Theory and Examples

In Exercises 51–54,

d. Over what intervals of x-values, if any, does the function y = f(x) increase as x increases? Decrease as x increases? How is this related to what you found in part (c)? (We will say more about this relationship in Section 4.3.)

y = −1/x

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To determine where the function y = -1/x is increasing or decreasing, we first need to find its derivative. The derivative, y', will help us understand the behavior of the function.

Calculate the derivative of y = -1/x. Using the power rule, rewrite the function as y = -x^(-1) and differentiate to get y' = x^(-2) = 1/x^2.

Analyze the sign of the derivative y' = 1/x^2. Since 1/x^2 is always positive for all x ≠ 0, the function does not change sign and is always positive.

Since the derivative y' = 1/x^2 is positive for all x ≠ 0, the function y = -1/x is decreasing on its entire domain, which is x < 0 and x > 0.

Relate this to part (c) by considering the critical points and intervals of increase or decrease. Since there are no critical points where the derivative is zero or undefined (other than x = 0, which is not in the domain), the function is decreasing on both intervals x < 0 and x > 0.

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주요 개념

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Increasing and Decreasing Functions

A function is increasing on an interval if, for any two numbers x1 and x2 within that interval, x1 < x2 implies f(x1) < f(x2). Conversely, a function is decreasing if x1 < x2 implies f(x1) > f(x2). Understanding these definitions helps determine where the function y = -1/x increases or decreases as x changes.

Derivative and Critical Points

The derivative of a function, f'(x), provides information about the function's rate of change. Critical points occur where f'(x) = 0 or is undefined, indicating potential changes in increasing or decreasing behavior. For y = -1/x, the derivative helps identify intervals of increase or decrease by analyzing where the derivative is positive or negative.

Behavior of Rational Functions

Rational functions, like y = -1/x, have unique behaviors due to their asymptotes and undefined points. Understanding how these functions behave near vertical asymptotes and across different intervals is crucial. For y = -1/x, the function's behavior as x approaches zero from either side is key to determining its increasing or decreasing nature.

Theory and ExamplesIn Exercises 51–54,d. Over what intervals of x-values, if any, does the function y = f(x) increase as x increases? Decrease as x increases? How is this related to what you found in part (c)? (We will say more about this relationship in Section 4.3.)y = −1/x