding where to evaluate the function, as changes in the interval endpoints will affect the average rate of change computation.
Delving into Function Evaluation
Function evaluation involves calculating the output of a function for specific input values. For the function \( r(t) = 4t^3 \), the evaluations occur at the boundaries of the interval:
  • At \( t = 2 \), evaluate the function: \( r(2) = 4(2)^3 = 32 \).
  • At \( t = 2+h \), the value is \( r(2+h) = 4(2+h)^3 \).
Properly evaluating a function at specific points is essential. It sets the stage for operations or analyses, like finding the average rate of change across an interval.
Mastering Expansion and Simplification
Expansion and simplification are vital techniques in algebra, essential for solving polynomial equations. In our problem, we expand the expression \((2+h)^3\):
  • Multiply the binomial step by step to obtain: \( (2+h)^3 = 8 + 12h + 6h^2 + h^3 \).
  • Next, scale by 4 to get \( r(2+h) = 32 + 48h + 24h^2 + 4h^3 \).
After expansion, simplification involves condensing the expression to its core components. Understanding these processes makes it feasible to handle more complex expressions and provides clarity when analyzing rate changes or transforming functions.

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Most popular questions from this chapter

For the following exercises, describe how the formula is a transformation of a toolkit function. Then sketch a graph of the transformation. $$ h(x)=-2|x-4|+3 $$

. Given \(f(x)=2 x^{2}+4 x\) and \(g(x)=\frac{1}{2 x},\) find \(f+g, f-g, f g,\) and \(\frac{f}{g}\) . Determine the domain for each function in interval notation.

For the following exercises, find functions \(f(x)\) and \(g(x)\) so the given function can be expressed as \(h(x)=f(g(x))\) $$ h(x)=(x-5)^{3} $$

For the following exercises, determine whether the function is odd, even, or neither. $$ f(x)=3 x^{4} $$

For the following exercises, describe how the graph of the function is a transformation of the graph of the original function \(f\) . $$ y=f(x+4)-1 $$
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