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Chapter 1: Problem 28

If \(f(x)=2 x+6,\) compute \(f(0)\) and \(f^{\prime}(0).\)

Short Answer

Expert verified
f(0) = 6, f'(0) = 2

Step by step solution

01

Identify the given function

The given function is: \[ f(x) = 2x + 6 \]
02

Evaluate the function at x = 0

To find \( f(0) \), substitute \( x = 0 \) into the function: \[ f(0) = 2(0) + 6 = 6 \]
03

Determine the derivative of the function

Find the derivative \( f'(x) \) with respect to \( x \). For the function \( f(x) = 2x + 6 \), the derivative \( f'(x) \) is: \[ f'(x) = 2 \]
04

Evaluate the derivative at x = 0

To find \( f'(0) \), use the derivative found in the previous step: \[ f'(0) = 2 \]

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Key Concepts

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

function evaluation
Function evaluation is a fundamental concept in mathematics. It involves finding the value of a function at a specific input value. To evaluate a function, you replace the variable with the given number and perform the arithmetic operations. In the exercise, we have the function:

\( f(x) = 2x + 6 \)

To evaluate the function at \(x = 0\), we substitute 0 into the function and get:

\( f(0) = 2(0) + 6 = 6 \).

Function evaluation helps us understand how the function behaves at specific points, offering insights into its graph and outputs.
derivative calculation
Derivative calculation is crucial in understanding how a function changes. The derivative of a function at a point tells us the rate at which the function's value is changing at that point. For a function like \(f(x) = 2x + 6\), we calculate the derivative using the differentiation rules.

The derivative of \( f(x) = 2x + 6 \) with respect to \(x\) is:

\( f'(x) = 2 \).

To evaluate the derivative at \( x = 0 \), we simply note that:

\( f'(0) = 2 \).

This result means that the function increases at a constant rate of 2 for each unit increase in \(x\). Understanding derivatives is essential in calculus for analyzing and predicting the behavior of functions.
linear functions
Linear functions are among the simplest types of functions. They are defined by expressions of the form \( f(x) = mx + b \), where \(m\) and \(b\) are constants. The function from our exercise, \( f(x) = 2x + 6 \), is a linear function.

Some key properties of linear functions include:
  • They produce straight-line graphs.
  • The slope (\(m\)) represents the rate of change of the function.
  • The y-intercept (\(b\)) is the function's value when \(x = 0\).

This particular function has a slope of 2 and a y-intercept of 6. The slope tells us that for every 1 unit increase in \(x\), the value of \(f(x)\) increases by 2 units. Linear functions are important in various applications like physics, economics, and engineering because of their simplicity and ease of use.

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Most popular questions from this chapter

Using the sum rule and the constant-multiple rule, show that for any functions \(f(x)\) and \(g(x).\) $$\frac{d}{d x}[f(x)-g(x)]=\frac{d}{d x} f(x)-\frac{d}{d x} g(x).$$

Use limits to compute \(f^{\prime}(x) .\) $$f(x)=\frac{x}{x+2}$$

Determine which of the following limits exist. Compute the limits that exist. $$\lim _{x \rightarrow 8} \frac{x^{2}+64}{x-8}$$

Determine whether each of the following functions is continuous and/or differentiable at \(x=1.\) $$f(x)=\frac{1}{x}$$

A particle is moving in a straight line in such a way that its position at time \(t\) (in seconds) is \(s(t)=t^{2}+3 t+2\) feet to the right of a reference point, for \(t \geq 0.\) (a) What is the velocity of the object when the time is 6 seconds? (b) Is the object moving toward the reference point when \(t=6 ?\) Explain your answer. (c) What is the object's velocity when the object is 6 feet from the reference point?
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