/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 53 Let \(f(x)=x^{2} .\) Find the av... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 4: Problem 53

Let \(f(x)=x^{2} .\) Find the average rate of change \(\Delta f / \Delta x\) on the interval \([a, x]\).

Short Answer

Expert verified
The average rate of change on \([a, x]\) is \(x + a\).

Step by step solution

01

Understand the Average Rate of Change Formula

The average rate of change of a function \( f(x) \) over an interval \([a, b]\) is given by \( \frac{f(b) - f(a)}{b - a}.\) Here, you need to apply this formula over the interval \([a, x]\).
02

Identify \( f(a) \) and \( f(x) \)

For the function \( f(x) = x^2 \), identify \( f(a) = a^2 \) and \( f(x) = x^2 \). These are the function values at the endpoints of your interval.
03

Calculate \( f(x) - f(a) \)

Compute the difference in function values: \( f(x) - f(a) = x^2 - a^2. \) This will be used as the numerator for the average rate of change formula.
04

Factor the Difference \( x^2 - a^2 \)

Notice that \( x^2 - a^2 \) is a difference of squares. It can be factored as \((x - a)(x + a)\).
05

Compute the Average Rate of Change

Now use the formula for the average rate of change: \[ \frac{x^2 - a^2}{x - a} = \frac{(x - a)(x + a)}{x - a}. \] Simplify this expression by canceling \(x - a\), which gives \( x + a \).
06

Conclude with the General Formula

After simplification, the average rate of change \( \frac{\Delta f}{\Delta x} \) on the interval \([a, x]\) is \( x + a \).

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Difference of Squares
The concept of the difference of squares is a fundamental algebraic technique used to simplify certain types of expressions. In the expression \(x^2 - a^2\), you will notice it is in the form of \(A^2 - B^2\), where \(A = x\) and \(B = a\). This specific form is what we call a "difference of squares."
  • The formula for factoring a difference of squares is: \(A^2 - B^2 = (A - B)(A + B)\).
  • When applied to \(x^2 - a^2\), it becomes \((x - a)(x + a)\).
Recognizing when an expression can be transformed into a difference of squares can simplify complex algebraic equations significantly. It is valuable in solving equations, simplifying complex expressions, and in this case, calculating the average rate of change.
Interval Notation
Interval notation is a shorthand way of expressing a range of values. It provides a clearer and more concise representation of intervals compared to inequality notation.
  • An interval \([a, x]\) denotes all the real numbers between \(a\) and \(x\), including the endpoints \(a\) and \(x\).
  • When you see brackets \([]\), it means that the endpoints are included, which is also known as a closed interval.
  • In other contexts, you might use parentheses \(()\) to imply the endpoints are not included, known as an open interval.
Using interval notation allows for uniformity and can simplify communication in mathematics by providing a consistent way to describe continuous values in a specific range.
Simplifying Expressions
The art of simplifying expressions is one of the building blocks of algebraic problem-solving. Simplification involves reducing complicated expressions into simpler or more manageable forms.
  • In the example \(\frac{(x - a)(x + a)}{x - a}\), simplification involves canceling terms across the numerator and the denominator.
  • Since \(x - a\) appears both in the numerator and the denominator, they can cancel each other out, simplifying the expression to \(x + a\).
  • This technique is especially useful in simplifying complex algebraic, trigonometric, or rational expressions.
Being skilled at simplifying expressions helps streamline calculations and can aid in finding solutions more efficiently. Plus, it enables deeper insights into algebraic structures and promotes a better understanding of underlying mathematical relationships.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

(a) Use a graphing utility to draw a graph of each function. (b) For each \(x\) -intercept, zoom in until you can estimate it accurately to the nearest one-tenth. (c) Use algebra to determine each \(x\) -intercept. If an intercept involves a radical, give that answer as well as a calculator approximation rounded to three decimal places. Check to see that your results are consistent with the graphical estimates obtained in part (b). $$N(t)=t^{7}+8 t^{4}+16 t$$

(a) Determine the \(x\) - and \(y\) -intercepts and the excluded regions for the graph of the given function. Specify your results using a sketch similar to Figure \(16(a) .\) In Exercises \(31-34\) you will first need to factor the polynomial. (b) Graph each function. $$y=(x-2)(x-1)(x+1)$$

The population \(y\) of a colony of bacteria after \(t\) hr is given by \(y=(t+12) /(0.0004 t+0.024) \quad\) where \(t \geq 0\) (a) Find the initial population (that is, the population when \(t=0 \mathrm{hr}\) ). (b) Determine the long-term behavior of the population (as in Example 7 ).

A 30 -in. piece of string is to be cut into two pieces. The first piece will be formed into the shape of an equilateral triangle and the second piece into a square. Find the length of the first piece if the combined area of the triangle and the square is to be as small as possible.

Let \(f(x)=\left(x^{5}+1\right) / x^{2}\) (a) Graph the function \(f\) using a viewing rectangle that extends from -4 to 4 in the \(x\) -direction and from -8 to 8 in the \(y\) -direction. (b) Add the graph of the curve \(y=x^{3}\) to your picture in part (a). Note that as \(|x|\) increases (that is, as \(x\) moves away from the origin), the graph of \(f\) looks more and more like the curve \(y=x^{3} .\) For additional perspective, first change the viewing rectangle so that \(y\) extends from -20 to \(20 .\) (Retain the \(x\) -settings for the moment.) Describe what you see. Next, adjust the viewing rectangle so that \(x\) extends from -10 to 10 and \(y\) extends from -100 to \(100 .\) Summarize your observations. (c) In the text we said that a line is an asymptote for a curve if the distance between the line and the curve approaches zero as we move further and further out along the curve. The work in part (b) illustrates that a curve can behave like an asymptote for another curve. In particular, part (b) illustrates that the distance between the curve \(y=x^{3}\) and the graph of the given function \(f\) approaches zero as we move further and further out along the graph of \(f .\) That is, the curve \(y=x^{3}\) is an "asymptote" for the graph of the given function \(f\). Complete the following two tables for a numerical perspective on this. In the tables, \(d\) denotes the vertical distance between the curve \(y=x^{3}\) and the graph of \(f:\) $$ d=\left|\frac{x^{5}+1}{x^{2}}-x^{3}\right| $$ $$\begin{array}{llllll} \hline x & 5 & 10 & 50 & 100 & 500 \\ \hline d & & & & \\ \hline & & & & \\ \hline x & -5 & -10 & -50 & -100 & -500 \\ \hline d & & & & \\ \hline \end{array}$$ (d) Parts (b) and (c) have provided both a graphical and a numerical perspective. For an algebraic perspective that ties together the previous results, verify the following identity, and then use it to explain why the results in parts (b) and (c) were inevitable: $$ \frac{x^{5}+1}{x^{2}}=x^{3}+\frac{1}{x^{2}} $$
See all solutions

Recommended explanations on Math Textbooks

Mechanics Maths

Read Explanation

Applied Mathematics

Read Explanation

Probability and Statistics

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Pure Maths

Read Explanation

Logic and Functions

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.