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Chapter 2: Problem 50

Compute the average rate of change of the given function over the interval \([x, x+h] .\) Here we assume \([x, x+h]\) is in the domain of the function. \(f(x)=x^{3}\)

Short Answer

Expert verified
The average rate of change of \( f(x)=x^3 \) over \([x, x+h]\) is \( 3x^2 + 3xh + h^2 \).

Step by step solution

01

Understand the Average Rate of Change

The average rate of change for a function between two points is calculated using the formula \( \frac{f(b) - f(a)}{b - a} \), where \( a \) and \( b \) are the endpoints of the interval. In this exercise, \( a = x \) and \( b = x + h \).
02

Calculate Function Values at Endpoints

First, calculate \( f(x) \), which is simply \( x^3 \). Then, calculate \( f(x+h) \) which is \( (x+h)^3 \).
03

Expand \( (x+h)^3 \)

Expand \( (x+h)^3 \) using the binomial theorem: \[ (x+h)^3 = x^3 + 3x^2h + 3xh^2 + h^3 \].
04

Find \( f(x+h) - f(x) \)

Substitute the values from Step 2:\[ f(x+h) - f(x) = (x^3 + 3x^2h + 3xh^2 + h^3) - x^3 \]. Simplify to:\[ 3x^2h + 3xh^2 + h^3 \].
05

Compute the Average Rate of Change

Using the difference from Step 4, the average rate of change is:\[ \frac{3x^2h + 3xh^2 + h^3}{h} \].
06

Simplify the Expression

Factor \( h \) out of the numerator:\[ \frac{h(3x^2 + 3xh + h^2)}{h} \]. Cancel \( h \) from the numerator and denominator to get:\[ 3x^2 + 3xh + h^2 \].

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Key Concepts

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

Binomial Theorem
The binomial theorem is a powerful mathematical tool used to expand expressions of the form \((a + b)^n\) where \(n\) is a non-negative integer. In simpler terms, it tells us how to multiply out powers of a sum.
  • It provides a formula that expresses \((a + b)^n\) as a sum involving terms of the form \( \binom{n}{k} a^{n-k}b^k \), where \(k\) is an integer between 0 and \(n\).
  • This is incredibly helpful when working with polynomial functions and helps simplify expressions that might otherwise require tedious distribution.
  • For example, using the binomial theorem, we can expand \((x+h)^3\) as:\[ (x+h)^3 = x^3 + 3x^2h + 3xh^2 + h^3 \] This expansion is crucial in solving for the average rate of change.
Function Values
Function values are simply the outputs of a function given specific inputs. For each value of the input \(x\), there is a corresponding function value \(f(x)\).
  • In polynomial functions like \(f(x) = x^3\), each substitution of \(x\) gives a new value.
  • When considering changes over intervals, we evaluate the function at different points to compare results.
  • For our average rate of change exercise, we find \(f(x)\) and \(f(x+h)\) by substituting \(x\) and \((x+h)\) into the function.
This method helps analyze changes in values, revealing how much the function's output increases or decreases between two points.
Interval of a Function
When discussing the interval of a function, we refer to the range of input values over which we evaluate the function. In our example, the interval is \([x, x+h]\).
  • An interval gives us a specific path along the function to study.
  • Different types of intervals include open intervals such as \((a, b)\), and closed intervals \([a, b]\).
  • For functions like \(f(x) = x^3\), we consider intervals to understand how the function behaves within certain boundaries.
By identifying and using such intervals, we can compute average rates of change and other properties relevant to the segment of the function we wish to analyze.
Polynomial Functions
Polynomial functions are mathematical expressions involving a sum of powers in one or more variables multiplied by coefficients. They have the general form:\[ P(x) = a_nx^n + a_{n-1}x^{n-1} + \ldots + a_1x + a_0 \]
  • Each term's power indicates the degree of the term, with the highest degree term defining the polynomial's degree
  • In our function \(f(x) = x^3\), we have a cubic polynomial, which is a polynomial of degree 3.
  • Polynomials are smooth and continuous, making them easy to differentiate and analyze, which is why they are frequently used in algebra and calculus.
Understanding polynomial functions is essential when calculating rates of change over specific intervals, as they provide the foundation for the necessary calculations.

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Most popular questions from this chapter

Graph the function. Find the zeros of each function and the \(x\) - and \(y\) -intercepts of each graph, if any exist. From the graph, determine the domain and range of each function, list the intervals on which the function is increasing, decreasing or constant, and find the relative and absolute extrema, if they exist. \(f(x)=|x|+4\)

Graph the function. Find the zeros of each function and the \(x\) - and \(y\) -intercepts of each graph, if any exist. From the graph, determine the domain and range of each function, list the intervals on which the function is increasing, decreasing or constant, and find the relative and absolute extrema, if they exist. . \(f(x)=-3|x|\)

Suppose \(C(x)=x^{2}-10 x+27, x \geq 0\) represents the costs, in hundreds of dollars, to produce \(x\) thousand pens. Find the number of pens which can be produced for no more than $$\$ 1100 .$$

Sketch the graph of the relation. \(R=\left\\{(x, y): x^{2}

Graph the function. Find the zeros of each function and the \(x\) - and \(y\) -intercepts of each graph, if any exist. From the graph, determine the domain and range of each function, list the intervals on which the function is increasing, decreasing or constant, and find the relative and absolute extrema, if they exist. \(f(x)=|x+4|\)
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