Ch. 4 - Applications of the Derivative

Briggs - Calculus: Early Transcendentals 3rd Edition

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Briggs 3rd Edition

Ch. 4 - Applications of the Derivative

Problem 4.3.62

Chapter 4, Problem 4.3.62

Sketching curves Sketch a graph of a function f that is continuous on (-∞,∞) and has the following properties.

f'(x) < 0 and f"(x) > 0 on (-∞,0); f'(x) > 0 and f"(x) < 0 on (0,∞)

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Start by understanding the properties of the function f. The function is continuous on (-∞, ∞), meaning there are no breaks, jumps, or holes in the graph.

Analyze the interval (-∞, 0). Here, f'(x) < 0 indicates that the function is decreasing, and f''(x) > 0 suggests that the function is concave up. This means the graph is sloping downwards but curving upwards, resembling a downward-sloping curve that is opening upwards.

Next, consider the interval (0, ∞). In this region, f'(x) > 0 implies the function is increasing, and f''(x) < 0 indicates that the function is concave down. This means the graph is sloping upwards but curving downwards, resembling an upward-sloping curve that is opening downwards.

Identify the behavior at x = 0. Since the function transitions from decreasing and concave up to increasing and concave down, x = 0 is likely a point of inflection where the concavity changes.

Combine these observations to sketch the graph. Start with a curve that decreases and is concave up on (-∞, 0), then smoothly transition at x = 0 to a curve that increases and is concave down on (0, ∞). Ensure the graph is continuous and reflects the described properties.

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Key Concepts

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

Derivative and Its Significance

The derivative of a function, denoted as f'(x), represents the rate of change of the function at a given point. If f'(x) < 0, the function is decreasing, while f'(x) > 0 indicates that the function is increasing. Understanding the sign of the derivative is crucial for determining the behavior of the function across different intervals.

Second Derivative and Concavity

The second derivative, f''(x), provides information about the concavity of the function. If f''(x) > 0, the function is concave up, suggesting that the slope of the function is increasing. Conversely, if f''(x) < 0, the function is concave down, indicating that the slope is decreasing. This concept is essential for sketching the graph accurately.

Continuity of Functions

A function is continuous on an interval if there are no breaks, jumps, or holes in its graph. For the function f to be continuous on (-∞, ∞), it must be defined for all x in that interval and must not have any discontinuities. This property ensures that the function can be graphed smoothly, which is vital for visualizing its behavior based on the given derivative conditions.

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Locating critical points Find the critical points of the following functions. Assume a is a nonzero constant.

ƒ(x) = 3x² - 4x + 2

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{Use of Tech} Absolute maxima and minima

a. Find the critical points of f on the given interval.

b. Determine the absolute extreme values of f on the given interval.

c. Use a graphing utility to confirm your conclusions.

f(x) = 2ᶻ sin x on [-2,6]

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Textbook Question

{Use of Tech} Finding roots with Newton’s method For the given function f and initial approximation x₀, use Newton’s method to approximate a root of f. Stop calculating approximations when two successive approximations agree to five digits to the right of the decimal point after rounding. Show your work by making a table similar to that in Example 1.

f(x) = cos⁻¹ x - x; x₀ = 0.75

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Textbook Question

17–83. Limits Evaluate the following limits. Use l’Hôpital’s Rule when it is convenient and applicable.

lim_x→ ∞ (e³ˣ ) / (3e³ˣ + 5)

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Use the following graphs to identify the points (if any) on the interval [a, b] at which the function has an absolute maximum or an absolute minimum value <IMAGE>

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Textbook Question

23–68. Indefinite integrals Determine the following indefinite integrals. Check your work by differentiation.

∫ (√x(2x⁶ - 4³√)dx

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Sketching curves Sketch a graph of a function f that is continuous on (-∞,∞) and has the following properties.f'(x) < 0 and f"(x) > 0 on (-∞,0); f'(x) > 0 and f"(x) < 0 on (0,∞)

Verified video answer for a similar problem:

Key Concepts

Derivative and Its Significance

Second Derivative and Concavity

Continuity of Functions

Sketching curves Sketch a graph of a function f that is continuous on (-∞,∞) and has the following properties.

f'(x) < 0 and f"(x) > 0 on (-∞,0); f'(x) > 0 and f"(x) < 0 on (0,∞)