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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]

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1
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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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

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


∫ ((1/x²) - (2/(x⁵⸍²))) dx

Textbook Question

Optimal popcorn box A small popcorn box is created from a 12" x 12" sheet of paperboard by first cutting out four shaded rectangles, each of length x and width x/2 (see figure). The remaining paperboard is folded along the solid lines to form a box. What dimensions of the box maximize the volume of the box? <IMAGE>

Textbook Question

Minimum painting surface A metal cistern in the shape of a right circular cylinder with volume V = 50 m³ needs to be painted each year to reduce corrosion. The paint is applied only to surfaces exposed to the elements (the outside cylinder wall and the circular top). Find the dimensions r and h of the cylinder that minimize the area of the painted surfaces.

Textbook Question

Change in elevation The elevation h (in feet above the ground) of a stone dropped from a height of 1000 ft is modeled by the equation h(t) = 1000 - 16t², where t is measured in seconds and air resistance is neglected. Approximate the change in elevation over the interval 5 ≤ t ≤ 5.7 (recall that Δh ≈ h' (a) Δt).

Textbook Question

60–81. Limits Evaluate the following limits. Use l’Hôpital’s Rule when needed. 

lim_x→∞ ln ((x +1) / (x-1))

Textbook Question

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) = 10x² / (x² + 3)