Textbook Question
Theory and Examples
In Exercises 51 and 52, give reasons for your answers.
Let f(x) = |x³ − 9x|.
a. Does f'(0) exist?
Ch. 4 - Applications of Derivatives
Hass15th EditionThomas' CalculusISBN: 9780137616077
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Table of contents
- Ch. 1 - Functions155
- Ch. 2 - Limits and Continuity240
- Ch. 3 - Derivatives337
- Ch. 4 - Applications of Derivatives293
- Ch. 5 - Integrals41
- Ch. 6 - Applications of Definite Integrals25
- Ch. 7 - Transcendental Functions489
- Ch. 8 - Techniques of Integration398
- Ch. 9 - First-Order Differential Equations60
- Ch. 10 - Infinite Sequences and Series119
- Ch. 11 - Parametric Equations and Polar Coordinates58
Chapter 4, Problem 4.3.68a
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).
Verified step by step guidance1
Start by understanding the behavior of the function around the local minimum and maximum. A local minimum at (1, 1) means that the derivative changes from negative to positive at x = 1, indicating a 'valley' in the graph. Similarly, a local maximum at (3, 3) means the derivative changes from positive to negative at x = 3, indicating a 'peak'.
Consider the general shape of the graph. Between x = 1 and x = 3, the function should increase to reach the local maximum at (3, 3). Before x = 1 and after x = 3, the function should decrease, reflecting the local minimum and maximum respectively.
Sketch the graph starting from the left of x = 1. The function should decrease until it reaches the point (1, 1). At this point, the slope of the tangent line is zero, indicating a local minimum.
From x = 1 to x = 3, sketch the graph such that it increases to reach the local maximum at (3, 3). At this point, the slope of the tangent line is zero again, indicating a local maximum.
Finally, sketch the graph beyond x = 3, where the function should decrease again. Ensure the graph reflects the change in slope at the local minimum and maximum points, creating a smooth curve that is differentiable everywhere.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Differentiable Function
A differentiable function is one that has a derivative at every point in its domain. This means the function is smooth and continuous, without any sharp corners or cusps. Understanding differentiability is crucial for sketching graphs, as it ensures the function's behavior can be predicted using its derivative.
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05:53
Finding Differentials
Local Minimum and Maximum
A local minimum is a point where the function value is lower than all nearby points, while a local maximum is where the function value is higher than all nearby points. These points are critical in graph sketching, as they indicate where the function changes direction, often corresponding to where the derivative equals zero.
Recommended video:
06:02The Second Derivative Test: Finding Local Extrema
Critical Points and Derivatives
Critical points occur where the derivative of a function is zero or undefined, indicating potential local minima or maxima. Analyzing the derivative helps determine the nature of these points, guiding the sketching of the function's graph by showing where it increases or decreases.
Recommended video:
04:50Critical Points
Related Practice
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
1
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Textbook Question
Identifying Extrema
In Exercises 63 and 64, the graph of f' is given. Assume that f has domain (-2, 2).
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a. Either use the graph to determine which intervals f is increasing on and which intervals f is decreasing on, or explain why this information cannot be determined from the graph.
Textbook Question
Finding Antiderivatives
In Exercises 1–16, find an antiderivative for each function. Do as many as you can mentally. Check your answers by differentiation.
(3/2)√x
Textbook Question
20.The U.S. Postal Service will accept a box for domestic shipment only if the sum of its length and girth (distance around) does not exceed 108 in. a.What dimensions will give a box with a square end the largest possible volume?
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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(x) = √(x² − 2x − 3), 3 ≤ x < ∞