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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 4.R.27
Chapter 4, Problem 4.R.27

24–34. Curve sketching Use the guidelines given in Section 4.4 to make a complete graph of the following functions on their domains or on the given interval. Use a graphing utility to check your work.


ƒ(x) = 4cos (π (x-1)) on [0, 2]

Verified step by step guidance
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Identify the function: We have ƒ(x) = 4cos(π(x-1)). This is a cosine function with a vertical stretch factor of 4 and a horizontal shift of 1 unit to the right.
Determine the domain: The problem specifies the interval [0, 2]. This means we will focus on sketching the graph of the function within this interval.
Find the critical points: Calculate the derivative of ƒ(x) to find where the slope is zero or undefined. The derivative is ƒ'(x) = -4πsin(π(x-1)). Set ƒ'(x) = 0 to find critical points.
Analyze the behavior at critical points: Evaluate the second derivative, ƒ''(x) = -4π²cos(π(x-1)), to determine concavity at the critical points. This helps in understanding the shape of the graph around these points.
Sketch the graph: Use the information from the critical points, concavity, and the behavior at the endpoints of the interval [0, 2] to sketch the graph. Check the graph using a graphing utility to ensure accuracy.

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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 of a function, such as its domain, range, and behavior at critical points. For the given function f(x) = 4cos(π(x-1)), understanding how the cosine function behaves, including its periodicity and amplitude, is essential for sketching its graph accurately.
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Derivatives Applied To Velocity

Critical Points and Intervals

Identifying critical points, where the derivative is zero or undefined, is crucial for understanding the function's behavior. These points help determine local maxima, minima, and points of inflection, which are vital for sketching the graph of f(x) over the specified interval [0, 2].
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Graphing Techniques

Graphing techniques involve using various methods to visualize a function, including plotting points, analyzing symmetry, and understanding transformations. For f(x) = 4cos(π(x-1)), recognizing its transformations from the basic cosine function will aid in accurately sketching the graph within the given interval.
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Related Practice
Textbook Question

Use the graphs of ƒ' and ƒ" to complete the following steps. <IMAGE>

a. Find the critical points of f and determine where f is increasing and where it is decreasing.

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Textbook Question

45–46. Linear approximation


a. Find the linear approximation to f at the given point a.

b. Use your answer from part (a) to estimate the given function value. Does your approximation underestimate or overestimate the exact function value?


ƒ(x) = x²⸍³ ; a =27; ƒ(29)


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Textbook Question

90–103. Indefinite integrals Determine the following indefinite integrals.



∫ (⁴√x³ + √x⁵) dx

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Textbook Question

Locating extrema Consider the graph of a function ƒ on the interval [-3, 3]. <IMAGE>

a . Give the approximate coordinates of the local maxima and minima of ƒ

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Textbook Question

104–107. Functions from derivatives Find the function f with the following properties.


ƒ'(t) = sin t + 2t; ƒ(0) = 5

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Textbook Question

Find the critical points of the following functions on the given intervals. Identify the absolute maximum and absolute minimum values (if they exist).

g(x) = x⁴ - 50x² on [-1, 5]

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