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Chapter 2: Problem 69

Give an example of:A linear function with derivative 2 at \(x=0\).

Short Answer

Expert verified
An example is \( f(x) = 2x + 1 \).

Step by step solution

01

Understand the Definition of a Linear Function

A linear function is of the form \( f(x) = mx + c \), where \( m \) is the slope and \( c \) is the y-intercept of the line.
02

Find the Derivative of a Linear Function

The derivative of a function \( f(x) = mx + c \) is a constant, \( f'(x) = m \). This derivative represents the rate of change of the function.
03

Set the Derivative Equal to the Given Value

We want the derivative of the function at \( x = 0 \) to be 2. Since the derivative of a linear function is constant, it will be 2 for all \( x \). Therefore, set \( m = 2 \).
04

Construct the Function

Using the slope \( m = 2 \) found in the previous step, the function becomes \( f(x) = 2x + c \). Use any value for \( c \) as it does not affect the derivative.
05

Verify the Function Satisfies the Condition

Check that the derivative is indeed \( 2 \) at \( x = 0 \). We know \( f'(x) = 2 \), which satisfies the condition given in the exercise.

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Key Concepts

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

Linear Function
A linear function is a fundamental concept in calculus, represented by the formula \( f(x) = mx + c \). This equation defines a straight line when plotted on a graph.
Here are some key points about linear functions:
  • **Slope (\(m\))**: It dictates the steepness and direction of the line.
  • **Y-intercept (\(c\))**: The point where the line crosses the y-axis.
Linear functions are widely used in mathematics to model relationships with a constant rate of change. They form the foundation for more complex mathematical concepts.
Derivative
In calculus, the derivative is a measure of how a function changes as its input changes. For a function \( f(x) \), the derivative is denoted \( f'(x) \).
In the context of linear functions, such as \( f(x) = mx + c \), the derivative is simply the coefficient \(m\). This is because a linear function changes at the same rate throughout its entire domain.
  • **Constant Derivative**: For linear functions, the derivative \( f'(x) = m \) is constant, indicating that the slope does not change.
  • **Meaning**: The derivative reflects the function's sensitivity to changes in \( x \).
The simplicity of derivatives in linear functions makes them an appealing starting point for understanding calculus.
Slope
The slope is a critical concept of linear functions, revealing how much the function's value increases or decreases as \( x \) changes.
The formula for slope in a linear function is the coefficient \(m\) in \( f(x) = mx + c \).
  • A positive slope (\(m>0\)) indicates the function is increasing.
  • A negative slope (\(m<0\)) shows the function is decreasing.
  • A zero slope means the function is constant and has a horizontal line.
Understanding slope helps to anticipate the behavior of the function without graphing. It's a straightforward yet crucial component of analyzing functions.
Rate of Change
Rate of change in calculus refers to how quickly one quantity changes in relation to another.
For a linear function, this is given by the slope \(m\).
The rate of change can be:
  • **Constant**: In linear functions there is a uniform rate of change.
  • **Identical to the derivative**: As it expresses the rate at which the dependent variable responds to changes in an independent variable.
Rate of change is important for interpreting real-world scenarios, such as speed in physics or economic trends, through a mathematical lens. It allows us to quantify changes and make predictions based on linear models.

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Most popular questions from this chapter

If \(g\) is an odd function and \(g^{\prime}(4)=5,\) what is \(g^{\prime}(-4) ?\)

An electric charge, \(Q,\) in a circuit is given as a function of time, \(t,\) by $$ Q=\left\\{\begin{array}{ll} C & \text { for } t \leq 0 \\ C e^{-t / \hbar C} & \text { for } t>0 \end{array}\right. $$ where \(C\) and \(R\) are positive constants. The electric current, \(I\), is the rate of change of charge, so 1 $$ I=\frac{d Q}{d t} $$ (a) Is the charge, \(Q,\) a continuous function of time? (b) Do you think the current, \(I\), is defined for all times, t? [Hint: To graph this function, take, for example, \(C=1 \text { and } R=1 .]\)

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Use algebra to evaluate the limits. $$\lim _{h \rightarrow 0} \frac{(2-h)^{3}-8}{h}$$

Are the statements true or false? Give an explanation for your answer. If \(f(a) \neq g(a),\) then \(f^{\prime}(a) \neq g^{\prime}(a)\).
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