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Ch. 4 - Applications of Derivatives
Hass - Thomas' Calculus 15th Edition
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Tutti i libri di testoHass 15th EditionCh. 4 - Applications of DerivativesProblema 4.2.28
Capitolo 4, Problema 4.2.28

Finding Functions from Derivatives


Suppose that f(0) = 5 and that f'(x) = 2 for all x. Must f(x) = 2x + 5 for all x? Give reasons for your answer.

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To determine if f(x) = 2x + 5 for all x, we need to consider the information given: f'(x) = 2 for all x. This means that the derivative of f(x) is constant, indicating that f(x) is a linear function.
A linear function with a constant derivative of 2 can be expressed in the form f(x) = 2x + C, where C is a constant.
We are also given that f(0) = 5. We can use this initial condition to find the value of the constant C.
Substitute x = 0 into the function f(x) = 2x + C to find C: f(0) = 2(0) + C = 5, which simplifies to C = 5.
Thus, the function f(x) = 2x + 5 satisfies both the derivative condition f'(x) = 2 and the initial condition f(0) = 5. Therefore, f(x) = 2x + 5 for all x.

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Derivative

The derivative of a function at a point measures the rate at which the function value changes as its input changes. It is the slope of the tangent line to the function's graph at that point. In this context, f'(x) = 2 indicates that the function has a constant rate of change, meaning it is linear.
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Derivatives

Initial Condition

An initial condition provides a specific value of the function at a particular point, which helps determine the constant of integration when finding the original function from its derivative. Here, f(0) = 5 is the initial condition that allows us to solve for the constant term in the linear function f(x).
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Initial Value Problems

Linear Function

A linear function is of the form f(x) = mx + b, where m is the slope and b is the y-intercept. Given f'(x) = 2, the function has a slope of 2. Using the initial condition f(0) = 5, we find the y-intercept b = 5, confirming that f(x) = 2x + 5 is indeed the function satisfying both the derivative and initial condition.
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Linearization
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Theory and Examples


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