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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 3.2.53d
Chapter 3, Problem 3.2.53d

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

Verified step by step guidance
1
To determine where the function y = f(x) = x³/3 is increasing or decreasing, we first need to find its derivative. The derivative, f'(x), will help us understand the behavior of the function.
Calculate the derivative of y = x³/3 with respect to x. Using the power rule, the derivative of x³ is 3x². Therefore, f'(x) = (3x²)/3 = x².
Analyze the sign of the derivative f'(x) = x². Since x² is always non-negative (x² ≥ 0 for all x), the derivative is zero at x = 0 and positive for all other x-values.
Determine the intervals of increase and decrease: Since f'(x) = x² is positive for all x ≠ 0, the function y = x³/3 is increasing on the intervals (-∞, 0) and (0, ∞). There are no intervals where the function is decreasing because the derivative is never negative.
Relate this to part (c): The critical point found in part (c) is x = 0, where the derivative is zero. This is a point of inflection, not a local maximum or minimum, as the function changes concavity but continues to increase on either side of x = 0.

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Key Concepts

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

Derivative

The derivative of a function measures how the function's output value changes as its input changes. For a function y = f(x), the derivative f'(x) provides the rate of change of y with respect to x. In this context, finding the derivative of y = x³/3 helps determine where the function is increasing or decreasing by analyzing the sign of f'(x).
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Derivatives

Critical Points

Critical points occur where the derivative of a function is zero or undefined. These points are potential locations where the function changes from increasing to decreasing or vice versa. For y = x³/3, finding the critical points involves solving f'(x) = 0, which helps identify intervals of increase or decrease.
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Critical Points

Increasing and Decreasing Intervals

A function is increasing on an interval if its derivative is positive throughout that interval, and decreasing if the derivative is negative. By analyzing the sign of the derivative f'(x) for y = x³/3, we can determine the intervals over which the function increases or decreases, providing insight into the function's behavior.
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Determining Where a Function is Increasing & Decreasing
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

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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:


d. When does it speed up and slow down?


s = t³ - 6t² + 7t, 0 ≤ t ≤ 4

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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 = −1/x

180
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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?

246
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Textbook Question

Average single-family home prices P (in thousands of dollars) in Sacramento, California, are shown in the accompanying figure from the beginning of 2006 through the end of 2015.



d. During what year did home prices drop most rapidly and what is an estimate of this rate?

161
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Textbook Question

Particle motion At time t, the position of a body moving along the s-axis is s = t³ − 6t² + 9t m.


c. Find the total distance traveled by the body from t = 0 to t = 2.

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