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Ch. 3 - Derivatives
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077Not the one you use?Change textbook
All textbooksHass 15th EditionCh. 3 - DerivativesProblem 3.2.52d
Chapter 3, Problem 3.2.52d

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

Verified step by step guidance
1
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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Key Concepts

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

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.
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Determining Where a Function is Increasing & Decreasing

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.
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Critical Points

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.
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Intro to Rational Functions
Related Practice
Textbook Question

Suppose that functions ƒ(x) and g(x) and their first derivatives have the following values at x = 0 and x = 1.


x ƒ(x) g(x) ƒ'(x) g'(x)

0 1 1 -3 1/2

1 3 5 1/2 -4


Find the first derivatives of the following combinations at the given value of x.


d. ƒ(g(x)), x = 0

Textbook Question

Analyzing Motion Using Graphs


[Technology Exercise] Exercises 31–34 give the position function s = f(t) of an object moving along the s-axis as a function of time t. Graph f together with the velocity function v(t) = ds/dt = f'(t) and the acceleration function a(t) = d²s/dt² = f''(t). Comment on the object’s behavior in relation to the signs and values of v and a. Include in your commentary such topics as the following:


c. When does it change direction?


s = t² - 3t + 2, 0 ≤ t ≤ 5

Textbook Question

Theory and Examples


In Exercises 51–54,


c. For what values of x, if any, is f' positive? Zero? Negative?


y = −x²

Textbook Question

Right circular cylinder The total surface area S of a right circular cylinder is related to the base radius r and height h by the equation S = 2πr² + 2πrh.


d. How is dr/dt related to dh/dt if S is constant?

Textbook Question

The folium of Descartes (See Figure 3.27)


c. Find the coordinates of the point A in Figure 3.29 where the folium has a vertical tangent line.


Textbook Question

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 = x³/3