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Ch. 4 - Applications of the Derivative
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 4 - Applications of the DerivativeProblem 37
Chapter 4, Problem 37

Use the guidelines of this section to make a complete graph of f.
f(x) = x + 2 cos x on [-2π,2π)

Verified step by step guidance
1
Step 1: Identify the domain of the function f(x) = x + 2 cos x, which is given as [-2π, 2π). This means we will analyze the function from -2π to just before 2π.
Step 2: Determine the critical points by finding the derivative of f(x). The derivative, f'(x), is 1 - 2 sin x. Set f'(x) = 0 to find critical points: 1 - 2 sin x = 0, which simplifies to sin x = 1/2. Solve for x within the domain [-2π, 2π).
Step 3: Evaluate the function f(x) at the critical points and endpoints of the interval to find local maxima, minima, and endpoints. Calculate f(x) at x = -2π, x = 2π, and any critical points found in Step 2.
Step 4: Analyze the concavity by finding the second derivative, f''(x) = -2 cos x. Determine where f''(x) is positive (indicating concave up) and where it is negative (indicating concave down) within the domain.
Step 5: Use the information from Steps 2-4 to sketch the graph of f(x). Plot the critical points, endpoints, and note the behavior of the function (increasing/decreasing, concave up/down) to create a complete graph of f(x) = x + 2 cos x on the interval [-2π, 2π).

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Key Concepts

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

Function Analysis

Function analysis involves examining the properties and behavior of a function, such as its domain, range, and continuity. For the function f(x) = x + 2 cos x, understanding how the cosine function oscillates and how it affects the linear component is crucial for graphing. This analysis helps identify key features like intercepts and asymptotic behavior.
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Graphing Techniques

Graphing techniques include methods for accurately plotting a function on a coordinate plane. This involves determining critical points, such as maxima, minima, and points of inflection, as well as understanding the overall shape of the graph. For f(x) = x + 2 cos x, recognizing the periodic nature of the cosine function and how it modifies the linear term is essential for creating a complete graph.
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Interval Notation

Interval notation is a mathematical notation used to represent a range of values. In this case, the interval [-2π, 2π) indicates that the graph should be plotted from -2π to just below 2π, including -2π but excluding 2π. Understanding this notation is important for correctly defining the domain of the function and ensuring the graph accurately reflects the specified range.
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Related Practice
Textbook Question

Use the guidelines of this section to make a complete graph of f.

f(x) = 2 - 2x2/3 + x4/3

Textbook Question

Evaluate and simplify y'.

y = (3t²−1 / 3t²+1)^−3

Textbook Question

Use linear approximations to estimate the following quantities. Choose a value of a to produce a small error.

ln 1.05

Textbook Question

Use the guidelines of this section to make a complete graph of f.

f(x) = tan⁻¹ (x²/√3)

Textbook Question

Use linear approximations to estimate the following quantities. Choose a value of a to produce a small error.

√146

Textbook Question

Demand and elasticity The economic advisor of a large tire store proposes the demand function D(p) = 1800/p-40, where D(p) is the number of tires of one brand and size that can be sold in one day at a price p.

c. Find the elasticity function on the domain of the demand function.