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Ch. 4 - Applications of Derivatives
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077Not the one you use?Change textbook
All textbooksHass 15th EditionCh. 4 - Applications of DerivativesProblem 4.1.52a
Chapter 4, Problem 4.1.52a

Theory and Examples


In Exercises 51 and 52, give reasons for your answers.


Let f(x) = |x³ − 9x|.


a. Does f'(0) exist?

Verified step by step guidance
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To determine if f'(0) exists, we need to check if the function f(x) = |x³ − 9x| is differentiable at x = 0. Differentiability requires the function to be continuous and have a defined derivative at that point.
First, check the continuity of f(x) at x = 0. Since f(x) is an absolute value function, it is continuous everywhere, including at x = 0.
Next, consider the definition of the derivative: f'(x) = lim (h -> 0) [(f(x + h) - f(x)) / h]. We need to evaluate this limit at x = 0.
To evaluate the derivative, consider the piecewise nature of the absolute value function. For x³ - 9x, identify the points where the expression inside the absolute value changes sign, which are the roots of x³ - 9x = 0.
Calculate the left-hand and right-hand derivatives at x = 0 by considering the limits from both sides. If these one-sided derivatives are equal, then f'(0) exists; otherwise, it does not.

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Key Concepts

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

Derivative at a Point

The derivative of a function at a point measures the rate at which the function's value changes as its input changes. For f'(0) to exist, the function must be differentiable at x = 0, meaning it must be continuous and have a defined slope at that point. If the function has a sharp corner or cusp at x = 0, the derivative does not exist.
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Critical Points

Absolute Value Function

The absolute value function, denoted as |x|, outputs the non-negative value of x. When applied to a function like f(x) = |x³ − 9x|, it can create points where the function is not smooth, such as cusps or corners, which can affect differentiability. Understanding how the absolute value impacts the function's graph is crucial for analyzing its derivative.
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Average Value of a Function

Piecewise Functions

A piecewise function is defined by different expressions over different intervals. The function f(x) = |x³ − 9x| can be expressed as a piecewise function, where the expression inside the absolute value changes sign. Analyzing the behavior of each piece separately helps determine the function's continuity and differentiability at critical points like x = 0.
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Related Practice
Textbook Question

Checking Antiderivative Formulas


Right, or wrong? Say which for each formula and give a brief reason for each answer.


∫√(2x + 1) dx = √(x² + x +C)

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Textbook Question

Theory and Examples


Sketch the graph of a differentiable function y = f(x) that has a local minimum at (1, 1) and a local maximum at (3, 3).

Textbook Question

Finding displacement from an antiderivative of velocity


a. Suppose that the velocity of a body moving along the s-axis is


ds/dt = v = 9.8t − 3.


i. Find the body’s displacement over the time interval from t = 1 to t = 3 given that s = 5 when t = 0.

Textbook Question

Checking Antiderivative Formulas


Right, or wrong? Say which for each formula and give a brief reason for each answer.


∫tanθ sec²θ dθ = sec³θ / 3 + C

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Textbook Question

Identifying Extrema


In Exercises 41–52:


a. Identify the function’s local extreme values in the given domain, and say where they occur.


f(t) = t³ − 3t², −∞ < t ≤ 3

Textbook Question

Identifying Extrema


In Exercises 41–52:


a. Identify the function’s local extreme values in the given domain, and say where they occur.


f(x) = √(x² − 2x − 3), 3 ≤ x < ∞