/*! 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 148 If \(f(x)=m x+b,\) find \(\frac{... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 4: Problem 148

If \(f(x)=m x+b,\) find \(\frac{f(x+h)-f(x)}{h}, h \neq 0\) (Section \(2.2,\) Example 8 )

Short Answer

Expert verified
The value is \(m\).

Step by step solution

01

Substitution

Substitute \(f(x)\) and \(f(x+h)\) into the equation. This will give \(\frac{m(x+h) + b - (mx + b)}{h}\).
02

Simplification

Simplify the expression in the numerator, while keeping track of the terms. You will get \(\frac{mh + mx + b - mx - b}{h}\). Note that the terms \(mx\), \(b\), \(-mx\) and \(-b\) will cancel out to give \(\frac{mh}{h}\).
03

Cancel Common Factors

Notice that \(h\) is a common factor in the numerator and the denominator. Cancel out these common factors. The expression simplifies to \(m\).

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.

Linear Function
A linear function is one of the simplest forms of function in mathematics, often represented as \(f(x) = mx + b\). This equation represents a line when plotted on a graph. Here, "\(m\)" is the slope of the line, and "\(b\)" is the y-intercept, which is where the line crosses the y-axis.
Linear functions have a constant rate of change which makes them very predictable and easy to analyze.
  • When you increase \(x\) by 1, the value of \(f(x)\) will increase by \(m\).
  • The graph of a linear function is always a straight line.
  • This function is also called an affine function when \(b eq 0\).
Slope of a Line
The slope of a line in a linear function is a measure of how steep the line is. In the context of our equation \(f(x) = mx + b\), the slope is represented by the coefficient \(m\).
The slope can be understood as the rise over run, describing how much \(f(x)\) changes as \(x\) changes. If you imagine walking along a line:
  • If the line goes up as you move from left to right, \(m\) is positive.
  • If the line goes down as you move from left to right, \(m\) is negative.
  • If the line is horizontal, \(m\) is zero, indicating no change.
Understanding the slope helps us interpret how two quantities are related. For example, an incline in a road is similar to a positive slope of a line.
Derivative Concept
The derivative is a fundamental concept in calculus that represents an instantaneous rate of change of a function concerning its variable. It is often seen as the slope of a function at a given point. The difference quotient you calculated \[\frac{f(x+h)-f(x)}{h}\]reflects the average rate of change of the function over an interval \(h\).
Consider it like calculating the slope over an infinitesimally small section of the curve, as \(h\) becomes very small.
For linear functions specifically, the derivative provides the exact slope of the line:
  • Since linear functions have constant slopes, the derivative for \(f(x) = mx + b\) is \(m\).
  • In non-linear functions, the derivative can vary depending on \(x\).
  • Calculating derivatives gives insight into how quickly changes occur within the system modeled by the function.

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

In Example I on page \(520,\) we used two data points and an exponential function to model the population of the United States from 1970 through 2010 . The data are shown again in the table. Use all five data points to solve Exercises \(66-70\). $$ \begin{array}{cc} {x, \text { Number of Years }} & {y, \text { U.S. Population }} \\ {\text { after } 1969} & {\text { (millions) }} \\ {1(1970)} & {203.3} \\ {11(1980)} & {226.5} \\ {21(1990)} & {248.7} \\ {31(2000)} & {281.4} \\ {41(2010)} & {308.7} \end{array} $$ Use your graphing utility's power regression option to obtain a model of the form \(y=a x^{b}\) that fits the data. How well does the correlation coefficient, \(r,\) indicate that the model fits the data?

After a \(60 \%\) price reduction, you purchase a computer for \(\$ 440 .\) What was the computer's price before the reduction?

Explain how to use your calculator to find \(\log _{14} 283\)

Exercises \(86-88\) will help you prepare for the material covered in the first section of the next chapter. $$ \text { Simplify: } \frac{17 \pi}{6}-2 \pi $$

The function \(P(t)=145 e^{-0.092 t}\) models a runner's pulse, \(P(t),\) in beats per minute, \(t\) minutes after a race, where \(0 \leq t \leq 15 .\) Graph the function using a graphing utility. TRACE along the graph and determine after how many minutes the runner's pulse will be 70 beats per minute. Round to the nearest tenth of a minute. Verify your observation algebraically.
See all solutions

Recommended explanations on Math Textbooks

Discrete Mathematics

Read Explanation

Geometry

Read Explanation

Statistics

Read Explanation

Applied Mathematics

Read Explanation

Calculus

Read Explanation

Decision Maths

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.