/*! 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 169 For the following exercises, use... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 12: Problem 169

For the following exercises, use the definition of derivative \(\lim _{h \rightarrow 0} \frac{f(x+h)-f(x)}{h}\) to calculate the derivative of each function. $$f(x)=-2 x+1$$

Short Answer

Expert verified
The derivative of \(-2x + 1\) is \(-2\).

Step by step solution

01

Write the Definition of the Derivative

The definition of the derivative of a function \( f(x) \) at a point \( x \) is given by \( \lim_{h \to 0} \frac{f(x+h) - f(x)}{h} \). Here, \( f(x) = -2x + 1 \).
02

Substitute into the Definition

Substitute \( f(x) = -2x + 1 \) into the definition to find \( f(x+h) \):\[ f(x+h) = -2(x+h) + 1 = -2x - 2h + 1 \]
03

Compute the Difference

Subtract \( f(x) \) from \( f(x+h) \):\[ f(x+h) - f(x) = (-2x - 2h + 1) - (-2x + 1) = -2h \]
04

Formulate the Derivative Expression

Substitute \( f(x+h) - f(x) = -2h \) into the derivative definition:\[ \lim_{h \to 0} \frac{-2h}{h} \]
05

Simplify the Expression

Simplify the expression by dividing both numerator and denominator by \( h \):\[ \lim_{h \to 0} -2 = -2 \]
06

Evaluate the Limit

Evaluate the limit for \( h \to 0 \). Since \(-2\) is constant and not dependent on \( h \), the limit simplifies to \(-2\).

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.

Understanding the Definition of Derivative
In calculus, the derivative of a function at a point tells us the rate at which the function value changes as its input changes. Imagine the derivative as a tool that helps us predict how fast or slow something is happening, like the speed of a car or the growth of a plant. The formal definition of the derivative is given as:\[ \lim _{h \rightarrow 0} \frac{f(x+h)-f(x)}{h} \]This formula may look a bit complex at first, but it's essentially finding the slope of the tangent line through a point on the function's curve.
  • Top Part : The \( f(x+h)-f(x) \) calculates the change in the function鈥檚 output.
  • Bottom Part : The \( h \) represents an incredibly small change in the input that approaches zero.
  • Whole Limit: As \( h \) gets very close to zero, the formula captures the instantaneous rate of change.
This approach helps us examine the behavior of the function around a particular point.
The Role of the Limit Process
The limit process is a key concept in the calculation of derivatives and many other areas of calculus. Using limits, we aim to understand what happens to a function as its input approaches a particular value.

In the context of the derivative, the limit \( \lim_{h \to 0} \frac{-2h}{h} \) examines how the ratio of the change in the function over the change in its input behaves as \( h \) gets infinitely small. Here, the tricky part lies in the simplification and evaluation of the limit.
  • Cancel the \( h \) : In the expression \( \frac{-2h}{h} \), we simplify by canceling out \( h \), leading to \( -2 \).
  • Evaluate the limit: By recognizing that \(-2\) is constant, the limit of a constant remains the constant itself, leading to a derivative of \(-2\).
This process shows the stability of the result even when \( h \) tends towards zero, affirming that the function is changing at a consistent rate of \(-2\).
Conducting Function Analysis
Function analysis is the detailed study of a function's behavior. It involves understanding the various properties of the function and how they relate to the derivative.

When analyzing the function \( f(x) = -2x + 1 \), we discover it is a linear function, which means it forms a straight line when plotted on a graph. The derivative calculation confirms a constant rate of change.
  • Slope: The derivative \(-2\) represents the slope of the line. In this context, it seems the line is decreasing as \( x \) increases.
  • Intercept: The \(+1\) in the original function acts as the y-intercept, showing where the line crosses the y-axis.
  • Consistency: Because linear functions have a constant rate of change, their derivatives are also constant.
Understanding function properties such as uniform change helps predict their behavior effectively, especially for real-world applications.

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

For the following exercises, evaluate the following limits. $$ f(x)=\left\\{\begin{array}{ll}{2 x^{2}+2 x+1,} & {x \leq 0} \\ {x-3,} & {x>0^{\prime}}\end{array} \lim _{x \rightarrow 0} f(x)\right. $$

For the following exercises, evaluate each limit using algebraic techniques. $$\lim _{h \rightarrow 0}\left(\frac{1}{h}-\frac{1}{h^{2}+h}\right)$$

For the following exercises, evaluate each limit using algebraic techniques. $$\lim _{x \rightarrow-5}\left(\frac{\frac{1}{5}+\frac{1}{x}}{10+2 x}\right)$$

For the following exercises, use technology to evaluate the limit. $$\lim _{x \rightarrow 0} \frac{\sin (x)(1-\cos (x))}{2 x^{2}}$$

For the following exercises, use numerical evidence to determine whether the limit exists at \(x=a\). If not, describe the behavior of the graph of the function at \(x=a\). $$f(x)=\frac{-x}{x^{2}-x-6} ; a=3$$
See all solutions

Recommended explanations on Math Textbooks

Applied Mathematics

Read Explanation

Discrete Mathematics

Read Explanation

Pure Maths

Read Explanation

Probability and Statistics

Read Explanation

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.