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Ch. 4 - Applications of Derivatives
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077Not the one you use?Change textbook
All textbooksHass 15th EditionCh. 4 - Applications of DerivativesProblem 27
Chapter 4, Problem 27

Finding Functions from Derivatives


Suppose that f(−1) = 3 and that f'(x) = 0 for all x. Must f(x) = 3 for all x? Give reasons for your answer.

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First, understand the meaning of the derivative f'(x) = 0 for all x. This implies that the function f(x) has a constant slope of 0, meaning it is a horizontal line.
Since the derivative represents the rate of change of the function, a derivative of 0 indicates that the function does not change as x changes. Therefore, f(x) must be a constant function.
Given that f(x) is a constant function, it must take the same value for all x.
We are provided with the information that f(−1) = 3. Since f(x) is constant, this value must be the same for all x.
Thus, we conclude that f(x) = 3 for all x, because a constant function with f(−1) = 3 implies that the constant value is 3 everywhere.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Constant Function

A constant function is a function that always returns the same value, no matter the input. In mathematical terms, if f'(x) = 0 for all x, it implies that the function f(x) does not change as x changes, indicating that f(x) is a constant function.
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Derivative

The derivative of a function at a point measures the rate at which the function's value changes as its input changes. If the derivative f'(x) is zero for all x, it means the function has no slope and is flat, suggesting that the function is constant across its domain.
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Derivatives

Initial Condition

An initial condition provides specific information about a function at a particular point, which helps determine the constant of integration when solving differential equations. In this problem, f(−1) = 3 serves as an initial condition, confirming that the constant value of the function f(x) is 3.
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Initial Value Problems
Related Practice
Textbook Question

As a result of a heavy rain, the volume of water in a reservoir increased by 1400 acre-ft in 24 hours. Show that at some instant during that period the reservoir’s volume was increasing at a rate in excess of 225,000 gal/min. (An acre-foot is 43,560 ft³, the volume that would cover 1 acre to the depth of 1 ft. A cubic foot holds 7.48 gal.) 

Textbook Question

Calculate the first derivatives of ƒ(𝓍) = 𝓍²/ (𝓍² + 1) and g(𝓍) = ―1/ (𝓍² + 1) . What can you conclude about the graphs of these functions?

Textbook Question

Finding Functions from Derivatives


Suppose that f'(x) = 2x for all x. Find f(2) if


a. f(0) = 0

Textbook Question

In Exercises 9–66, graph the function using appropriate methods from the graphing procedures presented just before Example 9, identifying the coordinates of any local extreme points and inflection points. Then find coordinates of absolute extreme points, if any.

y = (x² - 49) / (x² + 5x - 14)

Textbook Question

The Mean Value Theorem                                                                                                                                                                  

                                                                                                                                                                                        

 a. Show that the equation 𝓍⁴ + 2𝓍² ― 2 = 0 has exactly one solution on [0,1] .

         

[Technology Exercises] b.Find the solution to as many decimal places as you can.  

Textbook Question

Finding Functions from Derivatives


Suppose that f'(x) = 2x for all x. Find f(2) if


b. f(1) = 0