/*! 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 67 Find the average rate of change ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 3: Problem 67

Find the average rate of change of \(g(x)=x^{3}-4 x+7\) : (a) From -3 to -2 (b) From -1 to 1 (c) From 1 to 3

Short Answer

Expert verified
a) 15, b) -3, c) 9

Step by step solution

01

Identify the formula for the average rate of change

The average rate of change of a function between two points is given by: \[\text{Average Rate of Change} = \frac{g(b) - g(a)}{b - a}\]where \(g(x)\) is the function, and \(a\) and \(b\) are the two points.
02

Calculate the average rate of change from -3 to -2

First, find \(g(-3)\) and \(g(-2)\).\[g(-3) = (-3)^3 - 4(-3) + 7 = -27 + 12 + 7 = -8\]\[g(-2) = (-2)^3 - 4(-2) + 7 = -8 + 8 + 7 = 7\]Now apply the formula:\[\frac{g(-2) - g(-3)}{-2 - (-3)} = \frac{7 - (-8)}{-2 + 3} = \frac{15}{1} = 15\]
03

Calculate the average rate of change from -1 to 1

First, find \(g(-1)\) and \(g(1)\).\[g(-1) = (-1)^3 - 4(-1) + 7 = -1 + 4 + 7 = 10\]\[g(1) = (1)^3 - 4(1) + 7 = 1 - 4 + 7 = 4\]Now apply the formula:\[\frac{g(1) - g(-1)}{1 - (-1)} = \frac{4 - 10}{1 + 1} = \frac{-6}{2} = -3\]
04

Calculate the average rate of change from 1 to 3

First, find \(g(1)\) and \(g(3)\).\[g(1) = (1)^3 - 4(1) + 7 = 1 - 4 + 7 = 4\]\[g(3) = (3)^3 - 4(3) + 7 = 27 - 12 + 7 = 22\]Now apply the formula:\[\frac{g(3) - g(1)}{3 - 1} = \frac{22 - 4}{3 - 1} = \frac{18}{2} = 9\]

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 Quotient
The difference quotient is a critical concept in calculus that simplifies finding the average rate of change for a function. In essence, it measures the change in the function's value over a specified interval. The formula is: \[ \text{Difference Quotient} = \frac{g(b) - g(a)}{b - a} \] where \( g(x) \) represents the function, and \( a \) and \( b \) denote the two points in the interval. Simply put, it calculates how much the function's output values change when moving from \( a \) to \( b \). Remember, understanding the difference quotient is essential for delving deeper into calculus topics like derivatives.
Polynomial Functions
A polynomial function consists of coefficients and variables raised to non-negative integer powers. They take the form: \[ g(x) = a_n x^n + a_{n-1} x^{n-1} + \ldots + a_1 x + a_0 \] In our exercise, the function given is a third-degree polynomial: \[ g(x) = x^3 - 4x + 7 \]. Here are some key points to remember:
  • The degree is the highest power of the variable (3 in this case).
  • Polynomials are continuous and smooth functions.
  • They are straightforward to evaluate for given values of x.
Knowing the structure of polynomial functions helps in quickly finding values or calculating derivatives.
Function Evaluation
Function evaluation is a method of finding the function's output for specific input values. In our exercise, we evaluated the function \( g(x) \) at different points. For example, \[ g(-3) = (-3)^3 - 4(-3) + 7 = -27 + 12 + 7 = -8 \] Evaluating a function involves substituting the variable with the given number and simplifying the expression. Here are the steps to remember:
  • Substitute the input value in place of the variable.
  • Perform arithmetic operations, following the order of operations (PEMDAS/BODMAS).
  • Arrive at the final output or value.
Understanding function evaluation is crucial for computations and for interpreting mathematical models.
Step-by-Step Solutions
Breaking down problems into smaller, manageable steps ensures better understanding and reduces errors. In this exercise, each part was solved incrementally:
  • Step 1: Identify the formula for the average rate of change.
  • Step 2: Evaluate the function values at the given points.
  • Step 3: Apply the average rate of change formula.
Here鈥檚 how it was applied in part (a):
  • First, find \( g(-3) \) and \( g(-2) \): \[ g(-3) = -8 \] \[ g(-2) = 7 \]
  • Second, apply the formula: \[ \frac{g(-2) - g(-3)}{-2 - (-3)} = \frac{15}{1} = 15 \]
This structured approach aids in comprehension, ensuring that you don't miss any critical steps in calculations.

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

Show that a constant function \(f(x)=b\) has an average rate of change of \(0 .\) Compute the average rate of change of \(y=\sqrt{4-x^{2}}\) on the interval \([-2,2] .\) Explain how this can happen.

Ethan has $$\$ 60,000$$ to invest. He puts part of the money in a CD that earns \(3 \%\) simple interest per year and the rest in a mutual fund that earns \(8 \%\) simple interest per year. How much did he invest in each if his earned interest the first year was $$\$ 3700$$.

Find the domain of \(h(x)=\sqrt[4]{x+7}+7 x\).

The daily rental charge for a moving truck is \(\$ 40\) plus a mileage charge of \(\$ 0.80\) per mile. Express the cost \(C\) to rent a moving truck for one day as a function of the number \(x\) of miles driven.

The total worldwide digital music revenues \(R\), in billions of dollars, for the years 2012 through 2017 can be modeled by the function $$ R(x)=0.15 x^{2}-0.03 x+5.46 $$ where \(x\) is the number of years after 2012 . (a) Find \(R(0), R(3),\) and \(R(5)\) and explain what each value represents. (b) Find \(r(x)=R(x-2)\) (c) Find \(r(2), r(5)\) and \(r(7)\) and explain what each value represents. (d) In the model \(r=r(x),\) what does \(x\) represent? (e) Would there be an advantage in using the model \(r\) when estimating the projected revenues for a given year instead of the model \(R ?\)
See all solutions

Recommended explanations on Math Textbooks

Logic and Functions

Read Explanation

Applied Mathematics

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Geometry

Read Explanation

Probability and Statistics

Read Explanation

Calculus

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.