Ch. 4 - Applications of Derivatives

Hass15th EditionThomas' CalculusISBN: 9780137616077Non è quello che usi tu?Cambia libro di testo

Hass 15th Edition

Ch. 4 - Applications of Derivatives

Problema 27

Capitolo 4, Problema 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.

Guida verificata passo dopo passo

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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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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Exponential Functions

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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Percorso guidato

05:44

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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05:03

Initial Value Problems

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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.)

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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?

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Finding Functions from Derivatives

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

a. f(0) = 0

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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)

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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.

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Domanda del libro di testo

Finding Functions from Derivatives

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